Theorem pwsval 16146

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

Hypotheses

Ref	Expression
pwsval.y	|- Y = ( R ^s I )
pwsval.f	|- F = (Scalar ` R )

Assertion

Ref	Expression
pwsval	|- ( ( R e. V /\ I e. W ) -> Y = ( F X_s ( I X. { R } ) ) )

Proof of Theorem pwsval

Dummy variables i r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwsval.y	. 2 |- Y = ( R ^s I )
2		elex 3212	. . 3 |- ( R e. V -> R e. _V )
3		elex 3212	. . 3 |- ( I e. W -> I e. _V )
4		simpl 473	. . . . . . 7 |- ( ( r = R /\ i = I ) -> r = R )
5	4	fveq2d 6195	. . . . . 6 |- ( ( r = R /\ i = I ) -> (Scalar ` r ) = (Scalar ` R ) )
6		pwsval.f	. . . . . 6 |- F = (Scalar ` R )
7	5, 6	syl6eqr 2674	. . . . 5 |- ( ( r = R /\ i = I ) -> (Scalar ` r ) = F )
8		id 22	. . . . . 6 |- ( i = I -> i = I )
9		sneq 4187	. . . . . 6 |- ( r = R -> { r } = { R } )
10		xpeq12 5134	. . . . . 6 |- ( ( i = I /\ { r } = { R } ) -> ( i X. { r } ) = ( I X. { R } ) )
11	8, 9, 10	syl2anr 495	. . . . 5 |- ( ( r = R /\ i = I ) -> ( i X. { r } ) = ( I X. { R } ) )
12	7, 11	oveq12d 6668	. . . 4 |- ( ( r = R /\ i = I ) -> ( (Scalar ` r ) X_s ( i X. { r } ) ) = ( F X_s ( I X. { R } ) ) )
13		df-pws 16110	. . . 4 |- ^s = ( r e. _V , i e. _V |-> ( (Scalar ` r ) X_s ( i X. { r } ) ) )
14		ovex 6678	. . . 4 |- ( F X_s ( I X. { R } ) ) e. _V
15	12, 13, 14	ovmpt2a 6791	. . 3 |- ( ( R e. _V /\ I e. _V ) -> ( R ^s I ) = ( F X_s ( I X. { R } ) ) )
16	2, 3, 15	syl2an 494	. 2 |- ( ( R e. V /\ I e. W ) -> ( R ^s I ) = ( F X_s ( I X. { R } ) ) )
17	1, 16	syl5eq 2668	1 |- ( ( R e. V /\ I e. W ) -> Y = ( F X_s ( I X. { R } ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   X. cxp 5112   ` cfv 5888  (class class class)co 6650  Scalarcsca 15944   X__scprds 16106   ^_s cpws 16107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-pws 16110

This theorem is referenced by:  pwsbas  16147  pwsplusgval  16150  pwsmulrval  16151  pwsle  16152  pwsvscafval  16154  pwssca  16156  pwsmnd  17325  pws0g  17326  pwspjmhm  17368  pwsgrp  17527  pwsinvg  17528  pwscmn  18266  pwsabl  18267  pwsgsum  18378  pwsring  18615  pws1  18616  pwscrng  18617  pwsmgp  18618  pwslmod  18970  frlmpws  20094  frlmlss  20095  frlmpwsfi  20096  frlmbas  20099  frlmip  20117  pwstps  21433  resspwsds  22177  pwsxms  22337  pwsms  22338  rrxprds  23177  cnpwstotbnd  33596  repwsmet  33633  rrnequiv  33634