Theorem prdsplusgfval 13518

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 13517	. . 3 |- ( ph -> ( F .+ G ) = ( x e. I |-> ( ( F ` x ) ( +g ` ( R ` x ) ) ( G ` x ) ) ) )
10	9	fveq1d 5674	. 2 |- ( ph -> ( ( F .+ G ) ` J ) = ( ( x e. I |-> ( ( F ` x ) ( +g ` ( R ` x ) ) ( G ` x ) ) ) ` J ) )
11		eqid 2234	. . 3 |- ( x e. I |-> ( ( F ` x ) ( +g ` ( R ` x ) ) ( G ` x ) ) ) = ( x e. I |-> ( ( F ` x ) ( +g ` ( R ` x ) ) ( G ` x ) ) )
12		2fveq3 5677	. . . 4 |- ( x = J -> ( +g ` ( R ` x ) ) = ( +g ` ( R ` J ) ) )
13		fveq2 5672	. . . 4 |- ( x = J -> ( F ` x ) = ( F ` J ) )
14		fveq2 5672	. . . 4 |- ( x = J -> ( G ` x ) = ( G ` J ) )
15	12, 13, 14	oveq123d 6073	. . 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 5691	. . . . 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 5908	. . . . . . 7 |- ( ( R Fn I /\ I e. W ) -> R e. _V )
20	5, 4, 19	syl2anc 411	. . . . . 6 |- ( ph -> R e. _V )
21		fvexg 5691	. . . . . 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 13346	. . . . . 6 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
24	23	slotex 13260	. . . . 5 |- ( ( R ` J ) e. _V -> ( +g ` ( R ` J ) ) e. _V )
25	22, 24	syl 14	. . . 4 |- ( ph -> ( +g ` ( R ` J ) ) e. _V )
26		fvexg 5691	. . . . 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 6086	. . . 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 5773	. 2 |- ( ph -> ( ( x e. I |-> ( ( F ` x ) ( +g ` ( R ` x ) ) ( G ` x ) ) ) ` J ) = ( ( F ` J ) ( +g ` ( R ` J ) ) ( G ` J ) ) )
31	10, 30	eqtrd 2267	1 |- ( ph -> ( ( F .+ G ) ` J ) = ( ( F ` J ) ( +g ` ( R ` J ) ) ( G ` J ) ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   _Vcvv 2815   |-> cmpt 4173   Fn wfn 5349   ` cfv 5354  (class class class)co 6052   Basecbs 13233   +g cplusg 13311   X__scprds 13499

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-addcom 8232  ax-mulcom 8233  ax-addass 8234  ax-mulass 8235  ax-distr 8236  ax-i2m1 8237  ax-0lt1 8238  ax-1rid 8239  ax-0id 8240  ax-rnegex 8241  ax-cnre 8243  ax-pre-ltirr 8244  ax-pre-ltwlin 8245  ax-pre-lttrn 8246  ax-pre-apti 8247  ax-pre-ltadd 8248

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-tp 3699  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-map 6886  df-ixp 6936  df-sup 7277  df-pnf 8315  df-mnf 8316  df-xr 8317  df-ltxr 8318  df-le 8319  df-sub 8451  df-neg 8452  df-inn 9243  df-2 9301  df-3 9302  df-4 9303  df-5 9304  df-6 9305  df-7 9306  df-8 9307  df-9 9308  df-n0 9502  df-z 9583  df-dec 9716  df-uz 9860  df-fz 10349  df-struct 13235  df-ndx 13236  df-slot 13237  df-base 13239  df-plusg 13324  df-mulr 13325  df-sca 13327  df-vsca 13328  df-ip 13329  df-tset 13330  df-ple 13331  df-ds 13333  df-hom 13335  df-cco 13336  df-rest 13475  df-topn 13476  df-topgen 13494  df-pt 13495  df-prds 13501

This theorem is referenced by:  prdssgrpd  13649  prdsmndd  13682