Theorem pwsval 17527

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 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑌 = (𝐹Xs(𝐼 × {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 3478	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
3		elex 3478	. . 3 ⊢ (𝐼 ∈ 𝑊 → 𝐼 ∈ V)
4		simpl 487	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → 𝑟 = 𝑅)
5	4	fveq2d 6875	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → (Scalar‘𝑟) = (Scalar‘𝑅))
6		pwsval.f	. . . . . 6 ⊢ 𝐹 = (Scalar‘𝑅)
7	5, 6	eqtr4di 2818	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → (Scalar‘𝑟) = 𝐹)
8		id 23	. . . . . 6 ⊢ (𝑖 = 𝐼 → 𝑖 = 𝐼)
9		sneq 4595	. . . . . 6 ⊢ (𝑟 = 𝑅 → {𝑟} = {𝑅})
10		xpeq12 5676	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ {𝑟} = {𝑅}) → (𝑖 × {𝑟}) = (𝐼 × {𝑅}))
11	8, 9, 10	syl2anr 608	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → (𝑖 × {𝑟}) = (𝐼 × {𝑅}))
12	7, 11	oveq12d 7418	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → ((Scalar‘𝑟)Xs(𝑖 × {𝑟})) = (𝐹Xs(𝐼 × {𝑅})))
13		df-pws 17490	. . . 4 ⊢ ↑s = (𝑟 ∈ V, 𝑖 ∈ V ↦ ((Scalar‘𝑟)Xs(𝑖 × {𝑟})))
14		ovex 7433	. . . 4 ⊢ (𝐹Xs(𝐼 × {𝑅})) ∈ V
15	12, 13, 14	ovmpoa 7555	. . 3 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ V) → (𝑅 ↑s 𝐼) = (𝐹Xs(𝐼 × {𝑅})))
16	2, 3, 15	syl2an 607	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑅 ↑s 𝐼) = (𝐹Xs(𝐼 × {𝑅})))
17	1, 16	eqtrid 2812	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑌 = (𝐹Xs(𝐼 × {𝑅})))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  Vcvv 3457  {csn 4585   × cxp 5649  ‘cfv 6525  (class class class)co 7400  Scalarcsca 17301  X_scprds 17486   ↑_s cpws 17487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-nul 5260  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5105  df-opab 5167  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6481  df-fun 6527  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-pws 17490

This theorem is referenced by:  pwsbas  17528  pwsplusgval  17532  pwsmulrval  17533  pwsle  17534  pwsvscafval  17536  pwssca  17538  pwsmnd  18818  pws0g  18819  pwspjmhm  18877  pwsgrp  19106  pwsinvg  19107  pwscmn  19921  pwsabl  19922  pwsgsum  20040  pwsring  20393  pws1  20394  pwscrng  20395  pwsmgp  20396  pwslmod  21057  frlmpws  21857  frlmlss  21858  frlmpwsfi  21859  frlmbas  21862  frlmip  21885  pwstps  23744  resspwsds  24486  pwsxms  24646  pwsms  24647  rrxprds  25505  cnpwstotbnd  38303  repwsmet  38340  rrnequiv  38341