### Nuprl Lemma : groupoid-square1_wf

`∀[G:Groupoid]. ∀[x00,x10,x01,x11:cat-ob(cat(G))]. ∀[a:cat-arrow(cat(G)) x00 x10]. ∀[b:cat-arrow(cat(G)) x10 x11].`
`∀[c:cat-arrow(cat(G)) x00 x01].`
`  (groupoid-square1(G;x00;x10;x01;x11;a;b;c) ∈ {d:cat-arrow(cat(G)) x01 x11| a o b = c o d} )`

Proof

Definitions occuring in Statement :  groupoid-square1: `groupoid-square1(G;x00;x10;x01;x11;a;b;c)` 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)` uall: `∀[x:A]. B[x]` member: `t ∈ T` set: `{x:A| B[x]} ` apply: `f a`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` groupoid-square1: `groupoid-square1(G;x00;x10;x01;x11;a;b;c)` prop: `ℙ` cat-square-commutes: `x_y1 o y1_z = x_y2 o y2_z` true: `True` implies: `P `` Q` squash: `↓T` all: `∀x:A. B[x]` and: `P ∧ Q` uiff: `uiff(P;Q)` rev_uimplies: `rev_uimplies(P;Q)` uimplies: `b supposing a`
Lemmas referenced :  groupoid-right-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 isect_memberFormation introduction cut sqequalRule dependent_set_memberEquality applyEquality lemma_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis axiomEquality equalityTransitivity equalitySymmetry isect_memberEquality because_Cache natural_numberEquality independent_functionElimination lambdaEquality imageElimination universeEquality dependent_functionElimination productElimination imageMemberEquality baseClosed independent_isectElimination

Latex:
\mforall{}[G:Groupoid].  \mforall{}[x00,x10,x01,x11:cat-ob(cat(G))].  \mforall{}[a:cat-arrow(cat(G))  x00  x10].
\mforall{}[b:cat-arrow(cat(G))  x10  x11].  \mforall{}[c:cat-arrow(cat(G))  x00  x01].
(groupoid-square1(G;x00;x10;x01;x11;a;b;c)  \mmember{}  \{d:cat-arrow(cat(G))  x01  x11|  a  o  b  =  c  o  d\}  )

Date html generated: 2020_05_20-AM-07_56_16
Last ObjectModification: 2016_01_17-PM-02_16_56

Theory : small!categories

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