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Theorem sraval 19941
Description: Lemma for srabase 19943 through sravsca 19947. (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 3459 . . . 4 (𝑊𝑉𝑊 ∈ V)
21adantr 484 . . 3 ((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) → 𝑊 ∈ V)
3 fveq2 6645 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
43pweqd 4516 . . . . 5 (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 (Base‘𝑊))
5 id 22 . . . . . . . 8 (𝑤 = 𝑊𝑤 = 𝑊)
6 oveq1 7142 . . . . . . . . 9 (𝑤 = 𝑊 → (𝑤s 𝑠) = (𝑊s 𝑠))
76opeq2d 4772 . . . . . . . 8 (𝑤 = 𝑊 → ⟨(Scalar‘ndx), (𝑤s 𝑠)⟩ = ⟨(Scalar‘ndx), (𝑊s 𝑠)⟩)
85, 7oveq12d 7153 . . . . . . 7 (𝑤 = 𝑊 → (𝑤 sSet ⟨(Scalar‘ndx), (𝑤s 𝑠)⟩) = (𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑠)⟩))
9 fveq2 6645 . . . . . . . 8 (𝑤 = 𝑊 → (.r𝑤) = (.r𝑊))
109opeq2d 4772 . . . . . . 7 (𝑤 = 𝑊 → ⟨( ·𝑠 ‘ndx), (.r𝑤)⟩ = ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩)
118, 10oveq12d 7153 . . . . . 6 (𝑤 = 𝑊 → ((𝑤 sSet ⟨(Scalar‘ndx), (𝑤s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑤)⟩) = ((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩))
129opeq2d 4772 . . . . . 6 (𝑤 = 𝑊 → ⟨(·𝑖‘ndx), (.r𝑤)⟩ = ⟨(·𝑖‘ndx), (.r𝑊)⟩)
1311, 12oveq12d 7153 . . . . 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 5115 . . . 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 19937 . . . 4 subringAlg = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ (((𝑤 sSet ⟨(Scalar‘ndx), (𝑤s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑤)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑤)⟩)))
16 fvex 6658 . . . . . 6 (Base‘𝑊) ∈ V
1716pwex 5246 . . . . 5 𝒫 (Base‘𝑊) ∈ V
1817mptex 6963 . . . 4 (𝑠 ∈ 𝒫 (Base‘𝑊) ↦ (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩)) ∈ V
1914, 15, 18fvmpt 6745 . . 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 488 . . . . . . 7 (((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
2221oveq2d 7151 . . . . . 6 (((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) ∧ 𝑠 = 𝑆) → (𝑊s 𝑠) = (𝑊s 𝑆))
2322opeq2d 4772 . . . . 5 (((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) ∧ 𝑠 = 𝑆) → ⟨(Scalar‘ndx), (𝑊s 𝑠)⟩ = ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩)
2423oveq2d 7151 . . . 4 (((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) ∧ 𝑠 = 𝑆) → (𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑠)⟩) = (𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩))
2524oveq1d 7150 . . 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 7150 . 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 488 . . 3 ((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) → 𝑆 ⊆ (Base‘𝑊))
2816elpw2 5212 . . 3 (𝑆 ∈ 𝒫 (Base‘𝑊) ↔ 𝑆 ⊆ (Base‘𝑊))
2927, 28sylibr 237 . 2 ((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) → 𝑆 ∈ 𝒫 (Base‘𝑊))
30 ovexd 7170 . 2 ((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) → (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩) ∈ V)
3120, 26, 29, 30fvmptd 6752 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 399   = wceq 1538  wcel 2111  Vcvv 3441  wss 3881  𝒫 cpw 4497  cop 4531  cmpt 5110  cfv 6324  (class class class)co 7135  ndxcnx 16472   sSet csts 16473  Basecbs 16475  s cress 16476  .rcmulr 16558  Scalarcsca 16560   ·𝑠 cvsca 16561  ·𝑖cip 16562  subringAlg csra 19933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-sra 19937
This theorem is referenced by:  sralem  19942  srasca  19946  sravsca  19947  sraip  19948  rlmval2  19959
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