Theorem pwsplusgval 13314

Description: Value of addition in a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)

Hypotheses

Ref	Expression
pwsplusgval.y	|- Y = ( R ^s I )
pwsplusgval.b	|- B = ( Base ` Y )
pwsplusgval.r	|- ( ph -> R e. V )
pwsplusgval.i	|- ( ph -> I e. W )
pwsplusgval.f	|- ( ph -> F e. B )
pwsplusgval.g	|- ( ph -> G e. B )
pwsplusgval.a	|- .+ = ( +g ` R )
pwsplusgval.p	|- .+b = ( +g ` Y )

Assertion

Ref	Expression
pwsplusgval	|- ( ph -> ( F .+b G ) = ( F oF .+ G ) )

Proof of Theorem pwsplusgval

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2229	. . . 4 |- ( (Scalar ` R ) X_s ( I X. { R } ) ) = ( (Scalar ` R ) X_s ( I X. { R } ) )
2		eqid 2229	. . . 4 |- ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) = ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) )
3		pwsplusgval.r	. . . . 5 |- ( ph -> R e. V )
4		scaslid 13172	. . . . . 6 |- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
5	4	slotex 13045	. . . . 5 |- ( R e. V -> (Scalar ` R ) e. _V )
6	3, 5	syl 14	. . . 4 |- ( ph -> (Scalar ` R ) e. _V )
7		pwsplusgval.i	. . . 4 |- ( ph -> I e. W )
8		fnconstg 5519	. . . . 5 |- ( R e. V -> ( I X. { R } ) Fn I )
9	3, 8	syl 14	. . . 4 |- ( ph -> ( I X. { R } ) Fn I )
10		pwsplusgval.f	. . . . 5 |- ( ph -> F e. B )
11		pwsplusgval.b	. . . . . 6 |- B = ( Base ` Y )
12		pwsplusgval.y	. . . . . . . . 9 |- Y = ( R ^s I )
13		eqid 2229	. . . . . . . . 9 |- (Scalar ` R ) = (Scalar ` R )
14	12, 13	pwsval 13310	. . . . . . . 8 |- ( ( R e. V /\ I e. W ) -> Y = ( (Scalar ` R ) X_s ( I X. { R } ) ) )
15	3, 7, 14	syl2anc 411	. . . . . . 7 |- ( ph -> Y = ( (Scalar ` R ) X_s ( I X. { R } ) ) )
16	15	fveq2d 5627	. . . . . 6 |- ( ph -> ( Base ` Y ) = ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
17	11, 16	eqtrid 2274	. . . . 5 |- ( ph -> B = ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
18	10, 17	eleqtrd 2308	. . . 4 |- ( ph -> F e. ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
19		pwsplusgval.g	. . . . 5 |- ( ph -> G e. B )
20	19, 17	eleqtrd 2308	. . . 4 |- ( ph -> G e. ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
21		eqid 2229	. . . 4 |- ( +g ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) = ( +g ` ( (Scalar ` R ) X_s ( I X. { R } ) ) )
22	1, 2, 6, 7, 9, 18, 20, 21	prdsplusgval 13302	. . 3 |- ( ph -> ( F ( +g ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) G ) = ( x e. I |-> ( ( F ` x ) ( +g ` ( ( I X. { R } ) ` x ) ) ( G ` x ) ) ) )
23		fvconst2g 5846	. . . . . . . 8 |- ( ( R e. V /\ x e. I ) -> ( ( I X. { R } ) ` x ) = R )
24	3, 23	sylan 283	. . . . . . 7 |- ( ( ph /\ x e. I ) -> ( ( I X. { R } ) ` x ) = R )
25	24	fveq2d 5627	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( +g ` ( ( I X. { R } ) ` x ) ) = ( +g ` R ) )
26		pwsplusgval.a	. . . . . 6 |- .+ = ( +g ` R )
27	25, 26	eqtr4di 2280	. . . . 5 |- ( ( ph /\ x e. I ) -> ( +g ` ( ( I X. { R } ) ` x ) ) = .+ )
28	27	oveqd 6011	. . . 4 |- ( ( ph /\ x e. I ) -> ( ( F ` x ) ( +g ` ( ( I X. { R } ) ` x ) ) ( G ` x ) ) = ( ( F ` x ) .+ ( G ` x ) ) )
29	28	mpteq2dva 4173	. . 3 |- ( ph -> ( x e. I |-> ( ( F ` x ) ( +g ` ( ( I X. { R } ) ` x ) ) ( G ` x ) ) ) = ( x e. I |-> ( ( F ` x ) .+ ( G ` x ) ) ) )
30	22, 29	eqtrd 2262	. 2 |- ( ph -> ( F ( +g ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) G ) = ( x e. I |-> ( ( F ` x ) .+ ( G ` x ) ) ) )
31		pwsplusgval.p	. . . 4 |- .+b = ( +g ` Y )
32	15	fveq2d 5627	. . . 4 |- ( ph -> ( +g ` Y ) = ( +g ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
33	31, 32	eqtrid 2274	. . 3 |- ( ph -> .+b = ( +g ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
34	33	oveqd 6011	. 2 |- ( ph -> ( F .+b G ) = ( F ( +g ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) G ) )
35		fvexg 5642	. . . 4 |- ( ( F e. B /\ x e. I ) -> ( F ` x ) e. _V )
36	10, 35	sylan 283	. . 3 |- ( ( ph /\ x e. I ) -> ( F ` x ) e. _V )
37		fvexg 5642	. . . 4 |- ( ( G e. B /\ x e. I ) -> ( G ` x ) e. _V )
38	19, 37	sylan 283	. . 3 |- ( ( ph /\ x e. I ) -> ( G ` x ) e. _V )
39		eqid 2229	. . . . 5 |- ( Base ` R ) = ( Base ` R )
40	12, 39, 11, 3, 7, 10	pwselbas 13313	. . . 4 |- ( ph -> F : I --> ( Base ` R ) )
41	40	feqmptd 5680	. . 3 |- ( ph -> F = ( x e. I |-> ( F ` x ) ) )
42	12, 39, 11, 3, 7, 19	pwselbas 13313	. . . 4 |- ( ph -> G : I --> ( Base ` R ) )
43	42	feqmptd 5680	. . 3 |- ( ph -> G = ( x e. I |-> ( G ` x ) ) )
44	7, 36, 38, 41, 43	offval2 6224	. 2 |- ( ph -> ( F oF .+ G ) = ( x e. I |-> ( ( F ` x ) .+ ( G ` x ) ) ) )
45	30, 34, 44	3eqtr4d 2272	1 |- ( ph -> ( F .+b G ) = ( F oF .+ G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   _Vcvv 2799   {csn 3666   |-> cmpt 4144   X. cxp 4714   Fn wfn 5309   ` cfv 5314  (class class class)co 5994   oFcof 6206   Basecbs 13018   +g cplusg 13096  Scalarcsca 13099   X__scprds 13284   ^_s cpws 13285

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 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-addcom 8087  ax-mulcom 8088  ax-addass 8089  ax-mulass 8090  ax-distr 8091  ax-i2m1 8092  ax-0lt1 8093  ax-1rid 8094  ax-0id 8095  ax-rnegex 8096  ax-cnre 8098  ax-pre-ltirr 8099  ax-pre-ltwlin 8100  ax-pre-lttrn 8101  ax-pre-apti 8102  ax-pre-ltadd 8103

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 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-of 6208  df-1st 6276  df-2nd 6277  df-map 6787  df-ixp 6836  df-sup 7139  df-pnf 8171  df-mnf 8172  df-xr 8173  df-ltxr 8174  df-le 8175  df-sub 8307  df-neg 8308  df-inn 9099  df-2 9157  df-3 9158  df-4 9159  df-5 9160  df-6 9161  df-7 9162  df-8 9163  df-9 9164  df-n0 9358  df-z 9435  df-dec 9567  df-uz 9711  df-fz 10193  df-struct 13020  df-ndx 13021  df-slot 13022  df-base 13024  df-plusg 13109  df-mulr 13110  df-sca 13112  df-vsca 13113  df-ip 13114  df-tset 13115  df-ple 13116  df-ds 13118  df-hom 13120  df-cco 13121  df-rest 13260  df-topn 13261  df-topgen 13279  df-pt 13280  df-prds 13286  df-pws 13309

This theorem is referenced by:  pwssub  13632  psrgrp  14634