Nuprl Lemma : respects-equality-product
∀[A,A':Type]. ∀[B:A ⟶ Type]. ∀[B':A' ⟶ Type].
(respects-equality(A;A')
⇒ (∀a:Base. ((a ∈ A) ⇒ (a ∈ A') ⇒ respects-equality(B[a];B'[a])))
⇒ respects-equality(a:A × B[a];a:A' × B'[a]))
Proof
Definitions occuring in Statement :
respects-equality: respects-equality(S;T),
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
function: x:A ⟶ B[x],
product: x:A × B[x],
base: Base,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
respects-equality: respects-equality(S;T),
all: ∀x:A. B[x],
pi1: fst(t),
pi2: snd(t),
so_apply: x[s]
Lemmas referenced :
istype-base,
respects-equality_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
Error :lambdaFormation_alt,
sqequalHypSubstitution,
equalityTransitivity,
hypothesis,
equalitySymmetry,
Error :inhabitedIsType,
thin,
productElimination,
sqequalRule,
Error :equalityIsType1,
hypothesisEquality,
dependent_functionElimination,
independent_functionElimination,
applyLambdaEquality,
Error :dependent_pairEquality_alt,
Error :universeIsType,
applyEquality,
Error :equalityIstype,
Error :productIsType,
because_Cache,
sqequalBase,
Error :functionIsType,
extract_by_obid,
isectElimination,
Error :lambdaEquality_alt,
axiomEquality,
Error :functionIsTypeImplies,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
instantiate,
universeEquality,
baseApply,
closedConclusion,
baseClosed
Latex:
\mforall{}[A,A':Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[B':A' {}\mrightarrow{} Type].
(respects-equality(A;A')
{}\mRightarrow{} (\mforall{}a:Base. ((a \mmember{} A) {}\mRightarrow{} (a \mmember{} A') {}\mRightarrow{} respects-equality(B[a];B'[a])))
{}\mRightarrow{} respects-equality(a:A \mtimes{} B[a];a:A' \mtimes{} B'[a]))
Date html generated:
2019_06_20-AM-11_14_44
Last ObjectModification:
2018_11_22-PM-11_26_42
Theory : core_2
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