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

Theorem sraval 19116
 Description: Lemma for srabase 19118 through sravsca 19122. (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 3202 . . . 4 (𝑊𝑉𝑊 ∈ V)
21adantr 481 . . 3 ((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) → 𝑊 ∈ V)
3 fveq2 6158 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
43pweqd 4141 . . . . 5 (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 (Base‘𝑊))
5 id 22 . . . . . . . 8 (𝑤 = 𝑊𝑤 = 𝑊)
6 oveq1 6622 . . . . . . . . 9 (𝑤 = 𝑊 → (𝑤s 𝑠) = (𝑊s 𝑠))
76opeq2d 4384 . . . . . . . 8 (𝑤 = 𝑊 → ⟨(Scalar‘ndx), (𝑤s 𝑠)⟩ = ⟨(Scalar‘ndx), (𝑊s 𝑠)⟩)
85, 7oveq12d 6633 . . . . . . 7 (𝑤 = 𝑊 → (𝑤 sSet ⟨(Scalar‘ndx), (𝑤s 𝑠)⟩) = (𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑠)⟩))
9 fveq2 6158 . . . . . . . 8 (𝑤 = 𝑊 → (.r𝑤) = (.r𝑊))
109opeq2d 4384 . . . . . . 7 (𝑤 = 𝑊 → ⟨( ·𝑠 ‘ndx), (.r𝑤)⟩ = ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩)
118, 10oveq12d 6633 . . . . . 6 (𝑤 = 𝑊 → ((𝑤 sSet ⟨(Scalar‘ndx), (𝑤s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑤)⟩) = ((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩))
129opeq2d 4384 . . . . . 6 (𝑤 = 𝑊 → ⟨(·𝑖‘ndx), (.r𝑤)⟩ = ⟨(·𝑖‘ndx), (.r𝑊)⟩)
1311, 12oveq12d 6633 . . . . 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 4703 . . . 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 19112 . . . 4 subringAlg = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ (((𝑤 sSet ⟨(Scalar‘ndx), (𝑤s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑤)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑤)⟩)))
16 fvex 6168 . . . . . 6 (Base‘𝑊) ∈ V
1716pwex 4818 . . . . 5 𝒫 (Base‘𝑊) ∈ V
1817mptex 6451 . . . 4 (𝑠 ∈ 𝒫 (Base‘𝑊) ↦ (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩)) ∈ V
1914, 15, 18fvmpt 6249 . . 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 477 . . . . . . 7 (((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
2221oveq2d 6631 . . . . . 6 (((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) ∧ 𝑠 = 𝑆) → (𝑊s 𝑠) = (𝑊s 𝑆))
2322opeq2d 4384 . . . . 5 (((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) ∧ 𝑠 = 𝑆) → ⟨(Scalar‘ndx), (𝑊s 𝑠)⟩ = ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩)
2423oveq2d 6631 . . . 4 (((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) ∧ 𝑠 = 𝑆) → (𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑠)⟩) = (𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩))
2524oveq1d 6630 . . 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 6630 . 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 477 . . 3 ((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) → 𝑆 ⊆ (Base‘𝑊))
2816elpw2 4798 . . 3 (𝑆 ∈ 𝒫 (Base‘𝑊) ↔ 𝑆 ⊆ (Base‘𝑊))
2927, 28sylibr 224 . 2 ((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) → 𝑆 ∈ 𝒫 (Base‘𝑊))
30 ovexd 6645 . 2 ((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) → (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩) ∈ V)
3120, 26, 29, 30fvmptd 6255 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 384   = wceq 1480   ∈ wcel 1987  Vcvv 3190   ⊆ wss 3560  𝒫 cpw 4136  ⟨cop 4161   ↦ cmpt 4683  ‘cfv 5857  (class class class)co 6615  ndxcnx 15797   sSet csts 15798  Basecbs 15800   ↾s cress 15801  .rcmulr 15882  Scalarcsca 15884   ·𝑠 cvsca 15885  ·𝑖cip 15886  subringAlg csra 19108 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-sra 19112 This theorem is referenced by:  sralem  19117  srasca  19121  sravsca  19122  sraip  19123  rlmval2  19134
 Copyright terms: Public domain W3C validator