Theorem prdsmulrfval 13327

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 13326	. . 3 |- ( ph -> ( F .x. G ) = ( x e. I |-> ( ( F ` x ) ( .r ` ( R ` x ) ) ( G ` x ) ) ) )
10	9	fveq1d 5631	. 2 |- ( ph -> ( ( F .x. G ) ` J ) = ( ( x e. I |-> ( ( F ` x ) ( .r ` ( R ` x ) ) ( G ` x ) ) ) ` J ) )
11		eqid 2229	. . 3 |- ( x e. I |-> ( ( F ` x ) ( .r ` ( R ` x ) ) ( G ` x ) ) ) = ( x e. I |-> ( ( F ` x ) ( .r ` ( R ` x ) ) ( G ` x ) ) )
12		2fveq3 5634	. . . 4 |- ( x = J -> ( .r ` ( R ` x ) ) = ( .r ` ( R ` J ) ) )
13		fveq2 5629	. . . 4 |- ( x = J -> ( F ` x ) = ( F ` J ) )
14		fveq2 5629	. . . 4 |- ( x = J -> ( G ` x ) = ( G ` J ) )
15	12, 13, 14	oveq123d 6028	. . 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 5648	. . . . 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 5865	. . . . . . 7 |- ( ( R Fn I /\ I e. W ) -> R e. _V )
20	5, 4, 19	syl2anc 411	. . . . . 6 |- ( ph -> R e. _V )
21		fvexg 5648	. . . . . 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 13173	. . . . . 6 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
24	23	slotex 13067	. . . . 5 |- ( ( R ` J ) e. _V -> ( .r ` ( R ` J ) ) e. _V )
25	22, 24	syl 14	. . . 4 |- ( ph -> ( .r ` ( R ` J ) ) e. _V )
26		fvexg 5648	. . . . 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 6041	. . . 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 1271	. . 3 |- ( ph -> ( ( F ` J ) ( .r ` ( R ` J ) ) ( G ` J ) ) e. _V )
30	11, 15, 16, 29	fvmptd3 5730	. 2 |- ( ph -> ( ( x e. I |-> ( ( F ` x ) ( .r ` ( R ` x ) ) ( G ` x ) ) ) ` J ) = ( ( F ` J ) ( .r ` ( R ` J ) ) ( G ` J ) ) )
31	10, 30	eqtrd 2262	1 |- ( ph -> ( ( F .x. G ) ` J ) = ( ( F ` J ) ( .r ` ( R ` J ) ) ( G ` J ) ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   _Vcvv 2799   |-> cmpt 4145   Fn wfn 5313   ` cfv 5318  (class class class)co 6007   Basecbs 13040   .rcmulr 13119   X__scprds 13306

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-addcom 8107  ax-mulcom 8108  ax-addass 8109  ax-mulass 8110  ax-distr 8111  ax-i2m1 8112  ax-0lt1 8113  ax-1rid 8114  ax-0id 8115  ax-rnegex 8116  ax-cnre 8118  ax-pre-ltirr 8119  ax-pre-ltwlin 8120  ax-pre-lttrn 8121  ax-pre-apti 8122  ax-pre-ltadd 8123

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-tp 3674  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-map 6805  df-ixp 6854  df-sup 7159  df-pnf 8191  df-mnf 8192  df-xr 8193  df-ltxr 8194  df-le 8195  df-sub 8327  df-neg 8328  df-inn 9119  df-2 9177  df-3 9178  df-4 9179  df-5 9180  df-6 9181  df-7 9182  df-8 9183  df-9 9184  df-n0 9378  df-z 9455  df-dec 9587  df-uz 9731  df-fz 10213  df-struct 13042  df-ndx 13043  df-slot 13044  df-base 13046  df-plusg 13131  df-mulr 13132  df-sca 13134  df-vsca 13135  df-ip 13136  df-tset 13137  df-ple 13138  df-ds 13140  df-hom 13142  df-cco 13143  df-rest 13282  df-topn 13283  df-topgen 13301  df-pt 13302  df-prds 13308

This theorem is referenced by: (None)