Nuprl Lemma : rfun*_functionality
∀[f:ℝ ⟶ ℝ]. ∀[x,y:ℝ*]. ((∀[a,b:ℝ]. (f a) = (f b) supposing a = b) ⇒ x = y ⇒ f*(x) = f*(y))
Proof
Definitions occuring in Statement :
rfun*: f*(x),
req*: x = y,
real*: ℝ*,
req: x = y,
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
req*: x = y,
exists: ∃x:A. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
rfun*: f*(x),
nat: ℕ,
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
uimplies: b supposing a,
guard: {T},
real*: ℝ*
Lemmas referenced :
int_upper_wf,
all_wf,
req_wf,
rfun*_wf,
real_wf,
int_upper_subtype_nat,
req*_wf,
uall_wf,
isect_wf,
real*_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
hypothesisEquality,
cut,
hypothesis,
dependent_functionElimination,
sqequalRule,
introduction,
extract_by_obid,
isectElimination,
setElimination,
rename,
because_Cache,
lambdaEquality,
applyEquality,
functionExtensionality,
functionEquality,
independent_isectElimination
Latex:
\mforall{}[f:\mBbbR{} {}\mrightarrow{} \mBbbR{}]. \mforall{}[x,y:\mBbbR{}*]. ((\mforall{}[a,b:\mBbbR{}]. (f a) = (f b) supposing a = b) {}\mRightarrow{} x = y {}\mRightarrow{} f*(x) = f*(y))
Date html generated:
2018_05_22-PM-03_15_09
Last ObjectModification:
2017_10_06-PM-02_25_58
Theory : reals_2
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