Theorem prdsplusgfval 13486

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 13485	. . 3 |- ( ph -> ( F .+ G ) = ( x e. I |-> ( ( F ` x ) ( +g ` ( R ` x ) ) ( G ` x ) ) ) )
10	9	fveq1d 5671	. 2 |- ( ph -> ( ( F .+ G ) ` J ) = ( ( x e. I |-> ( ( F ` x ) ( +g ` ( R ` x ) ) ( G ` x ) ) ) ` J ) )
11		eqid 2232	. . 3 |- ( x e. I |-> ( ( F ` x ) ( +g ` ( R ` x ) ) ( G ` x ) ) ) = ( x e. I |-> ( ( F ` x ) ( +g ` ( R ` x ) ) ( G ` x ) ) )
12		2fveq3 5674	. . . 4 |- ( x = J -> ( +g ` ( R ` x ) ) = ( +g ` ( R ` J ) ) )
13		fveq2 5669	. . . 4 |- ( x = J -> ( F ` x ) = ( F ` J ) )
14		fveq2 5669	. . . 4 |- ( x = J -> ( G ` x ) = ( G ` J ) )
15	12, 13, 14	oveq123d 6070	. . 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 5688	. . . . 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 5905	. . . . . . 7 |- ( ( R Fn I /\ I e. W ) -> R e. _V )
20	5, 4, 19	syl2anc 411	. . . . . 6 |- ( ph -> R e. _V )
21		fvexg 5688	. . . . . 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 13314	. . . . . 6 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
24	23	slotex 13228	. . . . 5 |- ( ( R ` J ) e. _V -> ( +g ` ( R ` J ) ) e. _V )
25	22, 24	syl 14	. . . 4 |- ( ph -> ( +g ` ( R ` J ) ) e. _V )
26		fvexg 5688	. . . . 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 6083	. . . 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 1274	. . 3 |- ( ph -> ( ( F ` J ) ( +g ` ( R ` J ) ) ( G ` J ) ) e. _V )
30	11, 15, 16, 29	fvmptd3 5770	. 2 |- ( ph -> ( ( x e. I |-> ( ( F ` x ) ( +g ` ( R ` x ) ) ( G ` x ) ) ) ` J ) = ( ( F ` J ) ( +g ` ( R ` J ) ) ( G ` J ) ) )
31	10, 30	eqtrd 2265	1 |- ( ph -> ( ( F .+ G ) ` J ) = ( ( F ` J ) ( +g ` ( R ` J ) ) ( G ` J ) ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203   _Vcvv 2812   |-> cmpt 4170   Fn wfn 5346   ` cfv 5351  (class class class)co 6049   Basecbs 13201   +g cplusg 13279   X__scprds 13467

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-0lt1 8229  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-cnre 8234  ax-pre-ltirr 8235  ax-pre-ltwlin 8236  ax-pre-lttrn 8237  ax-pre-apti 8238  ax-pre-ltadd 8239

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-tp 3696  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-map 6883  df-ixp 6933  df-sup 7274  df-pnf 8306  df-mnf 8307  df-xr 8308  df-ltxr 8309  df-le 8310  df-sub 8442  df-neg 8443  df-inn 9234  df-2 9292  df-3 9293  df-4 9294  df-5 9295  df-6 9296  df-7 9297  df-8 9298  df-9 9299  df-n0 9493  df-z 9574  df-dec 9706  df-uz 9850  df-fz 10339  df-struct 13203  df-ndx 13204  df-slot 13205  df-base 13207  df-plusg 13292  df-mulr 13293  df-sca 13295  df-vsca 13296  df-ip 13297  df-tset 13298  df-ple 13299  df-ds 13301  df-hom 13303  df-cco 13304  df-rest 13443  df-topn 13444  df-topgen 13462  df-pt 13463  df-prds 13469

This theorem is referenced by:  prdssgrpd  13617  prdsmndd  13650