Theorem ghmlin 13780

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 2229	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
3		ghmlin.a	. . . . . 6 |- .+ = ( +g ` S )
4		ghmlin.b	. . . . . 6 |- .+^ = ( +g ` T )
5	1, 2, 3, 4	isghm 13775	. . . . 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 6023	. . . . 5 |- ( a = U -> ( F ` ( a .+ b ) ) = ( F ` ( U .+ b ) ) )
9		fveq2 5626	. . . . . 6 |- ( a = U -> ( F ` a ) = ( F ` U ) )
10	9	oveq1d 6015	. . . . 5 |- ( a = U -> ( ( F ` a ) .+^ ( F ` b ) ) = ( ( F ` U ) .+^ ( F ` b ) ) )
11	8, 10	eqeq12d 2244	. . . 4 |- ( a = U -> ( ( F ` ( a .+ b ) ) = ( ( F ` a ) .+^ ( F ` b ) ) <-> ( F ` ( U .+ b ) ) = ( ( F ` U ) .+^ ( F ` b ) ) ) )
12		oveq2 6008	. . . . . 6 |- ( b = V -> ( U .+ b ) = ( U .+ V ) )
13	12	fveq2d 5630	. . . . 5 |- ( b = V -> ( F ` ( U .+ b ) ) = ( F ` ( U .+ V ) ) )
14		fveq2 5626	. . . . . 6 |- ( b = V -> ( F ` b ) = ( F ` V ) )
15	14	oveq2d 6016	. . . . 5 |- ( b = V -> ( ( F ` U ) .+^ ( F ` b ) ) = ( ( F ` U ) .+^ ( F ` V ) ) )
16	13, 15	eqeq12d 2244	. . . 4 |- ( b = V -> ( ( F ` ( U .+ b ) ) = ( ( F ` U ) .+^ ( F ` b ) ) <-> ( F ` ( U .+ V ) ) = ( ( F ` U ) .+^ ( F ` V ) ) ) )
17	11, 16	rspc2v 2920	. . 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 1223	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 1002   = wceq 1395   e. wcel 2200   A.wral 2508   -->wf 5313   ` cfv 5317  (class class class)co 6000   Basecbs 13027   +g cplusg 13105   Grpcgrp 13528   GrpHom cghm 13772

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1re 8089  ax-addrcl 8092

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-inn 9107  df-ndx 13030  df-slot 13031  df-base 13033  df-ghm 13773

This theorem is referenced by:  ghmid  13781  ghminv  13782  ghmsub  13783  ghmmhm  13785  ghmrn  13789  resghm  13792  ghmpreima  13798  ghmnsgima  13800  ghmnsgpreima  13801  ghmf1o  13807  invghm  13861  rhmopp  14134