Nuprl Lemma : qreal-function
∀[f:ℝ ⟶ ℝ ⟶ ℝ]. f ∈ [ℝ] ⟶ [ℝ] ⟶ [ℝ] supposing ∀a1,a2,b1,b2:ℝ. ((a1 = a2) ⇒ (b1 = b2) ⇒ (f[a1;b1] = f[a2;b2]))
Proof
Definitions occuring in Statement :
qreal: [ℝ],
req: x = y,
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
qreal: [ℝ],
quotient: x,y:A//B[x; y],
and: P ∧ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
qreal_wf,
quotient-member-eq,
real_wf,
req_wf,
req-equiv,
equal-wf-base,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
functionExtensionality,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
functionEquality,
lemma_by_obid,
hypothesis,
sqequalRule,
pertypeElimination,
productElimination,
thin,
because_Cache,
isectElimination,
lambdaEquality,
hypothesisEquality,
independent_isectElimination,
dependent_functionElimination,
applyEquality,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
productEquality,
axiomEquality,
isect_memberEquality
Latex:
\mforall{}[f:\mBbbR{} {}\mrightarrow{} \mBbbR{} {}\mrightarrow{} \mBbbR{}]
f \mmember{} [\mBbbR{}] {}\mrightarrow{} [\mBbbR{}] {}\mrightarrow{} [\mBbbR{}] supposing \mforall{}a1,a2,b1,b2:\mBbbR{}. ((a1 = a2) {}\mRightarrow{} (b1 = b2) {}\mRightarrow{} (f[a1;b1] = f[a2;b2]))
Date html generated:
2016_05_18-AM-11_14_42
Last ObjectModification:
2015_12_27-PM-10_39_41
Theory : reals
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