Theorem cmpasso 15120

Description: The 9th "axiom" of a category: (o` T) is associative.

Hypotheses

Ref	Expression
cmpasso.1	|- M = dom D
cmpasso.2	|- D = (dom` T)
cmpasso.5	|- C = (cod` T)
cmpasso.6	|- R = (o` T)

Assertion

Ref	Expression
cmpasso	|- ((T e. Cat /\ (F e. M /\ G e. M /\ H e. M)) -> (((D` H) = (C` G) /\ (D` G) = (C` F)) -> (HR(GRF)) = ((HRG)RF)))

Proof of Theorem cmpasso

Step	Hyp	Ref	Expression
1		cmpasso.2	. . . 4 |- D = (dom` T)
2		cmpasso.5	. . . 4 |- C = (cod` T)
3		eqid 1884	. . . 4 |- (id` T) = (id` T)
4		cmpasso.6	. . . 4 |- R = (o` T)
5		cmpasso.1	. . . 4 |- M = dom D
6		eqid 1884	. . . 4 |- dom (id` T) = dom (id` T)
7	1, 2, 3, 4, 5, 6	cati 15102	. . 3 |- (T e. Cat -> ((<.<.D, C>., <.(id` T), R>.>. e. Ded /\ A.f e. M A.g e. M A.h e. M (((D` h) = (C` g) /\ (D` g) = (C` f)) -> (hR(gRf)) = ((hRg)Rf))) /\ (A.x e. dom (id` T)A.f e. M ((C` f) = x -> (((id` T)` x)Rf) = f) /\ A.x e. dom (id` T)A.f e. M ((D` f) = x -> (fR((id` T)` x)) = f))))
8		fveq2 4681	. . . . . . . . 9 |- (f = F -> (C` f) = (C` F))
9	8	eqeq2d 1895	. . . . . . . 8 |- (f = F -> ((D` g) = (C` f) <-> (D` g) = (C` F)))
10	9	anbi2d 678	. . . . . . 7 |- (f = F -> (((D` h) = (C` g) /\ (D` g) = (C` f)) <-> ((D` h) = (C` g) /\ (D` g) = (C` F))))
11		opreq2 4890	. . . . . . . . 9 |- (f = F -> (gRf) = (gRF))
12	11	opreq2d 4898	. . . . . . . 8 |- (f = F -> (hR(gRf)) = (hR(gRF)))
13		opreq2 4890	. . . . . . . 8 |- (f = F -> ((hRg)Rf) = ((hRg)RF))
14	12, 13	eqeq12d 1899	. . . . . . 7 |- (f = F -> ((hR(gRf)) = ((hRg)Rf) <-> (hR(gRF)) = ((hRg)RF)))
15	10, 14	imbi12d 688	. . . . . 6 |- (f = F -> ((((D` h) = (C` g) /\ (D` g) = (C` f)) -> (hR(gRf)) = ((hRg)Rf)) <-> (((D` h) = (C` g) /\ (D` g) = (C` F)) -> (hR(gRF)) = ((hRg)RF))))
16		fveq2 4681	. . . . . . . . 9 |- (g = G -> (C` g) = (C` G))
17	16	eqeq2d 1895	. . . . . . . 8 |- (g = G -> ((D` h) = (C` g) <-> (D` h) = (C` G)))
18		fveq2 4681	. . . . . . . . 9 |- (g = G -> (D` g) = (D` G))
19	18	eqeq1d 1892	. . . . . . . 8 |- (g = G -> ((D` g) = (C` F) <-> (D` G) = (C` F)))
20	17, 19	anbi12d 690	. . . . . . 7 |- (g = G -> (((D` h) = (C` g) /\ (D` g) = (C` F)) <-> ((D` h) = (C` G) /\ (D` G) = (C` F))))
21		opreq1 4889	. . . . . . . . 9 |- (g = G -> (gRF) = (GRF))
22	21	opreq2d 4898	. . . . . . . 8 |- (g = G -> (hR(gRF)) = (hR(GRF)))
23		opreq2 4890	. . . . . . . . 9 |- (g = G -> (hRg) = (hRG))
24	23	opreq1d 4897	. . . . . . . 8 |- (g = G -> ((hRg)RF) = ((hRG)RF))
25	22, 24	eqeq12d 1899	. . . . . . 7 |- (g = G -> ((hR(gRF)) = ((hRg)RF) <-> (hR(GRF)) = ((hRG)RF)))
26	20, 25	imbi12d 688	. . . . . 6 |- (g = G -> ((((D` h) = (C` g) /\ (D` g) = (C` F)) -> (hR(gRF)) = ((hRg)RF)) <-> (((D` h) = (C` G) /\ (D` G) = (C` F)) -> (hR(GRF)) = ((hRG)RF))))
27		fveq2 4681	. . . . . . . . 9 |- (h = H -> (D` h) = (D` H))
28	27	eqeq1d 1892	. . . . . . . 8 |- (h = H -> ((D` h) = (C` G) <-> (D` H) = (C` G)))
29	28	anbi1d 679	. . . . . . 7 |- (h = H -> (((D` h) = (C` G) /\ (D` G) = (C` F)) <-> ((D` H) = (C` G) /\ (D` G) = (C` F))))
30		opreq1 4889	. . . . . . . 8 |- (h = H -> (hR(GRF)) = (HR(GRF)))
31		opreq1 4889	. . . . . . . . 9 |- (h = H -> (hRG) = (HRG))
32	31	opreq1d 4897	. . . . . . . 8 |- (h = H -> ((hRG)RF) = ((HRG)RF))
33	30, 32	eqeq12d 1899	. . . . . . 7 |- (h = H -> ((hR(GRF)) = ((hRG)RF) <-> (HR(GRF)) = ((HRG)RF)))
34	29, 33	imbi12d 688	. . . . . 6 |- (h = H -> ((((D` h) = (C` G) /\ (D` G) = (C` F)) -> (hR(GRF)) = ((hRG)RF)) <-> (((D` H) = (C` G) /\ (D` G) = (C` F)) -> (HR(GRF)) = ((HRG)RF))))
35	15, 26, 34	rcla43v 2386	. . . . 5 |- ((F e. M /\ G e. M /\ H e. M) -> (A.f e. M A.g e. M A.h e. M (((D` h) = (C` g) /\ (D` g) = (C` f)) -> (hR(gRf)) = ((hRg)Rf)) -> (((D` H) = (C` G) /\ (D` G) = (C` F)) -> (HR(GRF)) = ((HRG)RF))))
36	35	com12 14	. . . 4 |- (A.f e. M A.g e. M A.h e. M (((D` h) = (C` g) /\ (D` g) = (C` f)) -> (hR(gRf)) = ((hRg)Rf)) -> ((F e. M /\ G e. M /\ H e. M) -> (((D` H) = (C` G) /\ (D` G) = (C` F)) -> (HR(GRF)) = ((HRG)RF))))
37	36	ad2antlr 441	. . 3 |- (((<.<.D, C>., <.(id` T), R>.>. e. Ded /\ A.f e. M A.g e. M A.h e. M (((D` h) = (C` g) /\ (D` g) = (C` f)) -> (hR(gRf)) = ((hRg)Rf))) /\ (A.x e. dom (id` T)A.f e. M ((C` f) = x -> (((id` T)` x)Rf) = f) /\ A.x e. dom (id` T)A.f e. M ((D` f) = x -> (fR((id` T)` x)) = f))) -> ((F e. M /\ G e. M /\ H e. M) -> (((D` H) = (C` G) /\ (D` G) = (C` F)) -> (HR(GRF)) = ((HRG)RF))))
38	7, 37	syl 12	. 2 |- (T e. Cat -> ((F e. M /\ G e. M /\ H e. M) -> (((D` H) = (C` G) /\ (D` G) = (C` F)) -> (HR(GRF)) = ((HRG)RF))))
39	38	imp 377	1 |- ((T e. Cat /\ (F e. M /\ G e. M /\ H e. M)) -> (((D` H) = (C` G) /\ (D` G) = (C` F)) -> (HR(GRF)) = ((HRG)RF)))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  <.cop 3046  dom cdm 3986  ` cfv 3998  (class class class)co 4884  domcdom_ 15059  codccod_ 15060  idcid_ 15061  oco_ 15062   Ded cded 15081   Cat ccat 15099

This theorem is referenced by:  dualcat2 15133  cmpassoh 15150

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fo 4012  df-fv 4014  df-opr 4886  df-1st 5020  df-2nd 5021  df-doma 15064  df-coda 15065  df-ida 15066  df-cmpa 15067  df-cat 15100

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