Theorem asclghm 15025

Description: The algebra scalar lifting function is a group homomorphism. (Contributed by Mario Carneiro, 4-Jul-2015.)

Hypotheses

Ref	Expression
asclf.a	|- A = (algSc ` W )
asclf.f	|- F = (Scalar ` W )
asclf.r	|- ( ph -> W e. Ring )
asclf.l	|- ( ph -> W e. LMod )

Assertion

Ref	Expression
asclghm	|- ( ph -> A e. ( F GrpHom W ) )

Proof of Theorem asclghm

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

Step	Hyp	Ref	Expression
1		eqid 2238	. 2 |- ( Base ` F ) = ( Base ` F )
2		eqid 2238	. 2 |- ( Base ` W ) = ( Base ` W )
3		eqid 2238	. 2 |- ( +g ` F ) = ( +g ` F )
4		eqid 2238	. 2 |- ( +g ` W ) = ( +g ` W )
5		asclf.l	. . . 4 |- ( ph -> W e. LMod )
6		asclf.f	. . . . 5 |- F = (Scalar ` W )
7	6	lmodring 14631	. . . 4 |- ( W e. LMod -> F e. Ring )
8	5, 7	syl 14	. . 3 |- ( ph -> F e. Ring )
9	8	ringgrpd 14309	. 2 |- ( ph -> F e. Grp )
10		asclf.r	. . 3 |- ( ph -> W e. Ring )
11	10	ringgrpd 14309	. 2 |- ( ph -> W e. Grp )
12		asclf.a	. . 3 |- A = (algSc ` W )
13	12, 6, 10, 5, 1, 2	asclf 15024	. 2 |- ( ph -> A : ( Base ` F ) --> ( Base ` W ) )
14	5	adantr 276	. . . 4 |- ( ( ph /\ ( x e. ( Base ` F ) /\ y e. ( Base ` F ) ) ) -> W e. LMod )
15		simprl 535	. . . 4 |- ( ( ph /\ ( x e. ( Base ` F ) /\ y e. ( Base ` F ) ) ) -> x e. ( Base ` F ) )
16		simprr 537	. . . 4 |- ( ( ph /\ ( x e. ( Base ` F ) /\ y e. ( Base ` F ) ) ) -> y e. ( Base ` F ) )
17		eqid 2238	. . . . . . 7 |- ( 1r ` W ) = ( 1r ` W )
18	2, 17	ringidcl 14325	. . . . . 6 |- ( W e. Ring -> ( 1r ` W ) e. ( Base ` W ) )
19	10, 18	syl 14	. . . . 5 |- ( ph -> ( 1r ` W ) e. ( Base ` W ) )
20	19	adantr 276	. . . 4 |- ( ( ph /\ ( x e. ( Base ` F ) /\ y e. ( Base ` F ) ) ) -> ( 1r ` W ) e. ( Base ` W ) )
21		eqid 2238	. . . . 5 |- ( .s ` W ) = ( .s ` W )
22	2, 4, 6, 21, 1, 3	lmodvsdir 14649	. . . 4 |- ( ( W e. LMod /\ ( x e. ( Base ` F ) /\ y e. ( Base ` F ) /\ ( 1r ` W ) e. ( Base ` W ) ) ) -> ( ( x ( +g ` F ) y ) ( .s ` W ) ( 1r ` W ) ) = ( ( x ( .s ` W ) ( 1r ` W ) ) ( +g ` W ) ( y ( .s ` W ) ( 1r ` W ) ) ) )
23	14, 15, 16, 20, 22	syl13anc 1280	. . 3 |- ( ( ph /\ ( x e. ( Base ` F ) /\ y e. ( Base ` F ) ) ) -> ( ( x ( +g ` F ) y ) ( .s ` W ) ( 1r ` W ) ) = ( ( x ( .s ` W ) ( 1r ` W ) ) ( +g ` W ) ( y ( .s ` W ) ( 1r ` W ) ) ) )
24	1, 3	grpcl 13813	. . . . . 6 |- ( ( F e. Grp /\ x e. ( Base ` F ) /\ y e. ( Base ` F ) ) -> ( x ( +g ` F ) y ) e. ( Base ` F ) )
25	24	3expb 1235	. . . . 5 |- ( ( F e. Grp /\ ( x e. ( Base ` F ) /\ y e. ( Base ` F ) ) ) -> ( x ( +g ` F ) y ) e. ( Base ` F ) )
26	9, 25	sylan 283	. . . 4 |- ( ( ph /\ ( x e. ( Base ` F ) /\ y e. ( Base ` F ) ) ) -> ( x ( +g ` F ) y ) e. ( Base ` F ) )
27	10	adantr 276	. . . 4 |- ( ( ph /\ ( x e. ( Base ` F ) /\ y e. ( Base ` F ) ) ) -> W e. Ring )
28	12, 6, 1, 21, 17, 26, 14, 27	asclvald 15022	. . 3 |- ( ( ph /\ ( x e. ( Base ` F ) /\ y e. ( Base ` F ) ) ) -> ( A ` ( x ( +g ` F ) y ) ) = ( ( x ( +g ` F ) y ) ( .s ` W ) ( 1r ` W ) ) )
29	12, 6, 1, 21, 17, 15, 14, 27	asclvald 15022	. . . 4 |- ( ( ph /\ ( x e. ( Base ` F ) /\ y e. ( Base ` F ) ) ) -> ( A ` x ) = ( x ( .s ` W ) ( 1r ` W ) ) )
30	12, 6, 1, 21, 17, 16, 14, 27	asclvald 15022	. . . 4 |- ( ( ph /\ ( x e. ( Base ` F ) /\ y e. ( Base ` F ) ) ) -> ( A ` y ) = ( y ( .s ` W ) ( 1r ` W ) ) )
31	29, 30	oveq12d 6103	. . 3 |- ( ( ph /\ ( x e. ( Base ` F ) /\ y e. ( Base ` F ) ) ) -> ( ( A ` x ) ( +g ` W ) ( A ` y ) ) = ( ( x ( .s ` W ) ( 1r ` W ) ) ( +g ` W ) ( y ( .s ` W ) ( 1r ` W ) ) ) )
32	23, 28, 31	3eqtr4d 2281	. 2 |- ( ( ph /\ ( x e. ( Base ` F ) /\ y e. ( Base ` F ) ) ) -> ( A ` ( x ( +g ` F ) y ) ) = ( ( A ` x ) ( +g ` W ) ( A ` y ) ) )
33	1, 2, 3, 4, 9, 11, 13, 32	isghmd 14055	1 |- ( ph -> A e. ( F GrpHom W ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   ` cfv 5377  (class class class)co 6085   Basecbs 13352   +g cplusg 13431  Scalarcsca 13434   .scvsca 13435   Grpcgrp 13805   GrpHom cghm 14043   1rcur 14262   Ringcrg 14300   LModclmod 14623  algSccascl 14998

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-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-ltadd 8295

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-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  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-nul 3521  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-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-plusg 13444  df-mulr 13445  df-sca 13447  df-vsca 13448  df-0g 13612  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-grp 13808  df-ghm 14044  df-mgp 14218  df-ur 14263  df-ring 14302  df-lmod 14625  df-ascl 15001

This theorem is used by:  asclinvg  15032  asclrhm  15033