Theorem pwsmulrval 13372

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 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 13229	. . . . . 6 |- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
5	4	slotex 13102	. . . . 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 5531	. . . . 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 13367	. . . . . . . 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 5639	. . . . . 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 |- ( .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 13361	. . 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 5863	. . . . . . . 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 5639	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( .r ` ( ( I X. { R } ) ` x ) ) = ( .r ` R ) )
26		pwsmulrval.a	. . . . . 6 |- .x. = ( .r ` R )
27	25, 26	eqtr4di 2280	. . . . 5 |- ( ( ph /\ x e. I ) -> ( .r ` ( ( I X. { R } ) ` x ) ) = .x. )
28	27	oveqd 6030	. . . 4 |- ( ( ph /\ x e. I ) -> ( ( F ` x ) ( .r ` ( ( I X. { R } ) ` x ) ) ( G ` x ) ) = ( ( F ` x ) .x. ( G ` x ) ) )
29	28	mpteq2dva 4177	. . 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 2262	. 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 5639	. . . 4 |- ( ph -> ( .r ` Y ) = ( .r ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
33	31, 32	eqtrid 2274	. . 3 |- ( ph -> .xb = ( .r ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
34	33	oveqd 6030	. 2 |- ( ph -> ( F .xb G ) = ( F ( .r ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) G ) )
35		fvexg 5654	. . . 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 5654	. . . 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 13370	. . . 4 |- ( ph -> F : I --> ( Base ` R ) )
41	40	feqmptd 5695	. . 3 |- ( ph -> F = ( x e. I |-> ( F ` x ) ) )
42	12, 39, 11, 3, 7, 19	pwselbas 13370	. . . 4 |- ( ph -> G : I --> ( Base ` R ) )
43	42	feqmptd 5695	. . 3 |- ( ph -> G = ( x e. I |-> ( G ` x ) ) )
44	7, 36, 38, 41, 43	offval2 6246	. 2 |- ( ph -> ( F oF .x. G ) = ( x e. I |-> ( ( F ` x ) .x. ( G ` x ) ) ) )
45	30, 34, 44	3eqtr4d 2272	1 |- ( ph -> ( F .xb G ) = ( F oF .x. G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   _Vcvv 2800   {csn 3667   |-> cmpt 4148   X. cxp 4721   Fn wfn 5319   ` cfv 5324  (class class class)co 6013   oFcof 6228   Basecbs 13075   .rcmulr 13154  Scalarcsca 13156   X__scprds 13341   ^_s cpws 13342

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 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-addcom 8125  ax-mulcom 8126  ax-addass 8127  ax-mulass 8128  ax-distr 8129  ax-i2m1 8130  ax-0lt1 8131  ax-1rid 8132  ax-0id 8133  ax-rnegex 8134  ax-cnre 8136  ax-pre-ltirr 8137  ax-pre-ltwlin 8138  ax-pre-lttrn 8139  ax-pre-apti 8140  ax-pre-ltadd 8141

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 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-tp 3675  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-of 6230  df-1st 6298  df-2nd 6299  df-map 6814  df-ixp 6863  df-sup 7177  df-pnf 8209  df-mnf 8210  df-xr 8211  df-ltxr 8212  df-le 8213  df-sub 8345  df-neg 8346  df-inn 9137  df-2 9195  df-3 9196  df-4 9197  df-5 9198  df-6 9199  df-7 9200  df-8 9201  df-9 9202  df-n0 9396  df-z 9473  df-dec 9605  df-uz 9749  df-fz 10237  df-struct 13077  df-ndx 13078  df-slot 13079  df-base 13081  df-plusg 13166  df-mulr 13167  df-sca 13169  df-vsca 13170  df-ip 13171  df-tset 13172  df-ple 13173  df-ds 13175  df-hom 13177  df-cco 13178  df-rest 13317  df-topn 13318  df-topgen 13336  df-pt 13337  df-prds 13343  df-pws 13366

This theorem is referenced by: (None)