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Theorem pwsval 17532
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 3498 . . 3 (𝑅𝑉𝑅 ∈ V)
3 elex 3498 . . 3 (𝐼𝑊𝐼 ∈ V)
4 simpl 482 . . . . . . 7 ((𝑟 = 𝑅𝑖 = 𝐼) → 𝑟 = 𝑅)
54fveq2d 6910 . . . . . 6 ((𝑟 = 𝑅𝑖 = 𝐼) → (Scalar‘𝑟) = (Scalar‘𝑅))
6 pwsval.f . . . . . 6 𝐹 = (Scalar‘𝑅)
75, 6eqtr4di 2792 . . . . 5 ((𝑟 = 𝑅𝑖 = 𝐼) → (Scalar‘𝑟) = 𝐹)
8 id 22 . . . . . 6 (𝑖 = 𝐼𝑖 = 𝐼)
9 sneq 4640 . . . . . 6 (𝑟 = 𝑅 → {𝑟} = {𝑅})
10 xpeq12 5713 . . . . . 6 ((𝑖 = 𝐼 ∧ {𝑟} = {𝑅}) → (𝑖 × {𝑟}) = (𝐼 × {𝑅}))
118, 9, 10syl2anr 597 . . . . 5 ((𝑟 = 𝑅𝑖 = 𝐼) → (𝑖 × {𝑟}) = (𝐼 × {𝑅}))
127, 11oveq12d 7448 . . . 4 ((𝑟 = 𝑅𝑖 = 𝐼) → ((Scalar‘𝑟)Xs(𝑖 × {𝑟})) = (𝐹Xs(𝐼 × {𝑅})))
13 df-pws 17495 . . . 4 s = (𝑟 ∈ V, 𝑖 ∈ V ↦ ((Scalar‘𝑟)Xs(𝑖 × {𝑟})))
14 ovex 7463 . . . 4 (𝐹Xs(𝐼 × {𝑅})) ∈ V
1512, 13, 14ovmpoa 7587 . . 3 ((𝑅 ∈ V ∧ 𝐼 ∈ V) → (𝑅s 𝐼) = (𝐹Xs(𝐼 × {𝑅})))
162, 3, 15syl2an 596 . 2 ((𝑅𝑉𝐼𝑊) → (𝑅s 𝐼) = (𝐹Xs(𝐼 × {𝑅})))
171, 16eqtrid 2786 1 ((𝑅𝑉𝐼𝑊) → 𝑌 = (𝐹Xs(𝐼 × {𝑅})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  Vcvv 3477  {csn 4630   × cxp 5686  cfv 6562  (class class class)co 7430  Scalarcsca 17300  Xscprds 17491  s cpws 17492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-pws 17495
This theorem is referenced by:  pwsbas  17533  pwsplusgval  17536  pwsmulrval  17537  pwsle  17538  pwsvscafval  17540  pwssca  17542  pwsmnd  18797  pws0g  18798  pwspjmhm  18855  pwsgrp  19082  pwsinvg  19083  pwscmn  19895  pwsabl  19896  pwsgsum  20014  pwsring  20337  pws1  20338  pwscrng  20339  pwsmgp  20340  pwslmod  20985  frlmpws  21787  frlmlss  21788  frlmpwsfi  21789  frlmbas  21792  frlmip  21815  pwstps  23653  resspwsds  24397  pwsxms  24560  pwsms  24561  rrxprds  25436  cnpwstotbnd  37783  repwsmet  37820  rrnequiv  37821
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