Theorem pwsmulrval 13402

Description: Value of multiplication 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 )
pwsmulrval.a	|- .x. = ( .r ` R )
pwsmulrval.p	|- .xb = ( .r ` Y )

Assertion

Ref	Expression
pwsmulrval	|- ( ph -> ( F .xb G ) = ( F oF .x. G ) )

Proof of Theorem pwsmulrval

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2230	. . . 4 |- ( (Scalar ` R ) X_s ( I X. { R } ) ) = ( (Scalar ` R ) X_s ( I X. { R } ) )
2		eqid 2230	. . . 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 13259	. . . . . 6 |- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
5	4	slotex 13132	. . . . 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 5537	. . . . 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 2230	. . . . . . . . 9 |- (Scalar ` R ) = (Scalar ` R )
14	12, 13	pwsval 13397	. . . . . . . 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 5646	. . . . . 6 |- ( ph -> ( Base ` Y ) = ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
17	11, 16	eqtrid 2275	. . . . 5 |- ( ph -> B = ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
18	10, 17	eleqtrd 2309	. . . 4 |- ( ph -> F e. ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
19		pwsplusgval.g	. . . . 5 |- ( ph -> G e. B )
20	19, 17	eleqtrd 2309	. . . 4 |- ( ph -> G e. ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
21		eqid 2230	. . . 4 |- ( .r ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) = ( .r ` ( (Scalar ` R ) X_s ( I X. { R } ) ) )
22	1, 2, 6, 7, 9, 18, 20, 21	prdsmulrval 13391	. . 3 |- ( ph -> ( F ( .r ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) G ) = ( x e. I |-> ( ( F ` x ) ( .r ` ( ( I X. { R } ) ` x ) ) ( G ` x ) ) ) )
23		fvconst2g 5871	. . . . . . . 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 5646	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( .r ` ( ( I X. { R } ) ` x ) ) = ( .r ` R ) )
26		pwsmulrval.a	. . . . . 6 |- .x. = ( .r ` R )
27	25, 26	eqtr4di 2281	. . . . 5 |- ( ( ph /\ x e. I ) -> ( .r ` ( ( I X. { R } ) ` x ) ) = .x. )
28	27	oveqd 6040	. . . 4 |- ( ( ph /\ x e. I ) -> ( ( F ` x ) ( .r ` ( ( I X. { R } ) ` x ) ) ( G ` x ) ) = ( ( F ` x ) .x. ( G ` x ) ) )
29	28	mpteq2dva 4180	. . 3 |- ( ph -> ( x e. I |-> ( ( F ` x ) ( .r ` ( ( I X. { R } ) ` x ) ) ( G ` x ) ) ) = ( x e. I |-> ( ( F ` x ) .x. ( G ` x ) ) ) )
30	22, 29	eqtrd 2263	. 2 |- ( ph -> ( F ( .r ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) G ) = ( x e. I |-> ( ( F ` x ) .x. ( G ` x ) ) ) )
31		pwsmulrval.p	. . . 4 |- .xb = ( .r ` Y )
32	15	fveq2d 5646	. . . 4 |- ( ph -> ( .r ` Y ) = ( .r ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
33	31, 32	eqtrid 2275	. . 3 |- ( ph -> .xb = ( .r ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
34	33	oveqd 6040	. 2 |- ( ph -> ( F .xb G ) = ( F ( .r ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) G ) )
35		fvexg 5661	. . . 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 5661	. . . 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 2230	. . . . 5 |- ( Base ` R ) = ( Base ` R )
40	12, 39, 11, 3, 7, 10	pwselbas 13400	. . . 4 |- ( ph -> F : I --> ( Base ` R ) )
41	40	feqmptd 5702	. . 3 |- ( ph -> F = ( x e. I |-> ( F ` x ) ) )
42	12, 39, 11, 3, 7, 19	pwselbas 13400	. . . 4 |- ( ph -> G : I --> ( Base ` R ) )
43	42	feqmptd 5702	. . 3 |- ( ph -> G = ( x e. I |-> ( G ` x ) ) )
44	7, 36, 38, 41, 43	offval2 6256	. 2 |- ( ph -> ( F oF .x. G ) = ( x e. I |-> ( ( F ` x ) .x. ( G ` x ) ) ) )
45	30, 34, 44	3eqtr4d 2273	1 |- ( ph -> ( F .xb G ) = ( F oF .x. G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2201   _Vcvv 2801   {csn 3670   |-> cmpt 4151   X. cxp 4725   Fn wfn 5323   ` cfv 5328  (class class class)co 6023   oFcof 6238   Basecbs 13105   .rcmulr 13184  Scalarcsca 13186   X__scprds 13371   ^_s cpws 13372

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 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-addcom 8137  ax-mulcom 8138  ax-addass 8139  ax-mulass 8140  ax-distr 8141  ax-i2m1 8142  ax-0lt1 8143  ax-1rid 8144  ax-0id 8145  ax-rnegex 8146  ax-cnre 8148  ax-pre-ltirr 8149  ax-pre-ltwlin 8150  ax-pre-lttrn 8151  ax-pre-apti 8152  ax-pre-ltadd 8153

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 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3655  df-sn 3676  df-pr 3677  df-tp 3678  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-of 6240  df-1st 6308  df-2nd 6309  df-map 6824  df-ixp 6873  df-sup 7188  df-pnf 8221  df-mnf 8222  df-xr 8223  df-ltxr 8224  df-le 8225  df-sub 8357  df-neg 8358  df-inn 9149  df-2 9207  df-3 9208  df-4 9209  df-5 9210  df-6 9211  df-7 9212  df-8 9213  df-9 9214  df-n0 9408  df-z 9485  df-dec 9617  df-uz 9761  df-fz 10249  df-struct 13107  df-ndx 13108  df-slot 13109  df-base 13111  df-plusg 13196  df-mulr 13197  df-sca 13199  df-vsca 13200  df-ip 13201  df-tset 13202  df-ple 13203  df-ds 13205  df-hom 13207  df-cco 13208  df-rest 13347  df-topn 13348  df-topgen 13366  df-pt 13367  df-prds 13373  df-pws 13396

This theorem is referenced by: (None)