Theorem ghomgrpilem1 10380

Description: Lemma for ghomgrpi 10382.

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 3730	. . . . . 6 |- (x = A -> (F` x) = (F` A))
2	1	opreq1d 3981	. . . . 5 |- (x = A -> ((F` x)H(F` y)) = ((F` A)H(F` y)))
3		opreq1 3974	. . . . . 6 |- (x = A -> (xGy) = (AGy))
4	3	fveq2d 3734	. . . . 5 |- (x = A -> (F` (xGy)) = (F` (AGy)))
5	2, 4	eqeq12d 1492	. . . 4 |- (x = A -> (((F` x)H(F` y)) = (F` (xGy)) <-> ((F` A)H(F` y)) = (F` (AGy))))
6	5	ralbidv 1666	. . 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 10379	. . . . . 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 699	. . . . 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 1862	. 2 |- (A e. X -> A.y e. X ((F` A)H(F` y)) = (F` (AGy)))
17		fveq2 3730	. . . . 5 |- (y = B -> (F` y) = (F` B))
18	17	opreq2d 3982	. . . 4 |- (y = B -> ((F` A)H(F` y)) = ((F` A)H(F` B)))
19		opreq2 3975	. . . . 5 |- (y = B -> (AGy) = (AGB))
20	19	fveq2d 3734	. . . 4 |- (y = B -> (F` (AGy)) = (F` (AGB)))
21	18, 20	eqeq12d 1492	. . 3 |- (y = B -> (((F` A)H(F` y)) = (F` (AGy)) <-> ((F` A)H(F` B)) = (F` (AGB))))
22	21	rcla4v 1876	. 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 472	1 |- ((A e. X /\ B e. X) -> ((F` A)H(F` B)) = (F` (AGB)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  A.wral 1648   X. cxp 3174  ran crn 3177   |` cres 3178  -->wf 3184  ` cfv 3188  (class class class)co 3969  Grpcgr 8030  Idcgi 8031  invcgn 8032   GrpHom cghom 10373

This theorem is referenced by:  ghomgrpilem2 10381

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-opr 3971  df-oprab 3972  df-ghom 10375

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