Nuprl Lemma : function-eq-implies
∀[A:Type]. ∀[B:Base].
∀[f,g:Base]. (function-eq(A;a.B[a];f;g) ⇒ {∀[a:A]. ((f a) = (g a) ∈ B[a])})
supposing base-type-family{i:l}(A;a.B[a])
Proof
Definitions occuring in Statement :
function-eq: function-eq(A;a.B[a];f;g),
base-type-family: base-type-family{i:l}(A;a.B[a]),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T},
so_apply: x[s],
implies: P ⇒ Q,
apply: f a,
base: Base,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
guard: {T},
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
implies: P ⇒ Q,
function-eq: function-eq(A;a.B[a];f;g),
so_lambda: λ2x.t[x],
label: ...$L... t,
so_apply: x[s],
prop: ℙ,
all: ∀x:A. B[x],
and: P ∧ Q,
squash: ↓T,
subtype_rel: A ⊆r B,
true: True
Lemmas referenced :
function-eq_wf,
base_wf,
base-type-family_wf,
base-type-family-implies,
and_wf,
equal_wf,
squash_wf,
true_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalHypSubstitution,
hypothesis,
hypothesisEquality,
extract_by_obid,
isectElimination,
thin,
cumulativity,
baseApply,
closedConclusion,
baseClosed,
independent_isectElimination,
lambdaEquality,
dependent_functionElimination,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
universeEquality,
pointwiseFunctionality,
dependent_set_memberEquality,
independent_pairFormation,
applyLambdaEquality,
setElimination,
rename,
productElimination,
applyEquality,
imageElimination,
hyp_replacement,
natural_numberEquality,
imageMemberEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:Base].
\mforall{}[f,g:Base]. (function-eq(A;a.B[a];f;g) {}\mRightarrow{} \{\mforall{}[a:A]. ((f a) = (g a))\})
supposing base-type-family\{i:l\}(A;a.B[a])
Date html generated:
2017_04_14-AM-07_29_12
Last ObjectModification:
2017_02_27-PM-02_57_09
Theory : per!type
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