Nuprl Lemma : rabs-neq-zero

`∀x:ℝ. (x ≠ r0 `` (r0 < |x|))`

Proof

Definitions occuring in Statement :  rneq: `x ≠ y` rless: `x < y` rabs: `|x|` int-to-real: `r(n)` real: `ℝ` all: `∀x:A. B[x]` implies: `P `` Q` natural_number: `\$n`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` rneq: `x ≠ y` or: `P ∨ Q` rless: `x < y` sq_exists: `∃x:{A| B[x]}` member: `t ∈ T` int-to-real: `r(n)` rabs: `|x|` uall: `∀[x:A]. B[x]` real: `ℝ` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` less_than: `a < b` less_than': `less_than'(a;b)` top: `Top` true: `True` squash: `↓T` not: `¬A` false: `False` prop: `ℙ` nat_plus: `ℕ+` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` bfalse: `ff` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` decidable: `Dec(P)` subtype_rel: `A ⊆r B`
Lemmas referenced :  absval_unfold lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int top_wf less_than_wf nat_plus_properties satisfiable-full-omega-tt intformand_wf intformless_wf itermConstant_wf itermVar_wf itermAdd_wf itermMultiply_wf int_formula_prop_and_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_var_lemma int_term_value_add_lemma int_term_value_mul_lemma int_formula_prop_wf eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot decidable__lt add-is-int-iff intformnot_wf itermMinus_wf int_formula_prop_not_lemma int_term_value_minus_lemma false_wf int-to-real_wf real_wf rabs_wf rneq_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation sqequalHypSubstitution unionElimination thin setElimination rename introduction dependent_set_memberEquality hypothesisEquality sqequalRule cut extract_by_obid isectElimination applyEquality hypothesis minusEquality natural_numberEquality because_Cache equalityElimination equalityTransitivity equalitySymmetry productElimination independent_isectElimination lessCases isect_memberFormation sqequalAxiom isect_memberEquality independent_pairFormation voidElimination voidEquality imageMemberEquality baseClosed imageElimination independent_functionElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination computeAll promote_hyp instantiate cumulativity addEquality multiplyEquality pointwiseFunctionality baseApply closedConclusion

Latex:
\mforall{}x:\mBbbR{}.  (x  \mneq{}  r0  {}\mRightarrow{}  (r0  <  |x|))

Date html generated: 2017_10_03-AM-08_28_25
Last ObjectModification: 2017_07_28-AM-07_25_13

Theory : reals

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