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Theorem pwsval 17439
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 3492 . . 3 (𝑅𝑉𝑅 ∈ V)
3 elex 3492 . . 3 (𝐼𝑊𝐼 ∈ V)
4 simpl 482 . . . . . . 7 ((𝑟 = 𝑅𝑖 = 𝐼) → 𝑟 = 𝑅)
54fveq2d 6895 . . . . . 6 ((𝑟 = 𝑅𝑖 = 𝐼) → (Scalar‘𝑟) = (Scalar‘𝑅))
6 pwsval.f . . . . . 6 𝐹 = (Scalar‘𝑅)
75, 6eqtr4di 2789 . . . . 5 ((𝑟 = 𝑅𝑖 = 𝐼) → (Scalar‘𝑟) = 𝐹)
8 id 22 . . . . . 6 (𝑖 = 𝐼𝑖 = 𝐼)
9 sneq 4638 . . . . . 6 (𝑟 = 𝑅 → {𝑟} = {𝑅})
10 xpeq12 5701 . . . . . 6 ((𝑖 = 𝐼 ∧ {𝑟} = {𝑅}) → (𝑖 × {𝑟}) = (𝐼 × {𝑅}))
118, 9, 10syl2anr 596 . . . . 5 ((𝑟 = 𝑅𝑖 = 𝐼) → (𝑖 × {𝑟}) = (𝐼 × {𝑅}))
127, 11oveq12d 7430 . . . 4 ((𝑟 = 𝑅𝑖 = 𝐼) → ((Scalar‘𝑟)Xs(𝑖 × {𝑟})) = (𝐹Xs(𝐼 × {𝑅})))
13 df-pws 17402 . . . 4 s = (𝑟 ∈ V, 𝑖 ∈ V ↦ ((Scalar‘𝑟)Xs(𝑖 × {𝑟})))
14 ovex 7445 . . . 4 (𝐹Xs(𝐼 × {𝑅})) ∈ V
1512, 13, 14ovmpoa 7566 . . 3 ((𝑅 ∈ V ∧ 𝐼 ∈ V) → (𝑅s 𝐼) = (𝐹Xs(𝐼 × {𝑅})))
162, 3, 15syl2an 595 . 2 ((𝑅𝑉𝐼𝑊) → (𝑅s 𝐼) = (𝐹Xs(𝐼 × {𝑅})))
171, 16eqtrid 2783 1 ((𝑅𝑉𝐼𝑊) → 𝑌 = (𝐹Xs(𝐼 × {𝑅})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105  Vcvv 3473  {csn 4628   × cxp 5674  cfv 6543  (class class class)co 7412  Scalarcsca 17207  Xscprds 17398  s cpws 17399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-pws 17402
This theorem is referenced by:  pwsbas  17440  pwsplusgval  17443  pwsmulrval  17444  pwsle  17445  pwsvscafval  17447  pwssca  17449  pwsmnd  18697  pws0g  18698  pwspjmhm  18750  pwsgrp  18975  pwsinvg  18976  pwscmn  19776  pwsabl  19777  pwsgsum  19895  pwsring  20216  pws1  20217  pwscrng  20218  pwsmgp  20219  pwslmod  20729  frlmpws  21528  frlmlss  21529  frlmpwsfi  21530  frlmbas  21533  frlmip  21556  pwstps  23367  resspwsds  24111  pwsxms  24274  pwsms  24275  rrxprds  25150  cnpwstotbnd  36981  repwsmet  37018  rrnequiv  37019
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