Theorem prdsplusgfval 13369

Description: Value of a structure product sum at a single coordinate. (Contributed by Stefan O'Rear, 10-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 )
prdsplusgval.p	|- .+ = ( +g ` Y )
prdsplusgfval.j	|- ( ph -> J e. I )

Assertion

Ref	Expression
prdsplusgfval	|- ( ph -> ( ( F .+ G ) ` J ) = ( ( F ` J ) ( +g ` ( R ` J ) ) ( G ` J ) ) )

Proof of Theorem prdsplusgfval

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		prdsplusgval.p	. . . 4 |- .+ = ( +g ` Y )
9	1, 2, 3, 4, 5, 6, 7, 8	prdsplusgval 13368	. . 3 |- ( ph -> ( F .+ G ) = ( x e. I |-> ( ( F ` x ) ( +g ` ( R ` x ) ) ( G ` x ) ) ) )
10	9	fveq1d 5641	. 2 |- ( ph -> ( ( F .+ G ) ` J ) = ( ( x e. I |-> ( ( F ` x ) ( +g ` ( R ` x ) ) ( G ` x ) ) ) ` J ) )
11		eqid 2231	. . 3 |- ( x e. I |-> ( ( F ` x ) ( +g ` ( R ` x ) ) ( G ` x ) ) ) = ( x e. I |-> ( ( F ` x ) ( +g ` ( R ` x ) ) ( G ` x ) ) )
12		2fveq3 5644	. . . 4 |- ( x = J -> ( +g ` ( R ` x ) ) = ( +g ` ( R ` J ) ) )
13		fveq2 5639	. . . 4 |- ( x = J -> ( F ` x ) = ( F ` J ) )
14		fveq2 5639	. . . 4 |- ( x = J -> ( G ` x ) = ( G ` J ) )
15	12, 13, 14	oveq123d 6039	. . 3 |- ( x = J -> ( ( F ` x ) ( +g ` ( R ` x ) ) ( G ` x ) ) = ( ( F ` J ) ( +g ` ( R ` J ) ) ( G ` J ) ) )
16		prdsplusgfval.j	. . 3 |- ( ph -> J e. I )
17		fvexg 5658	. . . . 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 5876	. . . . . . 7 |- ( ( R Fn I /\ I e. W ) -> R e. _V )
20	5, 4, 19	syl2anc 411	. . . . . 6 |- ( ph -> R e. _V )
21		fvexg 5658	. . . . . 6 |- ( ( R e. _V /\ J e. I ) -> ( R ` J ) e. _V )
22	20, 16, 21	syl2anc 411	. . . . 5 |- ( ph -> ( R ` J ) e. _V )
23		plusgslid 13197	. . . . . 6 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
24	23	slotex 13111	. . . . 5 |- ( ( R ` J ) e. _V -> ( +g ` ( R ` J ) ) e. _V )
25	22, 24	syl 14	. . . 4 |- ( ph -> ( +g ` ( R ` J ) ) e. _V )
26		fvexg 5658	. . . . 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 6052	. . . 4 |- ( ( ( F ` J ) e. _V /\ ( +g ` ( R ` J ) ) e. _V /\ ( G ` J ) e. _V ) -> ( ( F ` J ) ( +g ` ( R ` J ) ) ( G ` J ) ) e. _V )
29	18, 25, 27, 28	syl3anc 1273	. . 3 |- ( ph -> ( ( F ` J ) ( +g ` ( R ` J ) ) ( G ` J ) ) e. _V )
30	11, 15, 16, 29	fvmptd3 5740	. 2 |- ( ph -> ( ( x e. I |-> ( ( F ` x ) ( +g ` ( R ` x ) ) ( G ` x ) ) ) ` J ) = ( ( F ` J ) ( +g ` ( R ` J ) ) ( G ` J ) ) )
31	10, 30	eqtrd 2264	1 |- ( ph -> ( ( F .+ G ) ` J ) = ( ( F ` J ) ( +g ` ( R ` J ) ) ( G ` J ) ) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   _Vcvv 2802   |-> cmpt 4150   Fn wfn 5321   ` cfv 5326  (class class class)co 6018   Basecbs 13084   +g cplusg 13162   X__scprds 13350

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-tp 3677  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-map 6819  df-ixp 6868  df-sup 7183  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-z 9480  df-dec 9612  df-uz 9756  df-fz 10244  df-struct 13086  df-ndx 13087  df-slot 13088  df-base 13090  df-plusg 13175  df-mulr 13176  df-sca 13178  df-vsca 13179  df-ip 13180  df-tset 13181  df-ple 13182  df-ds 13184  df-hom 13186  df-cco 13187  df-rest 13326  df-topn 13327  df-topgen 13345  df-pt 13346  df-prds 13352

This theorem is referenced by:  prdssgrpd  13500  prdsmndd  13533