Theorem lmhmvsca 19045

Description: The pointwise scalar product of a linear function and a constant is linear, over a commutative ring. (Contributed by Mario Carneiro, 22-Sep-2015.)

Hypotheses

Ref	Expression
lmhmvsca.v	|- V = ( Base ` M )
lmhmvsca.s	|- .x. = ( .s ` N )
lmhmvsca.j	|- J = (Scalar ` N )
lmhmvsca.k	|- K = ( Base ` J )

Assertion

Ref	Expression
lmhmvsca	|- ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) -> ( ( V X. { A } ) oF .x. F ) e. ( M LMHom N ) )

Proof of Theorem lmhmvsca

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

Step	Hyp	Ref	Expression
1		lmhmvsca.v	. 2 |- V = ( Base ` M )
2		eqid 2622	. 2 |- ( .s ` M ) = ( .s ` M )
3		lmhmvsca.s	. 2 |- .x. = ( .s ` N )
4		eqid 2622	. 2 |- (Scalar ` M ) = (Scalar ` M )
5		lmhmvsca.j	. 2 |- J = (Scalar ` N )
6		eqid 2622	. 2 |- ( Base ` (Scalar ` M ) ) = ( Base ` (Scalar ` M ) )
7		lmhmlmod1 19033	. . 3 |- ( F e. ( M LMHom N ) -> M e. LMod )
8	7	3ad2ant3 1084	. 2 |- ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) -> M e. LMod )
9		lmhmlmod2 19032	. . 3 |- ( F e. ( M LMHom N ) -> N e. LMod )
10	9	3ad2ant3 1084	. 2 |- ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) -> N e. LMod )
11	4, 5	lmhmsca 19030	. . 3 |- ( F e. ( M LMHom N ) -> J = (Scalar ` M ) )
12	11	3ad2ant3 1084	. 2 |- ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) -> J = (Scalar ` M ) )
13		fvex 6201	. . . . . . 7 |- ( Base ` M ) e. _V
14	1, 13	eqeltri 2697	. . . . . 6 |- V e. _V
15	14	a1i 11	. . . . 5 |- ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) -> V e. _V )
16		simpl2 1065	. . . . 5 |- ( ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) /\ v e. V ) -> A e. K )
17		eqid 2622	. . . . . . . 8 |- ( Base ` N ) = ( Base ` N )
18	1, 17	lmhmf 19034	. . . . . . 7 |- ( F e. ( M LMHom N ) -> F : V --> ( Base ` N ) )
19	18	3ad2ant3 1084	. . . . . 6 |- ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) -> F : V --> ( Base ` N ) )
20	19	ffvelrnda 6359	. . . . 5 |- ( ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) /\ v e. V ) -> ( F ` v ) e. ( Base ` N ) )
21		fconstmpt 5163	. . . . . 6 |- ( V X. { A } ) = ( v e. V |-> A )
22	21	a1i 11	. . . . 5 |- ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) -> ( V X. { A } ) = ( v e. V |-> A ) )
23	19	feqmptd 6249	. . . . 5 |- ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) -> F = ( v e. V |-> ( F ` v ) ) )
24	15, 16, 20, 22, 23	offval2 6914	. . . 4 |- ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) -> ( ( V X. { A } ) oF .x. F ) = ( v e. V |-> ( A .x. ( F ` v ) ) ) )
25		eqidd 2623	. . . . 5 |- ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) -> ( u e. ( Base ` N ) |-> ( A .x. u ) ) = ( u e. ( Base ` N ) |-> ( A .x. u ) ) )
26		oveq2 6658	. . . . 5 |- ( u = ( F ` v ) -> ( A .x. u ) = ( A .x. ( F ` v ) ) )
27	20, 23, 25, 26	fmptco 6396	. . . 4 |- ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) -> ( ( u e. ( Base ` N ) |-> ( A .x. u ) ) o. F ) = ( v e. V |-> ( A .x. ( F ` v ) ) ) )
28	24, 27	eqtr4d 2659	. . 3 |- ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) -> ( ( V X. { A } ) oF .x. F ) = ( ( u e. ( Base ` N ) |-> ( A .x. u ) ) o. F ) )
29		simp2 1062	. . . . 5 |- ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) -> A e. K )
30		lmhmvsca.k	. . . . . 6 |- K = ( Base ` J )
31	17, 5, 3, 30	lmodvsghm 18924	. . . . 5 |- ( ( N e. LMod /\ A e. K ) -> ( u e. ( Base ` N ) |-> ( A .x. u ) ) e. ( N GrpHom N ) )
32	10, 29, 31	syl2anc 693	. . . 4 |- ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) -> ( u e. ( Base ` N ) |-> ( A .x. u ) ) e. ( N GrpHom N ) )
33		lmghm 19031	. . . . 5 |- ( F e. ( M LMHom N ) -> F e. ( M GrpHom N ) )
34	33	3ad2ant3 1084	. . . 4 |- ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) -> F e. ( M GrpHom N ) )
35		ghmco 17680	. . . 4 |- ( ( ( u e. ( Base ` N ) |-> ( A .x. u ) ) e. ( N GrpHom N ) /\ F e. ( M GrpHom N ) ) -> ( ( u e. ( Base ` N ) |-> ( A .x. u ) ) o. F ) e. ( M GrpHom N ) )
36	32, 34, 35	syl2anc 693	. . 3 |- ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) -> ( ( u e. ( Base ` N ) |-> ( A .x. u ) ) o. F ) e. ( M GrpHom N ) )
37	28, 36	eqeltrd 2701	. 2 |- ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) -> ( ( V X. { A } ) oF .x. F ) e. ( M GrpHom N ) )
38		simpl1 1064	. . . . . 6 |- ( ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) /\ ( x e. ( Base ` (Scalar ` M ) ) /\ y e. V ) ) -> J e. CRing )
39		simpl2 1065	. . . . . 6 |- ( ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) /\ ( x e. ( Base ` (Scalar ` M ) ) /\ y e. V ) ) -> A e. K )
40		simprl 794	. . . . . . 7 |- ( ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) /\ ( x e. ( Base ` (Scalar ` M ) ) /\ y e. V ) ) -> x e. ( Base ` (Scalar ` M ) ) )
41	12	fveq2d 6195	. . . . . . . . 9 |- ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) -> ( Base ` J ) = ( Base ` (Scalar ` M ) ) )
42	30, 41	syl5eq 2668	. . . . . . . 8 |- ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) -> K = ( Base ` (Scalar ` M ) ) )
43	42	adantr 481	. . . . . . 7 |- ( ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) /\ ( x e. ( Base ` (Scalar ` M ) ) /\ y e. V ) ) -> K = ( Base ` (Scalar ` M ) ) )
44	40, 43	eleqtrrd 2704	. . . . . 6 |- ( ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) /\ ( x e. ( Base ` (Scalar ` M ) ) /\ y e. V ) ) -> x e. K )
45		eqid 2622	. . . . . . 7 |- ( .r ` J ) = ( .r ` J )
46	30, 45	crngcom 18562	. . . . . 6 |- ( ( J e. CRing /\ A e. K /\ x e. K ) -> ( A ( .r ` J ) x ) = ( x ( .r ` J ) A ) )
47	38, 39, 44, 46	syl3anc 1326	. . . . 5 |- ( ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) /\ ( x e. ( Base ` (Scalar ` M ) ) /\ y e. V ) ) -> ( A ( .r ` J ) x ) = ( x ( .r ` J ) A ) )
48	47	oveq1d 6665	. . . 4 |- ( ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) /\ ( x e. ( Base ` (Scalar ` M ) ) /\ y e. V ) ) -> ( ( A ( .r ` J ) x ) .x. ( F ` y ) ) = ( ( x ( .r ` J ) A ) .x. ( F ` y ) ) )
49	10	adantr 481	. . . . 5 |- ( ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) /\ ( x e. ( Base ` (Scalar ` M ) ) /\ y e. V ) ) -> N e. LMod )
50	19	adantr 481	. . . . . 6 |- ( ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) /\ ( x e. ( Base ` (Scalar ` M ) ) /\ y e. V ) ) -> F : V --> ( Base ` N ) )
51		simprr 796	. . . . . 6 |- ( ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) /\ ( x e. ( Base ` (Scalar ` M ) ) /\ y e. V ) ) -> y e. V )
52	50, 51	ffvelrnd 6360	. . . . 5 |- ( ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) /\ ( x e. ( Base ` (Scalar ` M ) ) /\ y e. V ) ) -> ( F ` y ) e. ( Base ` N ) )
53	17, 5, 3, 30, 45	lmodvsass 18888	. . . . 5 |- ( ( N e. LMod /\ ( A e. K /\ x e. K /\ ( F ` y ) e. ( Base ` N ) ) ) -> ( ( A ( .r ` J ) x ) .x. ( F ` y ) ) = ( A .x. ( x .x. ( F ` y ) ) ) )
54	49, 39, 44, 52, 53	syl13anc 1328	. . . 4 |- ( ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) /\ ( x e. ( Base ` (Scalar ` M ) ) /\ y e. V ) ) -> ( ( A ( .r ` J ) x ) .x. ( F ` y ) ) = ( A .x. ( x .x. ( F ` y ) ) ) )
55	17, 5, 3, 30, 45	lmodvsass 18888	. . . . 5 |- ( ( N e. LMod /\ ( x e. K /\ A e. K /\ ( F ` y ) e. ( Base ` N ) ) ) -> ( ( x ( .r ` J ) A ) .x. ( F ` y ) ) = ( x .x. ( A .x. ( F ` y ) ) ) )
56	49, 44, 39, 52, 55	syl13anc 1328	. . . 4 |- ( ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) /\ ( x e. ( Base ` (Scalar ` M ) ) /\ y e. V ) ) -> ( ( x ( .r ` J ) A ) .x. ( F ` y ) ) = ( x .x. ( A .x. ( F ` y ) ) ) )
57	48, 54, 56	3eqtr3d 2664	. . 3 |- ( ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) /\ ( x e. ( Base ` (Scalar ` M ) ) /\ y e. V ) ) -> ( A .x. ( x .x. ( F ` y ) ) ) = ( x .x. ( A .x. ( F ` y ) ) ) )
58	1, 4, 2, 6	lmodvscl 18880	. . . . . 6 |- ( ( M e. LMod /\ x e. ( Base ` (Scalar ` M ) ) /\ y e. V ) -> ( x ( .s ` M ) y ) e. V )
59	58	3expb 1266	. . . . 5 |- ( ( M e. LMod /\ ( x e. ( Base ` (Scalar ` M ) ) /\ y e. V ) ) -> ( x ( .s ` M ) y ) e. V )
60	8, 59	sylan 488	. . . 4 |- ( ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) /\ ( x e. ( Base ` (Scalar ` M ) ) /\ y e. V ) ) -> ( x ( .s ` M ) y ) e. V )
61	14	a1i 11	. . . . 5 |- ( ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) /\ ( x e. ( Base ` (Scalar ` M ) ) /\ y e. V ) ) -> V e. _V )
62		ffn 6045	. . . . . . 7 |- ( F : V --> ( Base ` N ) -> F Fn V )
63	19, 62	syl 17	. . . . . 6 |- ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) -> F Fn V )
64	63	adantr 481	. . . . 5 |- ( ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) /\ ( x e. ( Base ` (Scalar ` M ) ) /\ y e. V ) ) -> F Fn V )
65	4, 6, 1, 2, 3	lmhmlin 19035	. . . . . . . 8 |- ( ( F e. ( M LMHom N ) /\ x e. ( Base ` (Scalar ` M ) ) /\ y e. V ) -> ( F ` ( x ( .s ` M ) y ) ) = ( x .x. ( F ` y ) ) )
66	65	3expb 1266	. . . . . . 7 |- ( ( F e. ( M LMHom N ) /\ ( x e. ( Base ` (Scalar ` M ) ) /\ y e. V ) ) -> ( F ` ( x ( .s ` M ) y ) ) = ( x .x. ( F ` y ) ) )
67	66	3ad2antl3 1225	. . . . . 6 |- ( ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) /\ ( x e. ( Base ` (Scalar ` M ) ) /\ y e. V ) ) -> ( F ` ( x ( .s ` M ) y ) ) = ( x .x. ( F ` y ) ) )
68	67	adantr 481	. . . . 5 |- ( ( ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) /\ ( x e. ( Base ` (Scalar ` M ) ) /\ y e. V ) ) /\ ( x ( .s ` M ) y ) e. V ) -> ( F ` ( x ( .s ` M ) y ) ) = ( x .x. ( F ` y ) ) )
69	61, 39, 64, 68	ofc1 6920	. . . 4 |- ( ( ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) /\ ( x e. ( Base ` (Scalar ` M ) ) /\ y e. V ) ) /\ ( x ( .s ` M ) y ) e. V ) -> ( ( ( V X. { A } ) oF .x. F ) ` ( x ( .s ` M ) y ) ) = ( A .x. ( x .x. ( F ` y ) ) ) )
70	60, 69	mpdan 702	. . 3 |- ( ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) /\ ( x e. ( Base ` (Scalar ` M ) ) /\ y e. V ) ) -> ( ( ( V X. { A } ) oF .x. F ) ` ( x ( .s ` M ) y ) ) = ( A .x. ( x .x. ( F ` y ) ) ) )
71		eqidd 2623	. . . . . 6 |- ( ( ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) /\ ( x e. ( Base ` (Scalar ` M ) ) /\ y e. V ) ) /\ y e. V ) -> ( F ` y ) = ( F ` y ) )
72	61, 39, 64, 71	ofc1 6920	. . . . 5 |- ( ( ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) /\ ( x e. ( Base ` (Scalar ` M ) ) /\ y e. V ) ) /\ y e. V ) -> ( ( ( V X. { A } ) oF .x. F ) ` y ) = ( A .x. ( F ` y ) ) )
73	51, 72	mpdan 702	. . . 4 |- ( ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) /\ ( x e. ( Base ` (Scalar ` M ) ) /\ y e. V ) ) -> ( ( ( V X. { A } ) oF .x. F ) ` y ) = ( A .x. ( F ` y ) ) )
74	73	oveq2d 6666	. . 3 |- ( ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) /\ ( x e. ( Base ` (Scalar ` M ) ) /\ y e. V ) ) -> ( x .x. ( ( ( V X. { A } ) oF .x. F ) ` y ) ) = ( x .x. ( A .x. ( F ` y ) ) ) )
75	57, 70, 74	3eqtr4d 2666	. 2 |- ( ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) /\ ( x e. ( Base ` (Scalar ` M ) ) /\ y e. V ) ) -> ( ( ( V X. { A } ) oF .x. F ) ` ( x ( .s ` M ) y ) ) = ( x .x. ( ( ( V X. { A } ) oF .x. F ) ` y ) ) )
76	1, 2, 3, 4, 5, 6, 8, 10, 12, 37, 75	islmhmd 19039	1 |- ( ( J e. CRing /\ A e. K /\ F e. ( M LMHom N ) ) -> ( ( V X. { A } ) oF .x. F ) e. ( M LMHom N ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   |-> cmpt 4729   X. cxp 5112   o. ccom 5118   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   Basecbs 15857   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   GrpHom cghm 17657   CRingccrg 18548   LModclmod 18863   LMHom clmhm 19019

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-ghm 17658  df-cmn 18195  df-mgp 18490  df-cring 18550  df-lmod 18865  df-lmhm 19022

This theorem is referenced by:  mendlmod  37763