Nuprl Lemma : rneq-function
∀f:ℝ ⟶ ℝ. ((∀x,y:ℝ. ((x = y) ⇒ (f[x] = f[y]))) ⇒ (∀x,y:ℝ. (f[x] ≠ f[y] ⇒ x ≠ y)))
Proof
Definitions occuring in Statement :
rneq: x ≠ y,
req: x = y,
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
or: P ∨ Q,
not: ¬A,
guard: {T},
false: False,
iff: P ⇐⇒ Q,
and: P ∧ Q
Lemmas referenced :
real-weak-Markov,
rneq_wf,
real_wf,
all_wf,
req_wf,
rneq-cases,
not_wf,
rneq_irreflexivity,
rneq_functionality,
req_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_isectElimination,
because_Cache,
hypothesis,
isectElimination,
applyEquality,
functionExtensionality,
sqequalRule,
lambdaEquality,
functionEquality,
independent_functionElimination,
unionElimination,
inlFormation,
inrFormation,
voidElimination,
productElimination
Latex:
\mforall{}f:\mBbbR{} {}\mrightarrow{} \mBbbR{}. ((\mforall{}x,y:\mBbbR{}. ((x = y) {}\mRightarrow{} (f[x] = f[y]))) {}\mRightarrow{} (\mforall{}x,y:\mBbbR{}. (f[x] \mneq{} f[y] {}\mRightarrow{} x \mneq{} y)))
Date html generated:
2017_10_03-AM-09_11_39
Last ObjectModification:
2017_06_26-PM-01_50_51
Theory : reals
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