Theorem ghomlin 10393

Description: Linearity of a group homomorphism. (Contributed by Paul Chapman, 3-Mar-2008.)

Hypothesis

Ref	Expression
ghomlin.1	|- X = ran G

Assertion

Ref	Expression
ghomlin	|- (((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) /\ (A e. X /\ B e. X)) -> ((F` A)H(F` B)) = (F` (AGB)))

Proof of Theorem ghomlin

Step	Hyp	Ref	Expression
1		ghomlin.1	. . . . 5 |- X = ran G
2		eqid 1475	. . . . 5 |- ran H = ran H
3	1, 2	elghom 10384	. . . 4 |- ((G e. Grp /\ H e. Grp) -> (F e. (G GrpHom H) <-> (F:X-->ran H /\ A.x e. X A.y e. X ((F` x)H(F` y)) = (F` (xGy)))))
4	3	biimp3a 919	. . 3 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (F:X-->ran H /\ A.x e. X A.y e. X ((F` x)H(F` y)) = (F` (xGy))))
5	4	pm3.27d 325	. 2 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> A.x e. X A.y e. X ((F` x)H(F` y)) = (F` (xGy)))
6		fveq2 3724	. . . . 5 |- (x = A -> (F` x) = (F` A))
7	6	opreq1d 3975	. . . 4 |- (x = A -> ((F` x)H(F` y)) = ((F` A)H(F` y)))
8		opreq1 3968	. . . . 5 |- (x = A -> (xGy) = (AGy))
9	8	fveq2d 3728	. . . 4 |- (x = A -> (F` (xGy)) = (F` (AGy)))
10	7, 9	eqeq12d 1489	. . 3 |- (x = A -> (((F` x)H(F` y)) = (F` (xGy)) <-> ((F` A)H(F` y)) = (F` (AGy))))
11		fveq2 3724	. . . . 5 |- (y = B -> (F` y) = (F` B))
12	11	opreq2d 3976	. . . 4 |- (y = B -> ((F` A)H(F` y)) = ((F` A)H(F` B)))
13		opreq2 3969	. . . . 5 |- (y = B -> (AGy) = (AGB))
14	13	fveq2d 3728	. . . 4 |- (y = B -> (F` (AGy)) = (F` (AGB)))
15	12, 14	eqeq12d 1489	. . 3 |- (y = B -> (((F` A)H(F` y)) = (F` (AGy)) <-> ((F` A)H(F` B)) = (F` (AGB))))
16	10, 15	rcla42v 1880	. 2 |- ((A e. X /\ B e. X) -> (A.x e. X A.y e. X ((F` x)H(F` y)) = (F` (xGy)) -> ((F` A)H(F` B)) = (F` (AGB))))
17	5, 16	mpan9 470	1 |- (((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) /\ (A e. X /\ B e. X)) -> ((F` A)H(F` B)) = (F` (AGB)))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  A.wral 1645  ran crn 3171  -->wf 3178  ` cfv 3182  (class class class)co 3963  Grpcgr 8033   GrpHom cghom 10378

This theorem is referenced by:  ghomid 10394  ghomf1olem 10396

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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