Nuprl Lemma : cat-square-commutes-sym
∀[C:SmallCategory]. ∀[x,y1,y2,z:cat-ob(C)]. ∀[x_y1:cat-arrow(C) x y1]. ∀[y1_z:cat-arrow(C) y1 z]. ∀[x_y2:cat-arrow(C) x
y2].
∀[y2_z:cat-arrow(C) y2 z].
uiff(x_y1 o y1_z = x_y2 o y2_z;x_y2 o y2_z = x_y1 o y1_z)
Proof
Definitions occuring in Statement :
cat-square-commutes: x_y1 o y1_z = x_y2 o y2_z,
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
apply: f a
Definitions unfolded in proof :
prop: ℙ,
cat-square-commutes: x_y1 o y1_z = x_y2 o y2_z,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
small-category_wf,
cat-ob_wf,
cat-arrow_wf,
cat-square-commutes_wf
Rules used in proof :
applyEquality,
equalityTransitivity,
because_Cache,
isect_memberEquality,
independent_pairEquality,
productElimination,
hypothesisEquality,
thin,
isectElimination,
lemma_by_obid,
axiomEquality,
sqequalRule,
hypothesis,
equalitySymmetry,
sqequalHypSubstitution,
independent_pairFormation,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[x,y1,y2,z:cat-ob(C)]. \mforall{}[x$_{y1}$:cat-arrow(C) x y1]. \mforall{}[y1\mbackslash{}\000Cff24_{z}$:cat-arrow(C) y1 z].
\mforall{}[x$_{y2}$:cat-arrow(C) x y2]. \mforall{}[y2$_{z}$:cat-arrow(C) y2 z]\000C.
uiff(x$_{y1}$ o y1$_{z}$ = x$_{y2}$ o \000Cy2$_{z}$;x$_{y2}$ o y2$_{z}$ = x$\mbackslash{}ff\000C5f{y1}$ o y1$_{z}$)
Date html generated:
2016_05_18-AM-11_54_13
Last ObjectModification:
2015_12_28-PM-02_22_59
Theory : small!categories
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