Nuprl Lemma : groupoid-square-commutes-iff

`∀[G:Groupoid]. ∀[x,y1,y2,z:cat-ob(cat(G))]. ∀[x_y1:cat-arrow(cat(G)) x y1]. ∀[y1_z:cat-arrow(cat(G)) y1 z].`
`∀[x_y2:cat-arrow(cat(G)) x y2]. ∀[y2_z:cat-arrow(cat(G)) y2 z].`
`  uiff(x_y1 o y1_z = x_y2 o y2_z;y2_z`
`  = (cat-comp(cat(G)) y2 x z groupoid-inv(G;x;y2;x_y2) (cat-comp(cat(G)) x y1 z x_y1 y1_z))`
`  ∈ (cat-arrow(cat(G)) y2 z))`

Proof

Definitions occuring in Statement :  groupoid-inv: `groupoid-inv(G;x;y;x_y)` groupoid-cat: `cat(G)` groupoid: `Groupoid` cat-square-commutes: `x_y1 o y1_z = x_y2 o y2_z` cat-comp: `cat-comp(C)` cat-arrow: `cat-arrow(C)` cat-ob: `cat-ob(C)` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` apply: `f a` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` cat-square-commutes: `x_y1 o y1_z = x_y2 o y2_z` prop: `ℙ` true: `True` squash: `↓T` all: `∀x:A. B[x]` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` implies: `P `` Q`
Lemmas referenced :  equal_wf cat-arrow_wf groupoid-cat_wf cat-comp_wf groupoid-inv_wf cat-square-commutes_wf cat-ob_wf groupoid_wf cat-comp-ident1 iff_weakening_equal squash_wf true_wf cat-comp-assoc groupoid_inv
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut independent_pairFormation sqequalHypSubstitution hypothesis thin hyp_replacement equalitySymmetry applyLambdaEquality extract_by_obid isectElimination applyEquality hypothesisEquality because_Cache sqequalRule axiomEquality productElimination independent_pairEquality isect_memberEquality equalityTransitivity natural_numberEquality lambdaEquality imageElimination dependent_functionElimination imageMemberEquality baseClosed independent_isectElimination independent_functionElimination universeEquality

Latex:
\mforall{}[G:Groupoid].  \mforall{}[x,y1,y2,z:cat-ob(cat(G))].  \mforall{}[x\$_{y1}\$:cat-arrow(cat(G))  x  y1].  \000C\mforall{}[y1\$_{z}\$:cat-arrow(cat(G))
y1
z].
\mforall{}[x\$_{y2}\$:cat-arrow(cat(G))  x  y2].  \mforall{}[y2\$_{z}\$:cat-arrow(cat\000C(G))  y2  z].
uiff(x\$_{y1}\$  o  y1\$_{z}\$  =  x\$_{y2}\$  o  \000Cy2\$_{z}\$;y2\$_{z}\$
=  (cat-comp(cat(G))  y2  x  z  groupoid-inv(G;x;y2;x\$_{y2}\$)  (cat-comp(cat(G))  x  y\000C1  z  x\$_{y1}\$  y1\$_{z}\$)))

Date html generated: 2017_10_05-AM-00_49_20
Last ObjectModification: 2017_07_28-AM-09_20_16

Theory : small!categories

Home Index