Theorem prdsvscacl 18968

Description: Pointwise scalar multiplication is closed in products of modules. (Contributed by Stefan O'Rear, 10-Jan-2015.)

Hypotheses

Ref	Expression
prdsvscacl.y	|- Y = ( S X_s R )
prdsvscacl.b	|- B = ( Base ` Y )
prdsvscacl.t	|- .x. = ( .s ` Y )
prdsvscacl.k	|- K = ( Base ` S )
prdsvscacl.s	|- ( ph -> S e. Ring )
prdsvscacl.i	|- ( ph -> I e. W )
prdsvscacl.r	|- ( ph -> R : I --> LMod )
prdsvscacl.f	|- ( ph -> F e. K )
prdsvscacl.g	|- ( ph -> G e. B )
prdsvscacl.sr	|- ( ( ph /\ x e. I ) -> (Scalar ` ( R ` x ) ) = S )

Assertion

Ref	Expression
prdsvscacl	|- ( ph -> ( F .x. G ) e. B )

Distinct variable groups:   x, B   x, F   x, G   x, I   x, K   x, R   x, S   ph, x   x, W   x, Y

Allowed substitution hint:   .x. ( x)

Proof of Theorem prdsvscacl

Step	Hyp	Ref	Expression
1		prdsvscacl.y	. . 3 |- Y = ( S X_s R )
2		prdsvscacl.b	. . 3 |- B = ( Base ` Y )
3		prdsvscacl.t	. . 3 |- .x. = ( .s ` Y )
4		prdsvscacl.k	. . 3 |- K = ( Base ` S )
5		prdsvscacl.s	. . 3 |- ( ph -> S e. Ring )
6		prdsvscacl.i	. . 3 |- ( ph -> I e. W )
7		prdsvscacl.r	. . . 4 |- ( ph -> R : I --> LMod )
8		ffn 6045	. . . 4 |- ( R : I --> LMod -> R Fn I )
9	7, 8	syl 17	. . 3 |- ( ph -> R Fn I )
10		prdsvscacl.f	. . 3 |- ( ph -> F e. K )
11		prdsvscacl.g	. . 3 |- ( ph -> G e. B )
12	1, 2, 3, 4, 5, 6, 9, 10, 11	prdsvscaval 16139	. 2 |- ( ph -> ( F .x. G ) = ( x e. I |-> ( F ( .s ` ( R ` x ) ) ( G ` x ) ) ) )
13	7	ffvelrnda 6359	. . . . 5 |- ( ( ph /\ x e. I ) -> ( R ` x ) e. LMod )
14	10	adantr 481	. . . . . 6 |- ( ( ph /\ x e. I ) -> F e. K )
15		prdsvscacl.sr	. . . . . . . 8 |- ( ( ph /\ x e. I ) -> (Scalar ` ( R ` x ) ) = S )
16	15	fveq2d 6195	. . . . . . 7 |- ( ( ph /\ x e. I ) -> ( Base ` (Scalar ` ( R ` x ) ) ) = ( Base ` S ) )
17	16, 4	syl6eqr 2674	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( Base ` (Scalar ` ( R ` x ) ) ) = K )
18	14, 17	eleqtrrd 2704	. . . . 5 |- ( ( ph /\ x e. I ) -> F e. ( Base ` (Scalar ` ( R ` x ) ) ) )
19	5	adantr 481	. . . . . 6 |- ( ( ph /\ x e. I ) -> S e. Ring )
20	6	adantr 481	. . . . . 6 |- ( ( ph /\ x e. I ) -> I e. W )
21	9	adantr 481	. . . . . 6 |- ( ( ph /\ x e. I ) -> R Fn I )
22	11	adantr 481	. . . . . 6 |- ( ( ph /\ x e. I ) -> G e. B )
23		simpr 477	. . . . . 6 |- ( ( ph /\ x e. I ) -> x e. I )
24	1, 2, 19, 20, 21, 22, 23	prdsbasprj 16132	. . . . 5 |- ( ( ph /\ x e. I ) -> ( G ` x ) e. ( Base ` ( R ` x ) ) )
25		eqid 2622	. . . . . 6 |- ( Base ` ( R ` x ) ) = ( Base ` ( R ` x ) )
26		eqid 2622	. . . . . 6 |- (Scalar ` ( R ` x ) ) = (Scalar ` ( R ` x ) )
27		eqid 2622	. . . . . 6 |- ( .s ` ( R ` x ) ) = ( .s ` ( R ` x ) )
28		eqid 2622	. . . . . 6 |- ( Base ` (Scalar ` ( R ` x ) ) ) = ( Base ` (Scalar ` ( R ` x ) ) )
29	25, 26, 27, 28	lmodvscl 18880	. . . . 5 |- ( ( ( R ` x ) e. LMod /\ F e. ( Base ` (Scalar ` ( R ` x ) ) ) /\ ( G ` x ) e. ( Base ` ( R ` x ) ) ) -> ( F ( .s ` ( R ` x ) ) ( G ` x ) ) e. ( Base ` ( R ` x ) ) )
30	13, 18, 24, 29	syl3anc 1326	. . . 4 |- ( ( ph /\ x e. I ) -> ( F ( .s ` ( R ` x ) ) ( G ` x ) ) e. ( Base ` ( R ` x ) ) )
31	30	ralrimiva 2966	. . 3 |- ( ph -> A. x e. I ( F ( .s ` ( R ` x ) ) ( G ` x ) ) e. ( Base ` ( R ` x ) ) )
32	1, 2, 5, 6, 9	prdsbasmpt 16130	. . 3 |- ( ph -> ( ( x e. I |-> ( F ( .s ` ( R ` x ) ) ( G ` x ) ) ) e. B <-> A. x e. I ( F ( .s ` ( R ` x ) ) ( G ` x ) ) e. ( Base ` ( R ` x ) ) ) )
33	31, 32	mpbird 247	. 2 |- ( ph -> ( x e. I |-> ( F ( .s ` ( R ` x ) ) ( G ` x ) ) ) e. B )
34	12, 33	eqeltrd 2701	1 |- ( ph -> ( F .x. G ) e. B )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   |-> cmpt 4729   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   X__scprds 16106   Ringcrg 18547   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  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-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-int 4476  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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  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-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-prds 16108  df-lmod 18865

This theorem is referenced by:  prdslmodd  18969