### Nuprl Lemma : cat-square-commutes-comp

`∀[C:SmallCategory]. ∀[x1,x2,x3,y1,y2,y3:cat-ob(C)]. ∀[x1_y1:cat-arrow(C) x1 y1]. ∀[x2_y2:cat-arrow(C) x2 y2].`
`∀[x3_y3:cat-arrow(C) x3 y3]. ∀[y1_y2:cat-arrow(C) y1 y2]. ∀[y2_y3:cat-arrow(C) y2 y3]. ∀[x1_x2:cat-arrow(C) x1 x2].`
`∀[x2_x3:cat-arrow(C) x2 x3].`
`  (x1_y1 o cat-comp(C) y1 y2 y3 y1_y2 y2_y3 = cat-comp(C) x1 x2 x3 x1_x2 x2_x3 o x3_y3) supposing `
`     (x1_y1 o y1_y2 = x1_x2 o x2_y2 and `
`     x2_y2 o y2_y3 = x2_x3 o x3_y3)`

Proof

Definitions occuring in Statement :  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)` small-category: `SmallCategory` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` apply: `f a`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` cat-square-commutes: `x_y1 o y1_z = x_y2 o y2_z` squash: `↓T` prop: `ℙ` all: `∀x:A. B[x]` true: `True` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` implies: `P `` Q`
Lemmas referenced :  equal_wf squash_wf true_wf cat-arrow_wf cat-comp-assoc cat-comp_wf iff_weakening_equal cat-square-commutes_wf cat-ob_wf small-category_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalHypSubstitution applyEquality thin lambdaEquality imageElimination extract_by_obid isectElimination hypothesisEquality equalityTransitivity hypothesis equalitySymmetry universeEquality dependent_functionElimination because_Cache natural_numberEquality sqequalRule imageMemberEquality baseClosed independent_isectElimination productElimination independent_functionElimination axiomEquality isect_memberEquality

Latex:
\mforall{}[C:SmallCategory].  \mforall{}[x1,x2,x3,y1,y2,y3:cat-ob(C)].  \mforall{}[x1\$_{y1}\$:cat-arrow(C)  x1  \000Cy1].  \mforall{}[x2\$_{y2}\$:cat-arrow(C)
x2
y2].
\mforall{}[x3\$_{y3}\$:cat-arrow(C)  x3  y3].  \mforall{}[y1\$_{y2}\$:cat-arrow(C)  y1\000C  y2].  \mforall{}[y2\$_{y3}\$:cat-arrow(C)  y2  y3].
\mforall{}[x1\$_{x2}\$:cat-arrow(C)  x1  x2].  \mforall{}[x2\$_{x3}\$:cat-arrow(C)  x2\000C  x3].
(x1\$_{y1}\$  o  cat-comp(C)  y1  y2  y3  y1\$_{y2}\$  y2\$_\mbackslash{}f\000Cf7by3}\$  =  cat-comp(C)  x1  x2  x3  x1\$_{x2}\$  x2\$_{x3}\$  o\000C  x3\$_{y3}\$)  supposing
(x1\$_{y1}\$  o  y1\$_{y2}\$  =  x1\$_{x2}\$  \000Co  x2\$_{y2}\$  and
x2\$_{y2}\$  o  y2\$_{y3}\$  =  x2\$_{x3}\$  o\000C  x3\$_{y3}\$)

Date html generated: 2017_10_05-AM-00_49_00
Last ObjectModification: 2017_07_28-AM-09_20_05

Theory : small!categories

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