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

Theorem sraval 21192
Description: Lemma for srabase 21195 through sravsca 21203. (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 3499 . . . 4 (𝑊𝑉𝑊 ∈ V)
21adantr 480 . . 3 ((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) → 𝑊 ∈ V)
3 fveq2 6907 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
43pweqd 4622 . . . . 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 4885 . . . . . . . 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 6907 . . . . . . . 8 (𝑤 = 𝑊 → (.r𝑤) = (.r𝑊))
109opeq2d 4885 . . . . . . 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 4885 . . . . . 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 5239 . . . 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 21190 . . . 4 subringAlg = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ (((𝑤 sSet ⟨(Scalar‘ndx), (𝑤s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑤)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑤)⟩)))
16 fvex 6920 . . . . . 6 (Base‘𝑊) ∈ V
1716pwex 5386 . . . . 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 4885 . . . . 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 5340 . . 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 1537  wcel 2106  Vcvv 3478  wss 3963  𝒫 cpw 4605  cop 4637  cmpt 5231  cfv 6563  (class class class)co 7431   sSet csts 17197  ndxcnx 17227  Basecbs 17245  s cress 17274  .rcmulr 17299  Scalarcsca 17301   ·𝑠 cvsca 17302  ·𝑖cip 17303  subringAlg csra 21188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-sra 21190
This theorem is referenced by:  sralem  21193  sralemOLD  21194  srasca  21201  srascaOLD  21202  sravsca  21203  sravscaOLD  21204  sraip  21205  rlmval2  21217  resssra  33617
  Copyright terms: Public domain W3C validator