Nuprl Lemma : groupoid-square2_wf
∀G:Groupoid. ∀x00,x10,x01,x11:cat-ob(cat(G)). ∀d:cat-arrow(cat(G)) x01 x11. ∀b:cat-arrow(cat(G)) x10 x11.
∀c:cat-arrow(cat(G)) x00 x01.
(groupoid-square2(G;x00;x10;x01;x11;b;c;d) ∈ {a:cat-arrow(cat(G)) x00 x10| a o b = c o d} )
Proof
Definitions occuring in Statement :
groupoid-square2: groupoid-square2(G;x00;x10;x01;x11;b;c;d),
groupoid-cat: cat(G),
groupoid: Groupoid,
cat-square-commutes: x_y1 o y1_z = x_y2 o y2_z,
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
all: ∀x:A. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
groupoid-square2: groupoid-square2(G;x00;x10;x01;x11;b;c;d),
uall: ∀[x:A]. B[x],
cat-square-commutes: x_y1 o y1_z = x_y2 o y2_z,
prop: ℙ,
true: True,
implies: P ⇒ Q,
squash: ↓T,
and: P ∧ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a
Lemmas referenced :
groupoid-left-cancellation,
cat-comp-ident,
groupoid_inv,
cat-comp-assoc,
true_wf,
squash_wf,
uiff_transitivity3,
cat-id_wf,
equal_wf,
groupoid_wf,
cat-ob_wf,
cat-arrow_wf,
cat-square-commutes_wf,
groupoid-inv_wf,
groupoid-cat_wf,
cat-comp_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalRule,
dependent_set_memberEquality,
applyEquality,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
natural_numberEquality,
independent_functionElimination,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
dependent_functionElimination,
productElimination,
imageMemberEquality,
baseClosed,
independent_isectElimination
Latex:
\mforall{}G:Groupoid. \mforall{}x00,x10,x01,x11:cat-ob(cat(G)). \mforall{}d:cat-arrow(cat(G)) x01 x11.
\mforall{}b:cat-arrow(cat(G)) x10 x11. \mforall{}c:cat-arrow(cat(G)) x00 x01.
(groupoid-square2(G;x00;x10;x01;x11;b;c;d) \mmember{} \{a:cat-arrow(cat(G)) x00 x10| a o b = c o d\} )
Date html generated:
2020_05_20-AM-07_56_22
Last ObjectModification:
2016_01_17-PM-02_16_54
Theory : small!categories
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