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Theorem sralem 21193
Description: Lemma for srabase 21195 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 2993 . . . . 5 (𝐸‘ndx) ≠ (Scalar‘ndx)
41, 3setsnid 17243 . . . 4 (𝐸𝑊) = (𝐸‘(𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩))
5 sralem.3 . . . . . 6 ( ·𝑠 ‘ndx) ≠ (𝐸‘ndx)
65necomi 2993 . . . . 5 (𝐸‘ndx) ≠ ( ·𝑠 ‘ndx)
71, 6setsnid 17243 . . . 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 2993 . . . . 5 (𝐸‘ndx) ≠ (·𝑖‘ndx)
101, 9setsnid 17243 . . . 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 2767 . . 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 481 . . . . 5 ((𝑊 ∈ V ∧ 𝜑) → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))
14 srapart.s . . . . . 6 (𝜑𝑆 ⊆ (Base‘𝑊))
15 sraval 21192 . . . . . 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 593 . . . . 5 ((𝑊 ∈ V ∧ 𝜑) → ((subringAlg ‘𝑊)‘𝑆) = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩))
1713, 16eqtrd 2775 . . . 4 ((𝑊 ∈ V ∧ 𝜑) → 𝐴 = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩))
1817fveq2d 6911 . . 3 ((𝑊 ∈ V ∧ 𝜑) → (𝐸𝐴) = (𝐸‘(((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩)))
1911, 18eqtr4id 2794 . 2 ((𝑊 ∈ V ∧ 𝜑) → (𝐸𝑊) = (𝐸𝐴))
201str0 17223 . . 3 ∅ = (𝐸‘∅)
21 fvprc 6899 . . . 4 𝑊 ∈ V → (𝐸𝑊) = ∅)
2221adantr 480 . . 3 ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝐸𝑊) = ∅)
23 fv2prc 6952 . . . . 5 𝑊 ∈ V → ((subringAlg ‘𝑊)‘𝑆) = ∅)
2412, 23sylan9eqr 2797 . . . 4 ((¬ 𝑊 ∈ V ∧ 𝜑) → 𝐴 = ∅)
2524fveq2d 6911 . . 3 ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝐸𝐴) = (𝐸‘∅))
2620, 22, 253eqtr4a 2801 . 2 ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝐸𝑊) = (𝐸𝐴))
2719, 26pm2.61ian 812 1 (𝜑 → (𝐸𝑊) = (𝐸𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wcel 2106  wne 2938  Vcvv 3478  wss 3963  c0 4339  cop 4637  cfv 6563  (class class class)co 7431   sSet csts 17197  Slot cslot 17215  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  ax-un 7754
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-oprab 7435  df-mpo 7436  df-sets 17198  df-slot 17216  df-sra 21190
This theorem is referenced by:  srabase  21195  sraaddg  21197  sramulr  21199  sratset  21206  srads  21209  cchhllem  28916
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