Nuprl Lemma : product-cat_wf
∀[A,B:SmallCategory]. (A × B ∈ SmallCategory)
Proof
Definitions occuring in Statement :
product-cat: A × B,
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
small-category: SmallCategory,
product-cat: A × B,
pi1: fst(t),
pi2: snd(t),
spreadn: spread4,
and: P ∧ Q,
all: ∀x:A. B[x],
cand: A c∧ B,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top
Lemmas referenced :
small-category_wf,
cat-ob_wf,
cat-arrow_wf,
cat-id_wf,
cat-comp_wf,
equal_wf,
squash_wf,
true_wf,
cat-comp-ident1,
iff_weakening_equal,
cat-comp-ident2,
cat-comp-assoc,
pi2_wf,
pi1_wf_top,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
dependent_set_memberEquality,
cut,
introduction,
extract_by_obid,
hypothesis,
dependent_pairEquality,
productEquality,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
functionEquality,
cumulativity,
universeEquality,
applyEquality,
functionExtensionality,
lambdaEquality,
productElimination,
sqequalRule,
because_Cache,
independent_pairEquality,
independent_pairFormation,
lambdaFormation,
imageElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}[A,B:SmallCategory]. (A \mtimes{} B \mmember{} SmallCategory)
Date html generated:
2020_05_20-AM-07_54_03
Last ObjectModification:
2017_07_28-AM-09_19_50
Theory : small!categories
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