Theorem ghmlin 14001

Description: A homomorphism of groups is linear. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Hypotheses

Ref	Expression
ghmlin.x	|- X = ( Base ` S )
ghmlin.a	|- .+ = ( +g ` S )
ghmlin.b	|- .+^ = ( +g ` T )

Assertion

Ref	Expression
ghmlin	|- ( ( F e. ( S GrpHom T ) /\ U e. X /\ V e. X ) -> ( F ` ( U .+ V ) ) = ( ( F ` U ) .+^ ( F ` V ) ) )

Proof of Theorem ghmlin

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ghmlin.x	. . . . . 6 |- X = ( Base ` S )
2		eqid 2234	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
3		ghmlin.a	. . . . . 6 |- .+ = ( +g ` S )
4		ghmlin.b	. . . . . 6 |- .+^ = ( +g ` T )
5	1, 2, 3, 4	isghm 13996	. . . . 5 |- ( F e. ( S GrpHom T ) <-> ( ( S e. Grp /\ T e. Grp ) /\ ( F : X --> ( Base ` T ) /\ A. a e. X A. b e. X ( F ` ( a .+ b ) ) = ( ( F ` a ) .+^ ( F ` b ) ) ) ) )
6	5	simprbi 275	. . . 4 |- ( F e. ( S GrpHom T ) -> ( F : X --> ( Base ` T ) /\ A. a e. X A. b e. X ( F ` ( a .+ b ) ) = ( ( F ` a ) .+^ ( F ` b ) ) ) )
7	6	simprd 114	. . 3 |- ( F e. ( S GrpHom T ) -> A. a e. X A. b e. X ( F ` ( a .+ b ) ) = ( ( F ` a ) .+^ ( F ` b ) ) )
8		fvoveq1 6081	. . . . 5 |- ( a = U -> ( F ` ( a .+ b ) ) = ( F ` ( U .+ b ) ) )
9		fveq2 5675	. . . . . 6 |- ( a = U -> ( F ` a ) = ( F ` U ) )
10	9	oveq1d 6073	. . . . 5 |- ( a = U -> ( ( F ` a ) .+^ ( F ` b ) ) = ( ( F ` U ) .+^ ( F ` b ) ) )
11	8, 10	eqeq12d 2249	. . . 4 |- ( a = U -> ( ( F ` ( a .+ b ) ) = ( ( F ` a ) .+^ ( F ` b ) ) <-> ( F ` ( U .+ b ) ) = ( ( F ` U ) .+^ ( F ` b ) ) ) )
12		oveq2 6066	. . . . . 6 |- ( b = V -> ( U .+ b ) = ( U .+ V ) )
13	12	fveq2d 5679	. . . . 5 |- ( b = V -> ( F ` ( U .+ b ) ) = ( F ` ( U .+ V ) ) )
14		fveq2 5675	. . . . . 6 |- ( b = V -> ( F ` b ) = ( F ` V ) )
15	14	oveq2d 6074	. . . . 5 |- ( b = V -> ( ( F ` U ) .+^ ( F ` b ) ) = ( ( F ` U ) .+^ ( F ` V ) ) )
16	13, 15	eqeq12d 2249	. . . 4 |- ( b = V -> ( ( F ` ( U .+ b ) ) = ( ( F ` U ) .+^ ( F ` b ) ) <-> ( F ` ( U .+ V ) ) = ( ( F ` U ) .+^ ( F ` V ) ) ) )
17	11, 16	rspc2v 2937	. . 3 |- ( ( U e. X /\ V e. X ) -> ( A. a e. X A. b e. X ( F ` ( a .+ b ) ) = ( ( F ` a ) .+^ ( F ` b ) ) -> ( F ` ( U .+ V ) ) = ( ( F ` U ) .+^ ( F ` V ) ) ) )
18	7, 17	mpan9 281	. 2 |- ( ( F e. ( S GrpHom T ) /\ ( U e. X /\ V e. X ) ) -> ( F ` ( U .+ V ) ) = ( ( F ` U ) .+^ ( F ` V ) ) )
19	18	3impb 1226	1 |- ( ( F e. ( S GrpHom T ) /\ U e. X /\ V e. X ) -> ( F ` ( U .+ V ) ) = ( ( F ` U ) .+^ ( F ` V ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2205   A.wral 2522   -->wf 5353   ` cfv 5357  (class class class)co 6058   Basecbs 13296   +g cplusg 13374   Grpcgrp 13755   GrpHom cghm 13993

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-inn 9255  df-ndx 13299  df-slot 13300  df-base 13302  df-ghm 13994

This theorem is referenced by:  ghmid  14002  ghminv  14003  ghmsub  14004  ghmmhm  14006  ghmrn  14010  resghm  14013  ghmpreima  14019  ghmnsgima  14021  ghmnsgpreima  14022  ghmf1o  14028  invghm  14082  rhmopp  14421