Theorem pwsmulrval 14211

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 2238	. . . 4 |- ( (Scalar ` R ) X_s ( I X. { R } ) ) = ( (Scalar ` R ) X_s ( I X. { R } ) )
2		eqid 2238	. . . 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 13509	. . . . . 6 |- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
5	4	slotex 13381	. . . . 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 5590	. . . . 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 2238	. . . . . . . . 9 |- (Scalar ` R ) = (Scalar ` R )
14	12, 13	pwsval 14206	. . . . . . . 8 |- ( ( R e. V /\ I e. W ) -> Y = ( (Scalar ` R ) X_s ( I X. { R } ) ) )
15	3, 7, 14	syl2anc 415	. . . . . . 7 |- ( ph -> Y = ( (Scalar ` R ) X_s ( I X. { R } ) ) )
16	15	fveq2d 5699	. . . . . 6 |- ( ph -> ( Base ` Y ) = ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
17	11, 16	eqtrid 2283	. . . . 5 |- ( ph -> B = ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
18	10, 17	eleqtrd 2317	. . . 4 |- ( ph -> F e. ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
19		pwsplusgval.g	. . . . 5 |- ( ph -> G e. B )
20	19, 17	eleqtrd 2317	. . . 4 |- ( ph -> G e. ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
21		eqid 2238	. . . 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 14187	. . 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 5929	. . . . . . . 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 5699	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( .r ` ( ( I X. { R } ) ` x ) ) = ( .r ` R ) )
26		pwsmulrval.a	. . . . . 6 |- .x. = ( .r ` R )
27	25, 26	eqtr4di 2289	. . . . 5 |- ( ( ph /\ x e. I ) -> ( .r ` ( ( I X. { R } ) ` x ) ) = .x. )
28	27	oveqd 6102	. . . 4 |- ( ( ph /\ x e. I ) -> ( ( F ` x ) ( .r ` ( ( I X. { R } ) ` x ) ) ( G ` x ) ) = ( ( F ` x ) .x. ( G ` x ) ) )
29	28	mpteq2dva 4221	. . 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 2271	. 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 5699	. . . 4 |- ( ph -> ( .r ` Y ) = ( .r ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
33	31, 32	eqtrid 2283	. . 3 |- ( ph -> .xb = ( .r ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
34	33	oveqd 6102	. 2 |- ( ph -> ( F .xb G ) = ( F ( .r ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) G ) )
35		fvexg 5714	. . . 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 5714	. . . 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 2238	. . . . 5 |- ( Base ` R ) = ( Base ` R )
40	12, 39, 11, 3, 7, 10	pwselbas 14209	. . . 4 |- ( ph -> F : I --> ( Base ` R ) )
41	40	feqmptd 5756	. . 3 |- ( ph -> F = ( x e. I |-> ( F ` x ) ) )
42	12, 39, 11, 3, 7, 19	pwselbas 14209	. . . 4 |- ( ph -> G : I --> ( Base ` R ) )
43	42	feqmptd 5756	. . 3 |- ( ph -> G = ( x e. I |-> ( G ` x ) ) )
44	7, 36, 38, 41, 43	offval2 6318	. 2 |- ( ph -> ( F oF .x. G ) = ( x e. I |-> ( ( F ` x ) .x. ( G ` x ) ) ) )
45	30, 34, 44	3eqtr4d 2281	1 |- ( ph -> ( F .xb G ) = ( F oF .x. G ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   {csn 3709   |-> cmpt 4192   X. cxp 4772   Fn wfn 5372   ` cfv 5377  (class class class)co 6085   oFcof 6300   Basecbs 13354   .rcmulr 13434  Scalarcsca 13436   X__scprds 14171   ^_s cpws 14204

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-tp 3717  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-of 6302  df-1st 6374  df-2nd 6375  df-map 6924  df-ixp 6981  df-sup 7324  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9306  df-2 9364  df-3 9365  df-4 9366  df-5 9367  df-6 9368  df-7 9369  df-8 9370  df-9 9371  df-n0 9566  df-z 9647  df-dec 9780  df-uz 9924  df-fz 10414  df-struct 13356  df-ndx 13357  df-slot 13358  df-base 13360  df-plusg 13446  df-mulr 13447  df-sca 13449  df-vsca 13450  df-ip 13451  df-tset 13452  df-ple 13453  df-ds 13455  df-hom 13457  df-cco 13458  df-rest 13597  df-topn 13598  df-topgen 13616  df-pt 13617  df-prds 14172  df-pws 14205

This theorem is used by: (None)