Theorem lmodvsghm 18924

Description: Scalar multiplication of the vector space by a fixed scalar is an automorphism of the additive group of vectors. (Contributed by Mario Carneiro, 5-May-2015.)

Hypotheses

Ref	Expression
lmodvsghm.v	|- V = ( Base ` W )
lmodvsghm.f	|- F = (Scalar ` W )
lmodvsghm.s	|- .x. = ( .s ` W )
lmodvsghm.k	|- K = ( Base ` F )

Assertion

Ref	Expression
lmodvsghm	|- ( ( W e. LMod /\ R e. K ) -> ( x e. V |-> ( R .x. x ) ) e. ( W GrpHom W ) )

Distinct variable groups:   x, K   x, R   x, .x.   x, V   x, W

Allowed substitution hint:   F( x)

Proof of Theorem lmodvsghm

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lmodvsghm.v	. 2 |- V = ( Base ` W )
2		eqid 2622	. 2 |- ( +g ` W ) = ( +g ` W )
3		lmodgrp 18870	. . 3 |- ( W e. LMod -> W e. Grp )
4	3	adantr 481	. 2 |- ( ( W e. LMod /\ R e. K ) -> W e. Grp )
5		lmodvsghm.f	. . . . 5 |- F = (Scalar ` W )
6		lmodvsghm.s	. . . . 5 |- .x. = ( .s ` W )
7		lmodvsghm.k	. . . . 5 |- K = ( Base ` F )
8	1, 5, 6, 7	lmodvscl 18880	. . . 4 |- ( ( W e. LMod /\ R e. K /\ x e. V ) -> ( R .x. x ) e. V )
9	8	3expa 1265	. . 3 |- ( ( ( W e. LMod /\ R e. K ) /\ x e. V ) -> ( R .x. x ) e. V )
10		eqid 2622	. . 3 |- ( x e. V |-> ( R .x. x ) ) = ( x e. V |-> ( R .x. x ) )
11	9, 10	fmptd 6385	. 2 |- ( ( W e. LMod /\ R e. K ) -> ( x e. V |-> ( R .x. x ) ) : V --> V )
12	1, 2, 5, 6, 7	lmodvsdi 18886	. . . . 5 |- ( ( W e. LMod /\ ( R e. K /\ y e. V /\ z e. V ) ) -> ( R .x. ( y ( +g ` W ) z ) ) = ( ( R .x. y ) ( +g ` W ) ( R .x. z ) ) )
13	12	3exp2 1285	. . . 4 |- ( W e. LMod -> ( R e. K -> ( y e. V -> ( z e. V -> ( R .x. ( y ( +g ` W ) z ) ) = ( ( R .x. y ) ( +g ` W ) ( R .x. z ) ) ) ) ) )
14	13	imp43 621	. . 3 |- ( ( ( W e. LMod /\ R e. K ) /\ ( y e. V /\ z e. V ) ) -> ( R .x. ( y ( +g ` W ) z ) ) = ( ( R .x. y ) ( +g ` W ) ( R .x. z ) ) )
15	1, 2	lmodvacl 18877	. . . . . 6 |- ( ( W e. LMod /\ y e. V /\ z e. V ) -> ( y ( +g ` W ) z ) e. V )
16	15	3expb 1266	. . . . 5 |- ( ( W e. LMod /\ ( y e. V /\ z e. V ) ) -> ( y ( +g ` W ) z ) e. V )
17	16	adantlr 751	. . . 4 |- ( ( ( W e. LMod /\ R e. K ) /\ ( y e. V /\ z e. V ) ) -> ( y ( +g ` W ) z ) e. V )
18		oveq2 6658	. . . . 5 |- ( x = ( y ( +g ` W ) z ) -> ( R .x. x ) = ( R .x. ( y ( +g ` W ) z ) ) )
19		ovex 6678	. . . . 5 |- ( R .x. ( y ( +g ` W ) z ) ) e. _V
20	18, 10, 19	fvmpt 6282	. . . 4 |- ( ( y ( +g ` W ) z ) e. V -> ( ( x e. V |-> ( R .x. x ) ) ` ( y ( +g ` W ) z ) ) = ( R .x. ( y ( +g ` W ) z ) ) )
21	17, 20	syl 17	. . 3 |- ( ( ( W e. LMod /\ R e. K ) /\ ( y e. V /\ z e. V ) ) -> ( ( x e. V |-> ( R .x. x ) ) ` ( y ( +g ` W ) z ) ) = ( R .x. ( y ( +g ` W ) z ) ) )
22		oveq2 6658	. . . . . 6 |- ( x = y -> ( R .x. x ) = ( R .x. y ) )
23		ovex 6678	. . . . . 6 |- ( R .x. y ) e. _V
24	22, 10, 23	fvmpt 6282	. . . . 5 |- ( y e. V -> ( ( x e. V |-> ( R .x. x ) ) ` y ) = ( R .x. y ) )
25		oveq2 6658	. . . . . 6 |- ( x = z -> ( R .x. x ) = ( R .x. z ) )
26		ovex 6678	. . . . . 6 |- ( R .x. z ) e. _V
27	25, 10, 26	fvmpt 6282	. . . . 5 |- ( z e. V -> ( ( x e. V |-> ( R .x. x ) ) ` z ) = ( R .x. z ) )
28	24, 27	oveqan12d 6669	. . . 4 |- ( ( y e. V /\ z e. V ) -> ( ( ( x e. V |-> ( R .x. x ) ) ` y ) ( +g ` W ) ( ( x e. V |-> ( R .x. x ) ) ` z ) ) = ( ( R .x. y ) ( +g ` W ) ( R .x. z ) ) )
29	28	adantl 482	. . 3 |- ( ( ( W e. LMod /\ R e. K ) /\ ( y e. V /\ z e. V ) ) -> ( ( ( x e. V |-> ( R .x. x ) ) ` y ) ( +g ` W ) ( ( x e. V |-> ( R .x. x ) ) ` z ) ) = ( ( R .x. y ) ( +g ` W ) ( R .x. z ) ) )
30	14, 21, 29	3eqtr4d 2666	. 2 |- ( ( ( W e. LMod /\ R e. K ) /\ ( y e. V /\ z e. V ) ) -> ( ( x e. V |-> ( R .x. x ) ) ` ( y ( +g ` W ) z ) ) = ( ( ( x e. V |-> ( R .x. x ) ) ` y ) ( +g ` W ) ( ( x e. V |-> ( R .x. x ) ) ` z ) ) )
31	1, 1, 2, 2, 4, 4, 11, 30	isghmd 17669	1 |- ( ( W e. LMod /\ R e. K ) -> ( x e. V |-> ( R .x. x ) ) e. ( W GrpHom W ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941  Scalarcsca 15944   .scvsca 15945   Grpcgrp 17422   GrpHom cghm 17657   LModclmod 18863

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-ghm 17658  df-lmod 18865

This theorem is referenced by:  gsumvsmul  18927  lmhmvsca  19045