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Theorem sraval 21174
Description: Lemma for srabase 21177 through sravsca 21185. (Contributed by Mario Carneiro, 27-Nov-2014.) (Revised by Thierry Arnoux, 16-Jun-2019.)
Assertion
Ref Expression
sraval ((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) → ((subringAlg ‘𝑊)‘𝑆) = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩))

Proof of Theorem sraval
Dummy variables 𝑠 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3501 . . . 4 (𝑊𝑉𝑊 ∈ V)
21adantr 480 . . 3 ((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) → 𝑊 ∈ V)
3 fveq2 6906 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
43pweqd 4617 . . . . 5 (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 (Base‘𝑊))
5 id 22 . . . . . . . 8 (𝑤 = 𝑊𝑤 = 𝑊)
6 oveq1 7438 . . . . . . . . 9 (𝑤 = 𝑊 → (𝑤s 𝑠) = (𝑊s 𝑠))
76opeq2d 4880 . . . . . . . 8 (𝑤 = 𝑊 → ⟨(Scalar‘ndx), (𝑤s 𝑠)⟩ = ⟨(Scalar‘ndx), (𝑊s 𝑠)⟩)
85, 7oveq12d 7449 . . . . . . 7 (𝑤 = 𝑊 → (𝑤 sSet ⟨(Scalar‘ndx), (𝑤s 𝑠)⟩) = (𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑠)⟩))
9 fveq2 6906 . . . . . . . 8 (𝑤 = 𝑊 → (.r𝑤) = (.r𝑊))
109opeq2d 4880 . . . . . . 7 (𝑤 = 𝑊 → ⟨( ·𝑠 ‘ndx), (.r𝑤)⟩ = ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩)
118, 10oveq12d 7449 . . . . . 6 (𝑤 = 𝑊 → ((𝑤 sSet ⟨(Scalar‘ndx), (𝑤s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑤)⟩) = ((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩))
129opeq2d 4880 . . . . . 6 (𝑤 = 𝑊 → ⟨(·𝑖‘ndx), (.r𝑤)⟩ = ⟨(·𝑖‘ndx), (.r𝑊)⟩)
1311, 12oveq12d 7449 . . . . 5 (𝑤 = 𝑊 → (((𝑤 sSet ⟨(Scalar‘ndx), (𝑤s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑤)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑤)⟩) = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩))
144, 13mpteq12dv 5233 . . . 4 (𝑤 = 𝑊 → (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ (((𝑤 sSet ⟨(Scalar‘ndx), (𝑤s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑤)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑤)⟩)) = (𝑠 ∈ 𝒫 (Base‘𝑊) ↦ (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩)))
15 df-sra 21172 . . . 4 subringAlg = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ (((𝑤 sSet ⟨(Scalar‘ndx), (𝑤s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑤)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑤)⟩)))
16 fvex 6919 . . . . . 6 (Base‘𝑊) ∈ V
1716pwex 5380 . . . . 5 𝒫 (Base‘𝑊) ∈ V
1817mptex 7243 . . . 4 (𝑠 ∈ 𝒫 (Base‘𝑊) ↦ (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩)) ∈ V
1914, 15, 18fvmpt 7016 . . 3 (𝑊 ∈ V → (subringAlg ‘𝑊) = (𝑠 ∈ 𝒫 (Base‘𝑊) ↦ (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩)))
202, 19syl 17 . 2 ((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) → (subringAlg ‘𝑊) = (𝑠 ∈ 𝒫 (Base‘𝑊) ↦ (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩)))
21 simpr 484 . . . . . . 7 (((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
2221oveq2d 7447 . . . . . 6 (((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) ∧ 𝑠 = 𝑆) → (𝑊s 𝑠) = (𝑊s 𝑆))
2322opeq2d 4880 . . . . 5 (((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) ∧ 𝑠 = 𝑆) → ⟨(Scalar‘ndx), (𝑊s 𝑠)⟩ = ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩)
2423oveq2d 7447 . . . 4 (((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) ∧ 𝑠 = 𝑆) → (𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑠)⟩) = (𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩))
2524oveq1d 7446 . . 3 (((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) ∧ 𝑠 = 𝑆) → ((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) = ((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩))
2625oveq1d 7446 . 2 (((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) ∧ 𝑠 = 𝑆) → (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩) = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩))
27 simpr 484 . . 3 ((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) → 𝑆 ⊆ (Base‘𝑊))
2816elpw2 5334 . . 3 (𝑆 ∈ 𝒫 (Base‘𝑊) ↔ 𝑆 ⊆ (Base‘𝑊))
2927, 28sylibr 234 . 2 ((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) → 𝑆 ∈ 𝒫 (Base‘𝑊))
30 ovexd 7466 . 2 ((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) → (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩) ∈ V)
3120, 26, 29, 30fvmptd 7023 1 ((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) → ((subringAlg ‘𝑊)‘𝑆) = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  wss 3951  𝒫 cpw 4600  cop 4632  cmpt 5225  cfv 6561  (class class class)co 7431   sSet csts 17200  ndxcnx 17230  Basecbs 17247  s cress 17274  .rcmulr 17298  Scalarcsca 17300   ·𝑠 cvsca 17301  ·𝑖cip 17302  subringAlg csra 21170
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-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  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-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-sra 21172
This theorem is referenced by:  sralem  21175  sralemOLD  21176  srasca  21183  srascaOLD  21184  sravsca  21185  sravscaOLD  21186  sraip  21187  rlmval2  21199  resssra  33638
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