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Theorem pwsval 17531
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 3501 . . 3 (𝑅𝑉𝑅 ∈ V)
3 elex 3501 . . 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 2795 . . . . 5 ((𝑟 = 𝑅𝑖 = 𝐼) → (Scalar‘𝑟) = 𝐹)
8 id 22 . . . . . 6 (𝑖 = 𝐼𝑖 = 𝐼)
9 sneq 4636 . . . . . 6 (𝑟 = 𝑅 → {𝑟} = {𝑅})
10 xpeq12 5710 . . . . . 6 ((𝑖 = 𝐼 ∧ {𝑟} = {𝑅}) → (𝑖 × {𝑟}) = (𝐼 × {𝑅}))
118, 9, 10syl2anr 597 . . . . 5 ((𝑟 = 𝑅𝑖 = 𝐼) → (𝑖 × {𝑟}) = (𝐼 × {𝑅}))
127, 11oveq12d 7449 . . . 4 ((𝑟 = 𝑅𝑖 = 𝐼) → ((Scalar‘𝑟)Xs(𝑖 × {𝑟})) = (𝐹Xs(𝐼 × {𝑅})))
13 df-pws 17494 . . . 4 s = (𝑟 ∈ V, 𝑖 ∈ V ↦ ((Scalar‘𝑟)Xs(𝑖 × {𝑟})))
14 ovex 7464 . . . 4 (𝐹Xs(𝐼 × {𝑅})) ∈ V
1512, 13, 14ovmpoa 7588 . . 3 ((𝑅 ∈ V ∧ 𝐼 ∈ V) → (𝑅s 𝐼) = (𝐹Xs(𝐼 × {𝑅})))
162, 3, 15syl2an 596 . 2 ((𝑅𝑉𝐼𝑊) → (𝑅s 𝐼) = (𝐹Xs(𝐼 × {𝑅})))
171, 16eqtrid 2789 1 ((𝑅𝑉𝐼𝑊) → 𝑌 = (𝐹Xs(𝐼 × {𝑅})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  {csn 4626   × cxp 5683  cfv 6561  (class class class)co 7431  Scalarcsca 17300  Xscprds 17490  s cpws 17491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-pws 17494
This theorem is referenced by:  pwsbas  17532  pwsplusgval  17535  pwsmulrval  17536  pwsle  17537  pwsvscafval  17539  pwssca  17541  pwsmnd  18785  pws0g  18786  pwspjmhm  18843  pwsgrp  19070  pwsinvg  19071  pwscmn  19881  pwsabl  19882  pwsgsum  20000  pwsring  20321  pws1  20322  pwscrng  20323  pwsmgp  20324  pwslmod  20968  frlmpws  21770  frlmlss  21771  frlmpwsfi  21772  frlmbas  21775  frlmip  21798  pwstps  23638  resspwsds  24382  pwsxms  24545  pwsms  24546  rrxprds  25423  cnpwstotbnd  37804  repwsmet  37841  rrnequiv  37842
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