Theorem pwsmulrval 13381

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 2231	. . . 4 |- ( (Scalar ` R ) X_s ( I X. { R } ) ) = ( (Scalar ` R ) X_s ( I X. { R } ) )
2		eqid 2231	. . . 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 13238	. . . . . 6 |- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
5	4	slotex 13111	. . . . 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 5534	. . . . 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 2231	. . . . . . . . 9 |- (Scalar ` R ) = (Scalar ` R )
14	12, 13	pwsval 13376	. . . . . . . 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 5643	. . . . . 6 |- ( ph -> ( Base ` Y ) = ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
17	11, 16	eqtrid 2276	. . . . 5 |- ( ph -> B = ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
18	10, 17	eleqtrd 2310	. . . 4 |- ( ph -> F e. ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
19		pwsplusgval.g	. . . . 5 |- ( ph -> G e. B )
20	19, 17	eleqtrd 2310	. . . 4 |- ( ph -> G e. ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
21		eqid 2231	. . . 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 13370	. . 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 5868	. . . . . . . 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 5643	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( .r ` ( ( I X. { R } ) ` x ) ) = ( .r ` R ) )
26		pwsmulrval.a	. . . . . 6 |- .x. = ( .r ` R )
27	25, 26	eqtr4di 2282	. . . . 5 |- ( ( ph /\ x e. I ) -> ( .r ` ( ( I X. { R } ) ` x ) ) = .x. )
28	27	oveqd 6035	. . . 4 |- ( ( ph /\ x e. I ) -> ( ( F ` x ) ( .r ` ( ( I X. { R } ) ` x ) ) ( G ` x ) ) = ( ( F ` x ) .x. ( G ` x ) ) )
29	28	mpteq2dva 4179	. . 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 2264	. 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 5643	. . . 4 |- ( ph -> ( .r ` Y ) = ( .r ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
33	31, 32	eqtrid 2276	. . 3 |- ( ph -> .xb = ( .r ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
34	33	oveqd 6035	. 2 |- ( ph -> ( F .xb G ) = ( F ( .r ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) G ) )
35		fvexg 5658	. . . 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 5658	. . . 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 2231	. . . . 5 |- ( Base ` R ) = ( Base ` R )
40	12, 39, 11, 3, 7, 10	pwselbas 13379	. . . 4 |- ( ph -> F : I --> ( Base ` R ) )
41	40	feqmptd 5699	. . 3 |- ( ph -> F = ( x e. I |-> ( F ` x ) ) )
42	12, 39, 11, 3, 7, 19	pwselbas 13379	. . . 4 |- ( ph -> G : I --> ( Base ` R ) )
43	42	feqmptd 5699	. . 3 |- ( ph -> G = ( x e. I |-> ( G ` x ) ) )
44	7, 36, 38, 41, 43	offval2 6251	. 2 |- ( ph -> ( F oF .x. G ) = ( x e. I |-> ( ( F ` x ) .x. ( G ` x ) ) ) )
45	30, 34, 44	3eqtr4d 2274	1 |- ( ph -> ( F .xb G ) = ( F oF .x. G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   _Vcvv 2802   {csn 3669   |-> cmpt 4150   X. cxp 4723   Fn wfn 5321   ` cfv 5326  (class class class)co 6018   oFcof 6233   Basecbs 13084   .rcmulr 13163  Scalarcsca 13165   X__scprds 13350   ^_s cpws 13351

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-of 6235  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  df-pws 13375

This theorem is referenced by: (None)