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Theorem pwsval 16067
Description: Value of a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
Hypotheses
Ref Expression
pwsval.y 𝑌 = (𝑅s 𝐼)
pwsval.f 𝐹 = (Scalar‘𝑅)
Assertion
Ref Expression
pwsval ((𝑅𝑉𝐼𝑊) → 𝑌 = (𝐹Xs(𝐼 × {𝑅})))

Proof of Theorem pwsval
Dummy variables 𝑖 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwsval.y . 2 𝑌 = (𝑅s 𝐼)
2 elex 3198 . . 3 (𝑅𝑉𝑅 ∈ V)
3 elex 3198 . . 3 (𝐼𝑊𝐼 ∈ V)
4 simpl 473 . . . . . . 7 ((𝑟 = 𝑅𝑖 = 𝐼) → 𝑟 = 𝑅)
54fveq2d 6152 . . . . . 6 ((𝑟 = 𝑅𝑖 = 𝐼) → (Scalar‘𝑟) = (Scalar‘𝑅))
6 pwsval.f . . . . . 6 𝐹 = (Scalar‘𝑅)
75, 6syl6eqr 2673 . . . . 5 ((𝑟 = 𝑅𝑖 = 𝐼) → (Scalar‘𝑟) = 𝐹)
8 id 22 . . . . . 6 (𝑖 = 𝐼𝑖 = 𝐼)
9 sneq 4158 . . . . . 6 (𝑟 = 𝑅 → {𝑟} = {𝑅})
10 xpeq12 5094 . . . . . 6 ((𝑖 = 𝐼 ∧ {𝑟} = {𝑅}) → (𝑖 × {𝑟}) = (𝐼 × {𝑅}))
118, 9, 10syl2anr 495 . . . . 5 ((𝑟 = 𝑅𝑖 = 𝐼) → (𝑖 × {𝑟}) = (𝐼 × {𝑅}))
127, 11oveq12d 6622 . . . 4 ((𝑟 = 𝑅𝑖 = 𝐼) → ((Scalar‘𝑟)Xs(𝑖 × {𝑟})) = (𝐹Xs(𝐼 × {𝑅})))
13 df-pws 16031 . . . 4 s = (𝑟 ∈ V, 𝑖 ∈ V ↦ ((Scalar‘𝑟)Xs(𝑖 × {𝑟})))
14 ovex 6632 . . . 4 (𝐹Xs(𝐼 × {𝑅})) ∈ V
1512, 13, 14ovmpt2a 6744 . . 3 ((𝑅 ∈ V ∧ 𝐼 ∈ V) → (𝑅s 𝐼) = (𝐹Xs(𝐼 × {𝑅})))
162, 3, 15syl2an 494 . 2 ((𝑅𝑉𝐼𝑊) → (𝑅s 𝐼) = (𝐹Xs(𝐼 × {𝑅})))
171, 16syl5eq 2667 1 ((𝑅𝑉𝐼𝑊) → 𝑌 = (𝐹Xs(𝐼 × {𝑅})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  {csn 4148   × cxp 5072  cfv 5847  (class class class)co 6604  Scalarcsca 15865  Xscprds 16027  s cpws 16028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-pws 16031
This theorem is referenced by:  pwsbas  16068  pwsplusgval  16071  pwsmulrval  16072  pwsle  16073  pwsvscafval  16075  pwssca  16077  pwsmnd  17246  pws0g  17247  pwspjmhm  17289  pwsgrp  17448  pwsinvg  17449  pwscmn  18187  pwsabl  18188  pwsgsum  18299  pwsring  18536  pws1  18537  pwscrng  18538  pwsmgp  18539  pwslmod  18889  frlmpws  20013  frlmlss  20014  frlmpwsfi  20015  frlmbas  20018  frlmip  20036  pwstps  21343  resspwsds  22087  pwsxms  22247  pwsms  22248  rrxprds  23085  cnpwstotbnd  33228  repwsmet  33265  rrnequiv  33266
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