Theorem ghmlin 14051

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 2238	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
3		ghmlin.a	. . . . . 6 |- .+ = ( +g ` S )
4		ghmlin.b	. . . . . 6 |- .+^ = ( +g ` T )
5	1, 2, 3, 4	isghm 14046	. . . . 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 6108	. . . . 5 |- ( a = U -> ( F ` ( a .+ b ) ) = ( F ` ( U .+ b ) ) )
9		fveq2 5695	. . . . . 6 |- ( a = U -> ( F ` a ) = ( F ` U ) )
10	9	oveq1d 6100	. . . . 5 |- ( a = U -> ( ( F ` a ) .+^ ( F ` b ) ) = ( ( F ` U ) .+^ ( F ` b ) ) )
11	8, 10	eqeq12d 2253	. . . 4 |- ( a = U -> ( ( F ` ( a .+ b ) ) = ( ( F ` a ) .+^ ( F ` b ) ) <-> ( F ` ( U .+ b ) ) = ( ( F ` U ) .+^ ( F ` b ) ) ) )
12		oveq2 6093	. . . . . 6 |- ( b = V -> ( U .+ b ) = ( U .+ V ) )
13	12	fveq2d 5699	. . . . 5 |- ( b = V -> ( F ` ( U .+ b ) ) = ( F ` ( U .+ V ) ) )
14		fveq2 5695	. . . . . 6 |- ( b = V -> ( F ` b ) = ( F ` V ) )
15	14	oveq2d 6101	. . . . 5 |- ( b = V -> ( ( F ` U ) .+^ ( F ` b ) ) = ( ( F ` U ) .+^ ( F ` V ) ) )
16	13, 15	eqeq12d 2253	. . . 4 |- ( b = V -> ( ( F ` ( U .+ b ) ) = ( ( F ` U ) .+^ ( F ` b ) ) <-> ( F ` ( U .+ V ) ) = ( ( F ` U ) .+^ ( F ` V ) ) ) )
17	11, 16	rspc2v 2943	. . 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 1230	1 |- ( ( F e. ( S GrpHom T ) /\ U e. X /\ V e. X ) -> ( F ` ( U .+ V ) ) = ( ( F ` U ) .+^ ( F ` V ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   -->wf 5373   ` cfv 5377  (class class class)co 6085   Basecbs 13352   +g cplusg 13431   Grpcgrp 13805   GrpHom cghm 14043

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-inn 9305  df-ndx 13355  df-slot 13356  df-base 13358  df-ghm 14044

This theorem is used by:  ghmid  14052  ghminv  14053  ghmsub  14054  ghmmhm  14056  ghmrn  14060  resghm  14063  ghmpreima  14069  ghmnsgima  14071  ghmnsgpreima  14072  ghmf1o  14078  invghm  14133  rhmopp  14483