### Nuprl Lemma : rmul-is-negative1

`∀x,y:ℝ.  (((x * y) < r0) `` (x ≠ r0 ∨ y ≠ r0))`

Proof

Definitions occuring in Statement :  rneq: `x ≠ y` rless: `x < y` rmul: `a * b` int-to-real: `r(n)` real: `ℝ` all: `∀x:A. B[x]` implies: `P `` Q` or: `P ∨ Q` natural_number: `\$n`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` rneq: `x ≠ y` member: `t ∈ T` uall: `∀[x:A]. B[x]` prop: `ℙ` int-to-real: `r(n)` rmul: `a * b` has-value: `(a)↓` exists: `∃x:A. B[x]` reg-seq-mul: `reg-seq-mul(x;y)` accelerate: `accelerate(k;f)` uimplies: `b supposing a` real: `ℝ` nat_plus: `ℕ+` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` false: `False` subtype_rel: `A ⊆r B` nat: `ℕ` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` int_upper: `{i...}` guard: `{T}` sq_type: `SQType(T)` nequal: `a ≠ b ∈ T ` true: `True` int_nzero: `ℤ-o` ge: `i ≥ j ` squash: `↓T` less_than: `a < b` le: `A ≤ B` lelt: `i ≤ j < k` int_seg: `{i..j-}` less_than': `less_than'(a;b)` absval: `|i|` cand: `A c∧ B` rev_uimplies: `rev_uimplies(P;Q)`
Lemmas referenced :  rless_wf rmul_wf int-to-real_wf real_wf real-has-value value-type-has-value int-value-type imax_wf absval_wf nat_plus_properties decidable__lt full-omega-unsat intformnot_wf intformless_wf itermConstant_wf istype-int int_formula_prop_not_lemma istype-void int_formula_prop_less_lemma int_term_value_constant_lemma int_formula_prop_wf istype-less_than rless-iff2 false_wf int_term_value_add_lemma int_formula_prop_le_lemma int_term_value_var_lemma int_term_value_mul_lemma int_formula_prop_eq_lemma int_formula_prop_and_lemma itermAdd_wf intformle_wf itermVar_wf itermMultiply_wf intformeq_wf intformand_wf add-is-int-iff int_upper_properties int_entire_a nequal_wf int_subtype_base subtype_base_sq mul_nzero mul_nat_plus istype-le decidable__le nat_properties imax_nat int_seg_properties int_seg_cases int_seg_subtype_special decidable__equal_int div_is_zero nat_plus_subtype_nat mul_preserves_le zero-div-rem int_term_value_minus_lemma itermMinus_wf absval_ubound div_absval_bound
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation_alt universeIsType cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis natural_numberEquality inhabitedIsType sqequalRule callbyvalueReduce productElimination intEquality independent_isectElimination multiplyEquality addEquality applyEquality setElimination rename dependent_set_memberEquality_alt dependent_functionElimination unionElimination approximateComputation independent_functionElimination dependent_pairFormation_alt lambdaEquality_alt isect_memberEquality_alt voidElimination equalityTransitivity equalitySymmetry because_Cache unionIsType inlFormation_alt inrFormation_alt independent_pairFormation int_eqEquality baseApply promote_hyp pointwiseFunctionality applyLambdaEquality sqequalBase baseClosed equalityIstype cumulativity instantiate closedConclusion divideEquality minusEquality productIsType imageElimination hypothesis_subsumption

Latex:
\mforall{}x,y:\mBbbR{}.    (((x  *  y)  <  r0)  {}\mRightarrow{}  (x  \mneq{}  r0  \mvee{}  y  \mneq{}  r0))

Date html generated: 2019_10_29-AM-10_05_18
Last ObjectModification: 2019_04_01-PM-11_22_07

Theory : reals

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