Theorem cmprtr 14760

Description: Composite of two right translations.

Hypotheses

Ref	Expression
trfun.2	|- F = (x e. X |-> (xGA))
trinv.1	|- X = ran G
cmprtr.1	|- G = (x e. X |-> (xGB))

Assertion

Ref	Expression
cmprtr	|- ((G e. Grp /\ A e. X /\ B e. X) -> (F o. G) = (x e. X |-> (xG(BGA))))

Distinct variable groups:   x,A   x,B   x,F   x,G   x,X

Proof of Theorem cmprtr

Step	Hyp	Ref	Expression
1		trinv.1	. . . . 5 |- X = ran G
2	1	eqimss2i 2669	. . . 4 |- ran G C_ X
3	2	a1i 8	. . 3 |- ((G e. Grp /\ A e. X /\ B e. X) -> ran G C_ X)
4		oprex 4907	. . . 4 |- (xGB) e. _V
5		oprex 4907	. . . 4 |- (aGA) e. _V
6		oprex 4907	. . . 4 |- ((xGB)GA) e. _V
7		opreq1 4889	. . . 4 |- (a = (xGB) -> (aGA) = ((xGB)GA))
8		cmprtr.1	. . . . 5 |- G = (x e. X |-> (xGB))
9		df-mpt 5006	. . . . 5 |- (x e. X |-> (xGB)) = {<.x, y>. | (x e. X /\ y = (xGB))}
10	8, 9	eqtri 1908	. . . 4 |- G = {<.x, y>. | (x e. X /\ y = (xGB))}
11		trfun.2	. . . . 5 |- F = (x e. X |-> (xGA))
12		df-mpt 5006	. . . . 5 |- (x e. X |-> (xGA)) = {<.x, y>. | (x e. X /\ y = (xGA))}
13		ax-17 1317	. . . . . 6 |- ((x e. X /\ y = (xGA)) -> A.a(x e. X /\ y = (xGA)))
14		ax-17 1317	. . . . . 6 |- ((x e. X /\ y = (xGA)) -> A.b(x e. X /\ y = (xGA)))
15		ax-17 1317	. . . . . 6 |- ((a e. X /\ b = (aGA)) -> A.x(a e. X /\ b = (aGA)))
16		ax-17 1317	. . . . . 6 |- ((a e. X /\ b = (aGA)) -> A.y(a e. X /\ b = (aGA)))
17		eleq1 1957	. . . . . . . 8 |- (x = a -> (x e. X <-> a e. X))
18	17	adantr 425	. . . . . . 7 |- ((x = a /\ y = b) -> (x e. X <-> a e. X))
19		simpr 350	. . . . . . . 8 |- ((x = a /\ y = b) -> y = b)
20		opreq1 4889	. . . . . . . . 9 |- (x = a -> (xGA) = (aGA))
21	20	adantr 425	. . . . . . . 8 |- ((x = a /\ y = b) -> (xGA) = (aGA))
22	19, 21	eqeq12d 1899	. . . . . . 7 |- ((x = a /\ y = b) -> (y = (xGA) <-> b = (aGA)))
23	18, 22	anbi12d 690	. . . . . 6 |- ((x = a /\ y = b) -> ((x e. X /\ y = (xGA)) <-> (a e. X /\ b = (aGA))))
24	13, 14, 15, 16, 23	cbvopab 3403	. . . . 5 |- {<.x, y>. | (x e. X /\ y = (xGA))} = {<.a, b>. | (a e. X /\ b = (aGA))}
25	11, 12, 24	3eqtri 1912	. . . 4 |- F = {<.a, b>. | (a e. X /\ b = (aGA))}
26		eqid 1884	. . . 4 |- {<.x, y>. | (x e. X /\ y = ((xGB)GA))} = {<.x, y>. | (x e. X /\ y = ((xGB)GA))}
27	4, 5, 6, 7, 10, 25, 26	fopabco 4805	. . 3 |- (ran G C_ X -> (F o. G) = {<.x, y>. | (x e. X /\ y = ((xGB)GA))})
28	3, 27	syl 12	. 2 |- ((G e. Grp /\ A e. X /\ B e. X) -> (F o. G) = {<.x, y>. | (x e. X /\ y = ((xGB)GA))})
29		ax-17 1317	. . 3 |- ((G e. Grp /\ A e. X /\ B e. X) -> A.x(G e. Grp /\ A e. X /\ B e. X))
30		ax-17 1317	. . 3 |- ((G e. Grp /\ A e. X /\ B e. X) -> A.y(G e. Grp /\ A e. X /\ B e. X))
31		simpl1 879	. . . . . 6 |- (((G e. Grp /\ A e. X /\ B e. X) /\ x e. X) -> G e. Grp)
32		simpr 350	. . . . . 6 |- (((G e. Grp /\ A e. X /\ B e. X) /\ x e. X) -> x e. X)
33		simpl3 881	. . . . . 6 |- (((G e. Grp /\ A e. X /\ B e. X) /\ x e. X) -> B e. X)
34		simpl2 880	. . . . . 6 |- (((G e. Grp /\ A e. X /\ B e. X) /\ x e. X) -> A e. X)
35	1	grpass 9327	. . . . . 6 |- ((G e. Grp /\ (x e. X /\ B e. X /\ A e. X)) -> ((xGB)GA) = (xG(BGA)))
36	31, 32, 33, 34, 35	syl13anc 1102	. . . . 5 |- (((G e. Grp /\ A e. X /\ B e. X) /\ x e. X) -> ((xGB)GA) = (xG(BGA)))
37	36	eqeq2d 1895	. . . 4 |- (((G e. Grp /\ A e. X /\ B e. X) /\ x e. X) -> (y = ((xGB)GA) <-> y = (xG(BGA))))
38	37	pm5.32da 711	. . 3 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((x e. X /\ y = ((xGB)GA)) <-> (x e. X /\ y = (xG(BGA)))))
39	29, 30, 38	opabbid 3399	. 2 |- ((G e. Grp /\ A e. X /\ B e. X) -> {<.x, y>. | (x e. X /\ y = ((xGB)GA))} = {<.x, y>. | (x e. X /\ y = (xG(BGA)))})
40		df-mpt 5006	. . . 4 |- (x e. X |-> (xG(BGA))) = {<.x, y>. | (x e. X /\ y = (xG(BGA)))}
41	40	eqcomi 1888	. . 3 |- {<.x, y>. | (x e. X /\ y = (xG(BGA)))} = (x e. X |-> (xG(BGA)))
42	41	a1i 8	. 2 |- ((G e. Grp /\ A e. X /\ B e. X) -> {<.x, y>. | (x e. X /\ y = (xG(BGA)))} = (x e. X |-> (xG(BGA))))
43	28, 39, 42	3eqtrd 1929	1 |- ((G e. Grp /\ A e. X /\ B e. X) -> (F o. G) = (x e. X |-> (xG(BGA))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   C_ wss 2593  {copab 3395  ran crn 3987   o. ccom 3990  (class class class)co 4884   e. cmpt 5004  Grpcgr 9311

This theorem is referenced by:  cmprtr2 14761

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-mpt 5006  df-grp 9316

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