### Nuprl Lemma : rnonzero-iff

`∀x:ℝ. (rnonzero(x) `⇐⇒` x ≠ r0)`

Proof

Definitions occuring in Statement :  rneq: `x ≠ y` rnonzero: `rnonzero(x)` int-to-real: `r(n)` real: `ℝ` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` natural_number: `\$n`
Definitions unfolded in proof :  int-to-real: `r(n)` rneq: `x ≠ y` rnonzero: `rnonzero(x)` rless: `x < y` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` exists: `∃x:A. B[x]` member: `t ∈ T` prop: `ℙ` uall: `∀[x:A]. B[x]` so_lambda: `λ2x.t[x]` real: `ℝ` subtype_rel: `A ⊆r B` nat: `ℕ` so_apply: `x[s]` rev_implies: `P `` Q` or: `P ∨ Q` sq_exists: `∃x:{A| B[x]}` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(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` nat_plus: `ℕ+` satisfiable_int_formula: `satisfiable_int_formula(fmla)` bfalse: `ff` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` decidable: `Dec(P)`
Lemmas referenced :  exists_wf nat_plus_wf less_than_wf absval_wf nat_wf absval_unfold lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int top_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 or_wf sq_exists_wf real_wf absval_ifthenelse assert_wf bnot_wf not_wf bool_cases iff_transitivity iff_weakening_uiff assert_of_bnot minus-is-int-iff
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity sqequalRule lambdaFormation independent_pairFormation sqequalHypSubstitution productElimination thin cut introduction extract_by_obid isectElimination hypothesis lambdaEquality natural_numberEquality applyEquality setElimination rename hypothesisEquality unionElimination dependent_pairFormation because_Cache minusEquality equalityElimination independent_isectElimination lessCases isect_memberFormation sqequalAxiom isect_memberEquality voidElimination voidEquality imageMemberEquality baseClosed imageElimination independent_functionElimination int_eqEquality intEquality dependent_functionElimination computeAll equalityTransitivity equalitySymmetry promote_hyp instantiate cumulativity dependent_set_memberEquality pointwiseFunctionality baseApply closedConclusion addEquality multiplyEquality impliesFunctionality inrFormation dependent_set_memberFormation inlFormation

Latex:
\mforall{}x:\mBbbR{}.  (rnonzero(x)  \mLeftarrow{}{}\mRightarrow{}  x  \mneq{}  r0)

Date html generated: 2017_10_03-AM-08_27_04
Last ObjectModification: 2017_07_28-AM-07_24_40

Theory : reals

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