Theorem ghgrpilem1 8097

Description: Lemma for ghgrpi 8101.

Hypotheses

Ref	Expression
ghgrpi.1	|- G e. Grp
ghgrpi.2	|- X = ran G
ghgrpi.3	|- F:X-onto->Y
ghgrpi.4	|- Y (_ A
ghgrpi.5	|- O Fn (A X. A)
ghgrpi.6	|- ((x e. X /\ y e. X) -> (F` (xGy)) = ((F` x)O(F` y)))
ghgrpi.7	|- H = (O |` (Y X. Y))

Assertion

Ref	Expression
ghgrpilem1	|- ((C e. X /\ D e. X) -> (F` (CGD)) = ((F` C)H(F` D)))

Distinct variable groups:   x,F,y   x,G,y   x,H,y   x,O,y   x,X,y   x,Y,y

Proof of Theorem ghgrpilem1

Step	Hyp	Ref	Expression
1		ghgrpi.6	. . . . 5 |- ((x e. X /\ y e. X) -> (F` (xGy)) = ((F` x)O(F` y)))
2	1	rgen2a 1697	. . . 4 |- A.x e. X A.y e. X (F` (xGy)) = ((F` x)O(F` y))
3		opreq1 3963	. . . . . . 7 |- (x = a -> (xGy) = (aGy))
4	3	fveq2d 3723	. . . . . 6 |- (x = a -> (F` (xGy)) = (F` (aGy)))
5		fveq2 3719	. . . . . . 7 |- (x = a -> (F` x) = (F` a))
6	5	opreq1d 3970	. . . . . 6 |- (x = a -> ((F` x)O(F` y)) = ((F` a)O(F` y)))
7	4, 6	eqeq12d 1487	. . . . 5 |- (x = a -> ((F` (xGy)) = ((F` x)O(F` y)) <-> (F` (aGy)) = ((F` a)O(F` y))))
8		opreq2 3964	. . . . . . 7 |- (y = b -> (aGy) = (aGb))
9	8	fveq2d 3723	. . . . . 6 |- (y = b -> (F` (aGy)) = (F` (aGb)))
10		fveq2 3719	. . . . . . 7 |- (y = b -> (F` y) = (F` b))
11	10	opreq2d 3971	. . . . . 6 |- (y = b -> ((F` a)O(F` y)) = ((F` a)O(F` b)))
12	9, 11	eqeq12d 1487	. . . . 5 |- (y = b -> ((F` (aGy)) = ((F` a)O(F` y)) <-> (F` (aGb)) = ((F` a)O(F` b))))
13	7, 12	cbvral2v 1800	. . . 4 |- (A.x e. X A.y e. X (F` (xGy)) = ((F` x)O(F` y)) <-> A.a e. X A.b e. X (F` (aGb)) = ((F` a)O(F` b)))
14	2, 13	mpbi 189	. . 3 |- A.a e. X A.b e. X (F` (aGb)) = ((F` a)O(F` b))
15		opreq1 3963	. . . . . 6 |- (a = C -> (aGb) = (CGb))
16	15	fveq2d 3723	. . . . 5 |- (a = C -> (F` (aGb)) = (F` (CGb)))
17		fveq2 3719	. . . . . 6 |- (a = C -> (F` a) = (F` C))
18	17	opreq1d 3970	. . . . 5 |- (a = C -> ((F` a)O(F` b)) = ((F` C)O(F` b)))
19	16, 18	eqeq12d 1487	. . . 4 |- (a = C -> ((F` (aGb)) = ((F` a)O(F` b)) <-> (F` (CGb)) = ((F` C)O(F` b))))
20		opreq2 3964	. . . . . 6 |- (b = D -> (CGb) = (CGD))
21	20	fveq2d 3723	. . . . 5 |- (b = D -> (F` (CGb)) = (F` (CGD)))
22		fveq2 3719	. . . . . 6 |- (b = D -> (F` b) = (F` D))
23	22	opreq2d 3971	. . . . 5 |- (b = D -> ((F` C)O(F` b)) = ((F` C)O(F` D)))
24	21, 23	eqeq12d 1487	. . . 4 |- (b = D -> ((F` (CGb)) = ((F` C)O(F` b)) <-> (F` (CGD)) = ((F` C)O(F` D))))
25	19, 24	rcla42v 1877	. . 3 |- ((C e. X /\ D e. X) -> (A.a e. X A.b e. X (F` (aGb)) = ((F` a)O(F` b)) -> (F` (CGD)) = ((F` C)O(F` D))))
26	14, 25	mpi 44	. 2 |- ((C e. X /\ D e. X) -> (F` (CGD)) = ((F` C)O(F` D)))
27		oprvalres 4028	. . . 4 |- (((F` C) e. Y /\ (F` D) e. Y) -> ((F` C)(O |` (Y X. Y))(F` D)) = ((F` C)O(F` D)))
28		ghgrpi.7	. . . . 5 |- H = (O |` (Y X. Y))
29	28	opreqi 3969	. . . 4 |- ((F` C)H(F` D)) = ((F` C)(O |` (Y X. Y))(F` D))
30	27, 29	syl5eq 1517	. . 3 |- (((F` C) e. Y /\ (F` D) e. Y) -> ((F` C)H(F` D)) = ((F` C)O(F` D)))
31		ghgrpi.3	. . . . . . 7 |- F:X-onto->Y
32		df-fo 3192	. . . . . . 7 |- (F:X-onto->Y <-> (F Fn X /\ ran F = Y))
33	31, 32	mpbi 189	. . . . . 6 |- (F Fn X /\ ran F = Y)
34	33	pm3.26i 320	. . . . 5 |- F Fn X
35		fnfvelrn 3808	. . . . 5 |- ((F Fn X /\ C e. X) -> (F` C) e. ran F)
36	34, 35	mpan 694	. . . 4 |- (C e. X -> (F` C) e. ran F)
37	33	pm3.27i 324	. . . 4 |- ran F = Y
38	36, 37	syl6eleq 1556	. . 3 |- (C e. X -> (F` C) e. Y)
39		fnfvelrn 3808	. . . . 5 |- ((F Fn X /\ D e. X) -> (F` D) e. ran F)
40	34, 39	mpan 694	. . . 4 |- (D e. X -> (F` D) e. ran F)
41	40, 37	syl6eleq 1556	. . 3 |- (D e. X -> (F` D) e. Y)
42	30, 38, 41	syl2an 454	. 2 |- ((C e. X /\ D e. X) -> ((F` C)H(F` D)) = ((F` C)O(F` D)))
43	26, 42	eqtr4d 1508	1 |- ((C e. X /\ D e. X) -> (F` (CGD)) = ((F` C)H(F` D)))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 955   e. wcel 957  A.wral 1643   (_ wss 2044   X. cxp 3164  ran crn 3167   |` cres 3168   Fn wfn 3173  -onto->wfo 3176  ` cfv 3178  (class class class)co 3958  Grpcgr 7995

This theorem is referenced by:  ghgrpilem3 8099  ghgrpilem4 8100  ghgrpi 8101

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-fo 3192  df-fv 3194  df-opr 3960

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