### Nuprl Lemma : radd-positive-implies

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

Proof

Definitions occuring in Statement :  rless: `x < y` radd: `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` rless: `x < y` sq_exists: `∃x:A [B[x]]` member: `t ∈ T` real: `ℝ` nat_plus: `ℕ+` uall: `∀[x:A]. B[x]` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` top: `Top` and: `P ∧ Q` prop: `ℙ` int-to-real: `r(n)` sq_type: `SQType(T)` guard: `{T}` subtype_rel: `A ⊆r B` int_nzero: `ℤ-o` true: `True` nequal: `a ≠ b ∈ T ` uiff: `uiff(P;Q)` nat: `ℕ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` le: `A ≤ B` less_than': `less_than'(a;b)` absval: `|i|` less_than: `a < b` squash: `↓T`
Lemmas referenced :  decidable__lt nat_plus_properties full-omega-unsat intformand_wf intformnot_wf intformless_wf itermConstant_wf itermMultiply_wf itermVar_wf istype-int int_formula_prop_and_lemma istype-void int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_mul_lemma int_term_value_var_lemma int_formula_prop_wf istype-less_than subtype_base_sq int_subtype_base decidable__equal_int intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma int-to-real_wf rless_wf radd_wf real_wf decidable__le intformle_wf int_formula_prop_le_lemma radd-approx divide_wfa nequal_wf istype-le div_rem_sum rem_bounds_absval absval_strict_ubound remainder_wfa absval_wf nat_wf set_subtype_base le_wf istype-false absval_pos
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation_alt sqequalHypSubstitution setElimination thin rename cut introduction extract_by_obid dependent_functionElimination natural_numberEquality applyEquality hypothesisEquality hypothesis dependent_set_memberEquality_alt multiplyEquality isectElimination unionElimination independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation_alt lambdaEquality_alt int_eqEquality isect_memberEquality_alt voidElimination sqequalRule independent_pairFormation universeIsType inlFormation_alt dependent_set_memberFormation_alt because_Cache instantiate cumulativity intEquality equalityTransitivity equalitySymmetry addEquality inhabitedIsType inrFormation_alt equalityIstype baseClosed sqequalBase productElimination imageElimination minusEquality

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

Date html generated: 2019_10_29-AM-10_00_07
Last ObjectModification: 2019_05_24-AM-11_13_35

Theory : reals

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