Theorem ghmlin 13321

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 2193	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
3		ghmlin.a	. . . . . 6 |- .+ = ( +g ` S )
4		ghmlin.b	. . . . . 6 |- .+^ = ( +g ` T )
5	1, 2, 3, 4	isghm 13316	. . . . 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 5942	. . . . 5 |- ( a = U -> ( F ` ( a .+ b ) ) = ( F ` ( U .+ b ) ) )
9		fveq2 5555	. . . . . 6 |- ( a = U -> ( F ` a ) = ( F ` U ) )
10	9	oveq1d 5934	. . . . 5 |- ( a = U -> ( ( F ` a ) .+^ ( F ` b ) ) = ( ( F ` U ) .+^ ( F ` b ) ) )
11	8, 10	eqeq12d 2208	. . . 4 |- ( a = U -> ( ( F ` ( a .+ b ) ) = ( ( F ` a ) .+^ ( F ` b ) ) <-> ( F ` ( U .+ b ) ) = ( ( F ` U ) .+^ ( F ` b ) ) ) )
12		oveq2 5927	. . . . . 6 |- ( b = V -> ( U .+ b ) = ( U .+ V ) )
13	12	fveq2d 5559	. . . . 5 |- ( b = V -> ( F ` ( U .+ b ) ) = ( F ` ( U .+ V ) ) )
14		fveq2 5555	. . . . . 6 |- ( b = V -> ( F ` b ) = ( F ` V ) )
15	14	oveq2d 5935	. . . . 5 |- ( b = V -> ( ( F ` U ) .+^ ( F ` b ) ) = ( ( F ` U ) .+^ ( F ` V ) ) )
16	13, 15	eqeq12d 2208	. . . 4 |- ( b = V -> ( ( F ` ( U .+ b ) ) = ( ( F ` U ) .+^ ( F ` b ) ) <-> ( F ` ( U .+ V ) ) = ( ( F ` U ) .+^ ( F ` V ) ) ) )
17	11, 16	rspc2v 2878	. . 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 1201	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 980   = wceq 1364   e. wcel 2164   A.wral 2472   -->wf 5251   ` cfv 5255  (class class class)co 5919   Basecbs 12621   +g cplusg 12698   Grpcgrp 13075   GrpHom cghm 13313

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-inn 8985  df-ndx 12624  df-slot 12625  df-base 12627  df-ghm 13314

This theorem is referenced by:  ghmid  13322  ghminv  13323  ghmsub  13324  ghmmhm  13326  ghmrn  13330  resghm  13333  ghmpreima  13339  ghmnsgima  13341  ghmnsgpreima  13342  ghmf1o  13348  invghm  13402  rhmopp  13675