Nuprl Lemma : prod-Leibniz-type
∀A,B:Type. (Leibniz-type{i:l}(A) ⇒ Leibniz-type{i:l}(B) ⇒ Leibniz-type{i:l}(A × B))
Proof
Definitions occuring in Statement :
Leibniz-type: Leibniz-type{i:l}(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
Leibniz-type: Leibniz-type{i:l}(T),
exists: ∃x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
or: P ∨ Q,
uall: ∀[x:A]. B[x],
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
pi1: fst(t),
pi2: snd(t),
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
not: ¬A,
false: False,
rev_implies: P ⇐ Q,
guard: {T}
Lemmas referenced :
pi1_wf_top,
istype-void,
pi2_wf,
subtype_rel_self,
Leibniz-type_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation_alt,
lambdaEquality_alt,
sqequalRule,
unionEquality,
applyEquality,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
isectElimination,
independent_pairEquality,
isect_memberEquality_alt,
voidElimination,
hypothesis,
universeIsType,
inhabitedIsType,
productIsType,
independent_pairFormation,
unionIsType,
because_Cache,
functionIsType,
instantiate,
equalityIstype,
cumulativity,
universeEquality,
unionElimination,
inlFormation_alt,
inrFormation_alt,
dependent_functionElimination,
independent_functionElimination,
applyLambdaEquality
Latex:
\mforall{}A,B:Type. (Leibniz-type\{i:l\}(A) {}\mRightarrow{} Leibniz-type\{i:l\}(B) {}\mRightarrow{} Leibniz-type\{i:l\}(A \mtimes{} B))
Date html generated:
2019_10_31-AM-07_25_57
Last ObjectModification:
2019_09_19-PM-06_34_10
Theory : constructive!algebra
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