### Nuprl Lemma : rinv_functionality

`∀[x,y:ℝ].  (rnonzero(x) `` (x = y) `` (rinv(x) = rinv(y)))`

Proof

Definitions occuring in Statement :  rinv: `rinv(x)` rnonzero: `rnonzero(x)` req: `x = y` real: `ℝ` uall: `∀[x:A]. B[x]` implies: `P `` Q`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` implies: `P `` Q` iff: `P `⇐⇒` Q` and: `P ∧ Q` uiff: `uiff(P;Q)` uimplies: `b supposing a` rnonzero: `rnonzero(x)` exists: `∃x:A. B[x]` rinv: `rinv(x)` subtype_rel: `A ⊆r B` nat_plus: `ℕ+` int_upper: `{i...}` so_lambda: `λ2x.t[x]` so_apply: `x[s]` prop: `ℙ` guard: `{T}` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` has-value: `(a)↓` real: `ℝ` nat: `ℕ` rev_uimplies: `rev_uimplies(P;Q)` le: `A ≤ B` rev_implies: `P `` Q` less_than': `less_than'(a;b)` true: `True` int_seg: `{i..j-}` lelt: `i ≤ j < k` less_than: `a < b` squash: `↓T` sq_stable: `SqStable(P)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` nequal: `a ≠ b ∈ T ` ge: `i ≥ j ` reg-seq-inv: `reg-seq-inv(x)` req: `x = y` reg-seq-adjust: `reg-seq-adjust(n;x)`
Rules used in proof :  cut introduction extract_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation hypothesis sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality lambdaFormation independent_functionElimination productElimination isectElimination independent_isectElimination sqequalRule dependent_pairFormation applyEquality intEquality because_Cache lambdaEquality natural_numberEquality setElimination rename setEquality applyLambdaEquality unionElimination int_eqEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll callbyvalueReduce dependent_set_memberEquality equalityTransitivity equalitySymmetry productEquality imageElimination promote_hyp imageMemberEquality baseClosed equalityElimination instantiate cumulativity multiplyEquality pointwiseFunctionality baseApply closedConclusion functionEquality functionExtensionality addEquality addLevel hyp_replacement levelHypothesis inrFormation lessCases sqequalAxiom inlFormation

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

Date html generated: 2017_10_02-PM-07_16_55
Last ObjectModification: 2017_07_28-AM-07_21_06

Theory : reals

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