Theorem pwsval 17498

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 3474	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
3		elex 3474	. . 3 ⊢ (𝐼 ∈ 𝑊 → 𝐼 ∈ V)
4		simpl 486	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → 𝑟 = 𝑅)
5	4	fveq2d 6867	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → (Scalar‘𝑟) = (Scalar‘𝑅))
6		pwsval.f	. . . . . 6 ⊢ 𝐹 = (Scalar‘𝑅)
7	5, 6	eqtr4di 2814	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → (Scalar‘𝑟) = 𝐹)
8		id 22	. . . . . 6 ⊢ (𝑖 = 𝐼 → 𝑖 = 𝐼)
9		sneq 4591	. . . . . 6 ⊢ (𝑟 = 𝑅 → {𝑟} = {𝑅})
10		xpeq12 5670	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ {𝑟} = {𝑅}) → (𝑖 × {𝑟}) = (𝐼 × {𝑅}))
11	8, 9, 10	syl2anr 606	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → (𝑖 × {𝑟}) = (𝐼 × {𝑅}))
12	7, 11	oveq12d 7410	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → ((Scalar‘𝑟)Xs(𝑖 × {𝑟})) = (𝐹Xs(𝐼 × {𝑅})))
13		df-pws 17461	. . . 4 ⊢ ↑s = (𝑟 ∈ V, 𝑖 ∈ V ↦ ((Scalar‘𝑟)Xs(𝑖 × {𝑟})))
14		ovex 7425	. . . 4 ⊢ (𝐹Xs(𝐼 × {𝑅})) ∈ V
15	12, 13, 14	ovmpoa 7547	. . 3 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ V) → (𝑅 ↑s 𝐼) = (𝐹Xs(𝐼 × {𝑅})))
16	2, 3, 15	syl2an 605	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑅 ↑s 𝐼) = (𝐹Xs(𝐼 × {𝑅})))
17	1, 16	eqtrid 2808	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑌 = (𝐹Xs(𝐼 × {𝑅})))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  Vcvv 3453  {csn 4581   × cxp 5643  ‘cfv 6517  (class class class)co 7392  Scalarcsca 17272  X_scprds 17457   ↑_s cpws 17458

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-iota 6473  df-fun 6519  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-pws 17461

This theorem is referenced by:  pwsbas  17499  pwsplusgval  17503  pwsmulrval  17504  pwsle  17505  pwsvscafval  17507  pwssca  17509  pwsmnd  18789  pws0g  18790  pwspjmhm  18847  pwsgrp  19077  pwsinvg  19078  pwscmn  19886  pwsabl  19887  pwsgsum  20005  pwsring  20351  pws1  20352  pwscrng  20353  pwsmgp  20354  pwslmod  21017  frlmpws  21782  frlmlss  21783  frlmpwsfi  21784  frlmbas  21787  frlmip  21810  pwstps  23670  resspwsds  24412  pwsxms  24572  pwsms  24573  rrxprds  25431  cnpwstotbnd  38260  repwsmet  38297  rrnequiv  38298