MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sraval Structured version   Visualization version   GIF version

Theorem sraval 19877
Description: Lemma for srabase 19879 through sravsca 19883. (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 3510 . . . 4 (𝑊𝑉𝑊 ∈ V)
21adantr 481 . . 3 ((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) → 𝑊 ∈ V)
3 fveq2 6663 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
43pweqd 4540 . . . . 5 (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 (Base‘𝑊))
5 id 22 . . . . . . . 8 (𝑤 = 𝑊𝑤 = 𝑊)
6 oveq1 7152 . . . . . . . . 9 (𝑤 = 𝑊 → (𝑤s 𝑠) = (𝑊s 𝑠))
76opeq2d 4802 . . . . . . . 8 (𝑤 = 𝑊 → ⟨(Scalar‘ndx), (𝑤s 𝑠)⟩ = ⟨(Scalar‘ndx), (𝑊s 𝑠)⟩)
85, 7oveq12d 7163 . . . . . . 7 (𝑤 = 𝑊 → (𝑤 sSet ⟨(Scalar‘ndx), (𝑤s 𝑠)⟩) = (𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑠)⟩))
9 fveq2 6663 . . . . . . . 8 (𝑤 = 𝑊 → (.r𝑤) = (.r𝑊))
109opeq2d 4802 . . . . . . 7 (𝑤 = 𝑊 → ⟨( ·𝑠 ‘ndx), (.r𝑤)⟩ = ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩)
118, 10oveq12d 7163 . . . . . 6 (𝑤 = 𝑊 → ((𝑤 sSet ⟨(Scalar‘ndx), (𝑤s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑤)⟩) = ((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩))
129opeq2d 4802 . . . . . 6 (𝑤 = 𝑊 → ⟨(·𝑖‘ndx), (.r𝑤)⟩ = ⟨(·𝑖‘ndx), (.r𝑊)⟩)
1311, 12oveq12d 7163 . . . . 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 5142 . . . 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 19873 . . . 4 subringAlg = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ (((𝑤 sSet ⟨(Scalar‘ndx), (𝑤s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑤)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑤)⟩)))
16 fvex 6676 . . . . . 6 (Base‘𝑊) ∈ V
1716pwex 5272 . . . . 5 𝒫 (Base‘𝑊) ∈ V
1817mptex 6977 . . . 4 (𝑠 ∈ 𝒫 (Base‘𝑊) ↦ (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩)) ∈ V
1914, 15, 18fvmpt 6761 . . 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 485 . . . . . . 7 (((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
2221oveq2d 7161 . . . . . 6 (((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) ∧ 𝑠 = 𝑆) → (𝑊s 𝑠) = (𝑊s 𝑆))
2322opeq2d 4802 . . . . 5 (((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) ∧ 𝑠 = 𝑆) → ⟨(Scalar‘ndx), (𝑊s 𝑠)⟩ = ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩)
2423oveq2d 7161 . . . 4 (((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) ∧ 𝑠 = 𝑆) → (𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑠)⟩) = (𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩))
2524oveq1d 7160 . . 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 7160 . 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 485 . . 3 ((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) → 𝑆 ⊆ (Base‘𝑊))
2816elpw2 5239 . . 3 (𝑆 ∈ 𝒫 (Base‘𝑊) ↔ 𝑆 ⊆ (Base‘𝑊))
2927, 28sylibr 235 . 2 ((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) → 𝑆 ∈ 𝒫 (Base‘𝑊))
30 ovexd 7180 . 2 ((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) → (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩) ∈ V)
3120, 26, 29, 30fvmptd 6767 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 396   = wceq 1528  wcel 2105  Vcvv 3492  wss 3933  𝒫 cpw 4535  cop 4563  cmpt 5137  cfv 6348  (class class class)co 7145  ndxcnx 16468   sSet csts 16469  Basecbs 16471  s cress 16472  .rcmulr 16554  Scalarcsca 16556   ·𝑠 cvsca 16557  ·𝑖cip 16558  subringAlg csra 19869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-sra 19873
This theorem is referenced by:  sralem  19878  srasca  19882  sravsca  19883  sraip  19884  rlmval2  19895
  Copyright terms: Public domain W3C validator