Theorem ghomgrpilem1 10385

Description: Lemma for ghomgrpi 10387.

Hypotheses

Ref	Expression
ghomgrpilem1.1	|- G e. Grp
ghomgrpilem1.2	|- H e. Grp
ghomgrpilem1.3	|- F e. (G GrpHom H)
ghomgrpilem1.4	|- X = ran G
ghomgrpilem1.5	|- U = (Id` G)
ghomgrpilem1.6	|- N = (inv` G)
ghomgrpilem1.7	|- W = ran H
ghomgrpilem1.8	|- T = (Id` H)
ghomgrpilem1.9	|- M = (inv` H)
ghomgrpilem1.10	|- Z = ran F
ghomgrpilem1.11	|- S = (H |` (Z X. Z))

Assertion

Ref	Expression
ghomgrpilem1	|- ((A e. X /\ B e. X) -> ((F` A)H(F` B)) = (F` (AGB)))

Proof of Theorem ghomgrpilem1

Step	Hyp	Ref	Expression
1		fveq2 3724	. . . . . 6 |- (x = A -> (F` x) = (F` A))
2	1	opreq1d 3975	. . . . 5 |- (x = A -> ((F` x)H(F` y)) = ((F` A)H(F` y)))
3		opreq1 3968	. . . . . 6 |- (x = A -> (xGy) = (AGy))
4	3	fveq2d 3728	. . . . 5 |- (x = A -> (F` (xGy)) = (F` (AGy)))
5	2, 4	eqeq12d 1489	. . . 4 |- (x = A -> (((F` x)H(F` y)) = (F` (xGy)) <-> ((F` A)H(F` y)) = (F` (AGy))))
6	5	ralbidv 1663	. . 3 |- (x = A -> (A.y e. X ((F` x)H(F` y)) = (F` (xGy)) <-> A.y e. X ((F` A)H(F` y)) = (F` (AGy))))
7		ghomgrpilem1.3	. . . . 5 |- F e. (G GrpHom H)
8		ghomgrpilem1.1	. . . . . 6 |- G e. Grp
9		ghomgrpilem1.2	. . . . . 6 |- H e. Grp
10		ghomgrpilem1.4	. . . . . . 7 |- X = ran G
11		ghomgrpilem1.7	. . . . . . 7 |- W = ran H
12	10, 11	elghom 10384	. . . . . 6 |- ((G e. Grp /\ H e. Grp) -> (F e. (G GrpHom H) <-> (F:X-->W /\ A.x e. X A.y e. X ((F` x)H(F` y)) = (F` (xGy)))))
13	8, 9, 12	mp2an 697	. . . . 5 |- (F e. (G GrpHom H) <-> (F:X-->W /\ A.x e. X A.y e. X ((F` x)H(F` y)) = (F` (xGy))))
14	7, 13	mpbi 189	. . . 4 |- (F:X-->W /\ A.x e. X A.y e. X ((F` x)H(F` y)) = (F` (xGy)))
15	14	pm3.27i 324	. . 3 |- A.x e. X A.y e. X ((F` x)H(F` y)) = (F` (xGy))
16	6, 15	vtoclri 1859	. 2 |- (A e. X -> A.y e. X ((F` A)H(F` y)) = (F` (AGy)))
17		fveq2 3724	. . . . 5 |- (y = B -> (F` y) = (F` B))
18	17	opreq2d 3976	. . . 4 |- (y = B -> ((F` A)H(F` y)) = ((F` A)H(F` B)))
19		opreq2 3969	. . . . 5 |- (y = B -> (AGy) = (AGB))
20	19	fveq2d 3728	. . . 4 |- (y = B -> (F` (AGy)) = (F` (AGB)))
21	18, 20	eqeq12d 1489	. . 3 |- (y = B -> (((F` A)H(F` y)) = (F` (AGy)) <-> ((F` A)H(F` B)) = (F` (AGB))))
22	21	rcla4v 1873	. 2 |- (B e. X -> (A.y e. X ((F` A)H(F` y)) = (F` (AGy)) -> ((F` A)H(F` B)) = (F` (AGB))))
23	16, 22	mpan9 470	1 |- ((A e. X /\ B e. X) -> ((F` A)H(F` B)) = (F` (AGB)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645   X. cxp 3168  ran crn 3171   |` cres 3172  -->wf 3178  ` cfv 3182  (class class class)co 3963  Grpcgr 8033  Idcgi 8034  invcgn 8035   GrpHom cghom 10378

This theorem is referenced by:  ghomgrpilem2 10386

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-opr 3965  df-oprab 3966  df-ghom 10380

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