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

Theorem sralem 20767
Description: Lemma for srabase 20769 and similar theorems. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
Hypotheses
Ref Expression
srapart.a (𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))
srapart.s (𝜑𝑆 ⊆ (Base‘𝑊))
sralem.1 𝐸 = Slot (𝐸‘ndx)
sralem.2 (Scalar‘ndx) ≠ (𝐸‘ndx)
sralem.3 ( ·𝑠 ‘ndx) ≠ (𝐸‘ndx)
sralem.4 (·𝑖‘ndx) ≠ (𝐸‘ndx)
Assertion
Ref Expression
sralem (𝜑 → (𝐸𝑊) = (𝐸𝐴))

Proof of Theorem sralem
StepHypRef Expression
1 sralem.1 . . . . 5 𝐸 = Slot (𝐸‘ndx)
2 sralem.2 . . . . . 6 (Scalar‘ndx) ≠ (𝐸‘ndx)
32necomi 2996 . . . . 5 (𝐸‘ndx) ≠ (Scalar‘ndx)
41, 3setsnid 17129 . . . 4 (𝐸𝑊) = (𝐸‘(𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩))
5 sralem.3 . . . . . 6 ( ·𝑠 ‘ndx) ≠ (𝐸‘ndx)
65necomi 2996 . . . . 5 (𝐸‘ndx) ≠ ( ·𝑠 ‘ndx)
71, 6setsnid 17129 . . . 4 (𝐸‘(𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩)) = (𝐸‘((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩))
8 sralem.4 . . . . . 6 (·𝑖‘ndx) ≠ (𝐸‘ndx)
98necomi 2996 . . . . 5 (𝐸‘ndx) ≠ (·𝑖‘ndx)
101, 9setsnid 17129 . . . 4 (𝐸‘((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩)) = (𝐸‘(((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩))
114, 7, 103eqtri 2765 . . 3 (𝐸𝑊) = (𝐸‘(((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩))
12 srapart.a . . . . . 6 (𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))
1312adantl 483 . . . . 5 ((𝑊 ∈ V ∧ 𝜑) → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))
14 srapart.s . . . . . 6 (𝜑𝑆 ⊆ (Base‘𝑊))
15 sraval 20766 . . . . . 6 ((𝑊 ∈ V ∧ 𝑆 ⊆ (Base‘𝑊)) → ((subringAlg ‘𝑊)‘𝑆) = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩))
1614, 15sylan2 594 . . . . 5 ((𝑊 ∈ V ∧ 𝜑) → ((subringAlg ‘𝑊)‘𝑆) = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩))
1713, 16eqtrd 2773 . . . 4 ((𝑊 ∈ V ∧ 𝜑) → 𝐴 = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩))
1817fveq2d 6885 . . 3 ((𝑊 ∈ V ∧ 𝜑) → (𝐸𝐴) = (𝐸‘(((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩)))
1911, 18eqtr4id 2792 . 2 ((𝑊 ∈ V ∧ 𝜑) → (𝐸𝑊) = (𝐸𝐴))
201str0 17109 . . 3 ∅ = (𝐸‘∅)
21 fvprc 6873 . . . 4 𝑊 ∈ V → (𝐸𝑊) = ∅)
2221adantr 482 . . 3 ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝐸𝑊) = ∅)
23 fv2prc 6926 . . . . 5 𝑊 ∈ V → ((subringAlg ‘𝑊)‘𝑆) = ∅)
2412, 23sylan9eqr 2795 . . . 4 ((¬ 𝑊 ∈ V ∧ 𝜑) → 𝐴 = ∅)
2524fveq2d 6885 . . 3 ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝐸𝐴) = (𝐸‘∅))
2620, 22, 253eqtr4a 2799 . 2 ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝐸𝑊) = (𝐸𝐴))
2719, 26pm2.61ian 811 1 (𝜑 → (𝐸𝑊) = (𝐸𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wcel 2107  wne 2941  Vcvv 3475  wss 3946  c0 4320  cop 4630  cfv 6535  (class class class)co 7396   sSet csts 17083  Slot cslot 17101  ndxcnx 17113  Basecbs 17131  s cress 17160  .rcmulr 17185  Scalarcsca 17187   ·𝑠 cvsca 17188  ·𝑖cip 17189  subringAlg csra 20758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-iun 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-ov 7399  df-oprab 7400  df-mpo 7401  df-sets 17084  df-slot 17102  df-sra 20762
This theorem is referenced by:  srabase  20769  sraaddg  20771  sramulr  20773  sratset  20780  srads  20783  cchhllem  28111
  Copyright terms: Public domain W3C validator