### Nuprl Lemma : rnonzero_functionality

`∀x,y:ℝ.  ((x = y) `` (rnonzero(x) `⇐⇒` rnonzero(y)))`

Proof

Definitions occuring in Statement :  rnonzero: `rnonzero(x)` req: `x = y` real: `ℝ` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` implies: `P `` Q`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` uall: `∀[x:A]. B[x]` real: `ℝ` rnonzero: `rnonzero(x)` exists: `∃x:A. B[x]` uimplies: `b supposing a` nat_plus: `ℕ+` subtype_rel: `A ⊆r B` nat: `ℕ` decidable: `Dec(P)` or: `P ∨ Q` less_than: `a < b` squash: `↓T` and: `P ∧ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` less_than': `less_than'(a;b)` true: `True` req: `x = y` le: `A ≤ B` uiff: `uiff(P;Q)` rev_uimplies: `rev_uimplies(P;Q)` ge: `i ≥ j ` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  rnonzero_wf req_wf real_wf rnonzero-lemma1 nat_plus_properties decidable__le absval_wf less_than_wf nat_wf satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf mul_nat_plus itermMultiply_wf int_term_value_mul_lemma subtract_wf multiply-is-int-iff false_wf decidable__equal_int intformeq_wf itermAdd_wf itermSubtract_wf int_formula_prop_eq_lemma int_term_value_add_lemma int_term_value_subtract_lemma and_wf equal_wf le_wf le_functionality le_weakening int-triangle-inequality mul_cancel_in_le decidable__lt add-is-int-iff req_inversion
Rules used in proof :  cut sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation introduction extract_by_obid sqequalHypSubstitution isectElimination thin setElimination rename hypothesisEquality hypothesis productElimination independent_isectElimination dependent_functionElimination natural_numberEquality applyEquality dependent_set_memberEquality lambdaEquality sqequalRule unionElimination imageElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll imageMemberEquality baseClosed because_Cache independent_functionElimination multiplyEquality addEquality pointwiseFunctionality equalityTransitivity equalitySymmetry promote_hyp baseApply closedConclusion setEquality hyp_replacement Error :applyLambdaEquality

Latex:
\mforall{}x,y:\mBbbR{}.    ((x  =  y)  {}\mRightarrow{}  (rnonzero(x)  \mLeftarrow{}{}\mRightarrow{}  rnonzero(y)))

Date html generated: 2016_10_26-AM-09_04_12
Last ObjectModification: 2016_07_12-AM-08_14_38

Theory : reals

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