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Theorem sraval 20934
Description: Lemma for srabase 20937 through sravsca 20945. (Contributed by Mario Carneiro, 27-Nov-2014.) (Revised by Thierry Arnoux, 16-Jun-2019.)
Assertion
Ref Expression
sraval ((π‘Š ∈ 𝑉 ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) β†’ ((subringAlg β€˜π‘Š)β€˜π‘†) = (((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘Š)⟩))
Proof of Theorem sraval
Dummy variables 𝑠 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3491 . . . 4 (π‘Š ∈ 𝑉 β†’ π‘Š ∈ V)
21adantr 479 . . 3 ((π‘Š ∈ 𝑉 ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) β†’ π‘Š ∈ V)
3 fveq2 6890 . . . . . 6 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘Š))
43pweqd 4618 . . . . 5 (𝑀 = π‘Š β†’ 𝒫 (Baseβ€˜π‘€) = 𝒫 (Baseβ€˜π‘Š))
5 id 22 . . . . . . . 8 (𝑀 = π‘Š β†’ 𝑀 = π‘Š)
6 oveq1 7418 . . . . . . . . 9 (𝑀 = π‘Š β†’ (𝑀 β†Ύs 𝑠) = (π‘Š β†Ύs 𝑠))
76opeq2d 4879 . . . . . . . 8 (𝑀 = π‘Š β†’ ⟨(Scalarβ€˜ndx), (𝑀 β†Ύs 𝑠)⟩ = ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑠)⟩)
85, 7oveq12d 7429 . . . . . . 7 (𝑀 = π‘Š β†’ (𝑀 sSet ⟨(Scalarβ€˜ndx), (𝑀 β†Ύs 𝑠)⟩) = (π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑠)⟩))
9 fveq2 6890 . . . . . . . 8 (𝑀 = π‘Š β†’ (.rβ€˜π‘€) = (.rβ€˜π‘Š))
109opeq2d 4879 . . . . . . 7 (𝑀 = π‘Š β†’ ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘€)⟩ = ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩)
118, 10oveq12d 7429 . . . . . 6 (𝑀 = π‘Š β†’ ((𝑀 sSet ⟨(Scalarβ€˜ndx), (𝑀 β†Ύs 𝑠)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘€)⟩) = ((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑠)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩))
129opeq2d 4879 . . . . . 6 (𝑀 = π‘Š β†’ ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘€)⟩ = ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘Š)⟩)
1311, 12oveq12d 7429 . . . . 5 (𝑀 = π‘Š β†’ (((𝑀 sSet ⟨(Scalarβ€˜ndx), (𝑀 β†Ύs 𝑠)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘€)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘€)⟩) = (((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑠)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘Š)⟩))
144, 13mpteq12dv 5238 . . . 4 (𝑀 = π‘Š β†’ (𝑠 ∈ 𝒫 (Baseβ€˜π‘€) ↦ (((𝑀 sSet ⟨(Scalarβ€˜ndx), (𝑀 β†Ύs 𝑠)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘€)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘€)⟩)) = (𝑠 ∈ 𝒫 (Baseβ€˜π‘Š) ↦ (((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑠)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘Š)⟩)))
15 df-sra 20930 . . . 4 subringAlg = (𝑀 ∈ V ↦ (𝑠 ∈ 𝒫 (Baseβ€˜π‘€) ↦ (((𝑀 sSet ⟨(Scalarβ€˜ndx), (𝑀 β†Ύs 𝑠)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘€)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘€)⟩)))
16 fvex 6903 . . . . . 6 (Baseβ€˜π‘Š) ∈ V
1716pwex 5377 . . . . 5 𝒫 (Baseβ€˜π‘Š) ∈ V
1817mptex 7226 . . . 4 (𝑠 ∈ 𝒫 (Baseβ€˜π‘Š) ↦ (((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑠)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘Š)⟩)) ∈ V
1914, 15, 18fvmpt 6997 . . 3 (π‘Š ∈ V β†’ (subringAlg β€˜π‘Š) = (𝑠 ∈ 𝒫 (Baseβ€˜π‘Š) ↦ (((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑠)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘Š)⟩)))
202, 19syl 17 . 2 ((π‘Š ∈ 𝑉 ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) β†’ (subringAlg β€˜π‘Š) = (𝑠 ∈ 𝒫 (Baseβ€˜π‘Š) ↦ (((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑠)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘Š)⟩)))
21 simpr 483 . . . . . . 7 (((π‘Š ∈ 𝑉 ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) ∧ 𝑠 = 𝑆) β†’ 𝑠 = 𝑆)
2221oveq2d 7427 . . . . . 6 (((π‘Š ∈ 𝑉 ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) ∧ 𝑠 = 𝑆) β†’ (π‘Š β†Ύs 𝑠) = (π‘Š β†Ύs 𝑆))
2322opeq2d 4879 . . . . 5 (((π‘Š ∈ 𝑉 ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) ∧ 𝑠 = 𝑆) β†’ ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑠)⟩ = ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩)
2423oveq2d 7427 . . . 4 (((π‘Š ∈ 𝑉 ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) ∧ 𝑠 = 𝑆) β†’ (π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑠)⟩) = (π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩))
2524oveq1d 7426 . . 3 (((π‘Š ∈ 𝑉 ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) ∧ 𝑠 = 𝑆) β†’ ((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑠)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) = ((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩))
2625oveq1d 7426 . 2 (((π‘Š ∈ 𝑉 ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) ∧ 𝑠 = 𝑆) β†’ (((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑠)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘Š)⟩) = (((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘Š)⟩))
27 simpr 483 . . 3 ((π‘Š ∈ 𝑉 ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) β†’ 𝑆 βŠ† (Baseβ€˜π‘Š))
2816elpw2 5344 . . 3 (𝑆 ∈ 𝒫 (Baseβ€˜π‘Š) ↔ 𝑆 βŠ† (Baseβ€˜π‘Š))
2927, 28sylibr 233 . 2 ((π‘Š ∈ 𝑉 ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) β†’ 𝑆 ∈ 𝒫 (Baseβ€˜π‘Š))
30 ovexd 7446 . 2 ((π‘Š ∈ 𝑉 ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) β†’ (((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘Š)⟩) ∈ V)
3120, 26, 29, 30fvmptd 7004 1 ((π‘Š ∈ 𝑉 ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) β†’ ((subringAlg β€˜π‘Š)β€˜π‘†) = (((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘Š)⟩))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  Vcvv 3472   βŠ† wss 3947  π’« cpw 4601  βŸ¨cop 4633   ↦ cmpt 5230  β€˜cfv 6542  (class class class)co 7411   sSet csts 17100  ndxcnx 17130  Basecbs 17148   β†Ύs cress 17177  .rcmulr 17202  Scalarcsca 17204   ·𝑠 cvsca 17205  Β·π‘–cip 17206  subringAlg csra 20926
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-sra 20930
This theorem is referenced by:  sralem  20935  sralemOLD  20936  srasca  20943  srascaOLD  20944  sravsca  20945  sravscaOLD  20946  sraip  20947  rlmval2  20961  resssra  32962
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