Theorem prdsmulrfval 13392

Description: Value of a structure product's ring product at a single coordinate. (Contributed by Mario Carneiro, 11-Jan-2015.)

Hypotheses

Ref	Expression
prdsbasmpt.y	|- Y = ( S X_s R )
prdsbasmpt.b	|- B = ( Base ` Y )
prdsbasmpt.s	|- ( ph -> S e. V )
prdsbasmpt.i	|- ( ph -> I e. W )
prdsbasmpt.r	|- ( ph -> R Fn I )
prdsplusgval.f	|- ( ph -> F e. B )
prdsplusgval.g	|- ( ph -> G e. B )
prdsmulrval.t	|- .x. = ( .r ` Y )
prdsmulrfval.j	|- ( ph -> J e. I )

Assertion

Ref	Expression
prdsmulrfval	|- ( ph -> ( ( F .x. G ) ` J ) = ( ( F ` J ) ( .r ` ( R ` J ) ) ( G ` J ) ) )

Proof of Theorem prdsmulrfval

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prdsbasmpt.y	. . . 4 |- Y = ( S X_s R )
2		prdsbasmpt.b	. . . 4 |- B = ( Base ` Y )
3		prdsbasmpt.s	. . . 4 |- ( ph -> S e. V )
4		prdsbasmpt.i	. . . 4 |- ( ph -> I e. W )
5		prdsbasmpt.r	. . . 4 |- ( ph -> R Fn I )
6		prdsplusgval.f	. . . 4 |- ( ph -> F e. B )
7		prdsplusgval.g	. . . 4 |- ( ph -> G e. B )
8		prdsmulrval.t	. . . 4 |- .x. = ( .r ` Y )
9	1, 2, 3, 4, 5, 6, 7, 8	prdsmulrval 13391	. . 3 |- ( ph -> ( F .x. G ) = ( x e. I |-> ( ( F ` x ) ( .r ` ( R ` x ) ) ( G ` x ) ) ) )
10	9	fveq1d 5644	. 2 |- ( ph -> ( ( F .x. G ) ` J ) = ( ( x e. I |-> ( ( F ` x ) ( .r ` ( R ` x ) ) ( G ` x ) ) ) ` J ) )
11		eqid 2230	. . 3 |- ( x e. I |-> ( ( F ` x ) ( .r ` ( R ` x ) ) ( G ` x ) ) ) = ( x e. I |-> ( ( F ` x ) ( .r ` ( R ` x ) ) ( G ` x ) ) )
12		2fveq3 5647	. . . 4 |- ( x = J -> ( .r ` ( R ` x ) ) = ( .r ` ( R ` J ) ) )
13		fveq2 5642	. . . 4 |- ( x = J -> ( F ` x ) = ( F ` J ) )
14		fveq2 5642	. . . 4 |- ( x = J -> ( G ` x ) = ( G ` J ) )
15	12, 13, 14	oveq123d 6044	. . 3 |- ( x = J -> ( ( F ` x ) ( .r ` ( R ` x ) ) ( G ` x ) ) = ( ( F ` J ) ( .r ` ( R ` J ) ) ( G ` J ) ) )
16		prdsmulrfval.j	. . 3 |- ( ph -> J e. I )
17		fvexg 5661	. . . . 5 |- ( ( F e. B /\ J e. I ) -> ( F ` J ) e. _V )
18	6, 16, 17	syl2anc 411	. . . 4 |- ( ph -> ( F ` J ) e. _V )
19		fnex 5879	. . . . . . 7 |- ( ( R Fn I /\ I e. W ) -> R e. _V )
20	5, 4, 19	syl2anc 411	. . . . . 6 |- ( ph -> R e. _V )
21		fvexg 5661	. . . . . 6 |- ( ( R e. _V /\ J e. I ) -> ( R ` J ) e. _V )
22	20, 16, 21	syl2anc 411	. . . . 5 |- ( ph -> ( R ` J ) e. _V )
23		mulrslid 13238	. . . . . 6 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
24	23	slotex 13132	. . . . 5 |- ( ( R ` J ) e. _V -> ( .r ` ( R ` J ) ) e. _V )
25	22, 24	syl 14	. . . 4 |- ( ph -> ( .r ` ( R ` J ) ) e. _V )
26		fvexg 5661	. . . . 5 |- ( ( G e. B /\ J e. I ) -> ( G ` J ) e. _V )
27	7, 16, 26	syl2anc 411	. . . 4 |- ( ph -> ( G ` J ) e. _V )
28		ovexg 6057	. . . 4 |- ( ( ( F ` J ) e. _V /\ ( .r ` ( R ` J ) ) e. _V /\ ( G ` J ) e. _V ) -> ( ( F ` J ) ( .r ` ( R ` J ) ) ( G ` J ) ) e. _V )
29	18, 25, 27, 28	syl3anc 1273	. . 3 |- ( ph -> ( ( F ` J ) ( .r ` ( R ` J ) ) ( G ` J ) ) e. _V )
30	11, 15, 16, 29	fvmptd3 5743	. 2 |- ( ph -> ( ( x e. I |-> ( ( F ` x ) ( .r ` ( R ` x ) ) ( G ` x ) ) ) ` J ) = ( ( F ` J ) ( .r ` ( R ` J ) ) ( G ` J ) ) )
31	10, 30	eqtrd 2263	1 |- ( ph -> ( ( F .x. G ) ` J ) = ( ( F ` J ) ( .r ` ( R ` J ) ) ( G ` J ) ) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2201   _Vcvv 2801   |-> cmpt 4151   Fn wfn 5323   ` cfv 5328  (class class class)co 6023   Basecbs 13105   .rcmulr 13184   X__scprds 13371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-addcom 8137  ax-mulcom 8138  ax-addass 8139  ax-mulass 8140  ax-distr 8141  ax-i2m1 8142  ax-0lt1 8143  ax-1rid 8144  ax-0id 8145  ax-rnegex 8146  ax-cnre 8148  ax-pre-ltirr 8149  ax-pre-ltwlin 8150  ax-pre-lttrn 8151  ax-pre-apti 8152  ax-pre-ltadd 8153

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3655  df-sn 3676  df-pr 3677  df-tp 3678  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-map 6824  df-ixp 6873  df-sup 7188  df-pnf 8221  df-mnf 8222  df-xr 8223  df-ltxr 8224  df-le 8225  df-sub 8357  df-neg 8358  df-inn 9149  df-2 9207  df-3 9208  df-4 9209  df-5 9210  df-6 9211  df-7 9212  df-8 9213  df-9 9214  df-n0 9408  df-z 9485  df-dec 9617  df-uz 9761  df-fz 10249  df-struct 13107  df-ndx 13108  df-slot 13109  df-base 13111  df-plusg 13196  df-mulr 13197  df-sca 13199  df-vsca 13200  df-ip 13201  df-tset 13202  df-ple 13203  df-ds 13205  df-hom 13207  df-cco 13208  df-rest 13347  df-topn 13348  df-topgen 13366  df-pt 13367  df-prds 13373

This theorem is referenced by: (None)