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Theorem sralem 21175
Description: Lemma for srabase 21177 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 2995 . . . . 5 (𝐸‘ndx) ≠ (Scalar‘ndx)
41, 3setsnid 17245 . . . 4 (𝐸𝑊) = (𝐸‘(𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩))
5 sralem.3 . . . . . 6 ( ·𝑠 ‘ndx) ≠ (𝐸‘ndx)
65necomi 2995 . . . . 5 (𝐸‘ndx) ≠ ( ·𝑠 ‘ndx)
71, 6setsnid 17245 . . . 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 2995 . . . . 5 (𝐸‘ndx) ≠ (·𝑖‘ndx)
101, 9setsnid 17245 . . . 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 2769 . . 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 21174 . . . . . 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 2777 . . . 4 ((𝑊 ∈ V ∧ 𝜑) → 𝐴 = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩))
1817fveq2d 6910 . . 3 ((𝑊 ∈ V ∧ 𝜑) → (𝐸𝐴) = (𝐸‘(((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩)))
1911, 18eqtr4id 2796 . 2 ((𝑊 ∈ V ∧ 𝜑) → (𝐸𝑊) = (𝐸𝐴))
201str0 17226 . . 3 ∅ = (𝐸‘∅)
21 fvprc 6898 . . . 4 𝑊 ∈ V → (𝐸𝑊) = ∅)
2221adantr 480 . . 3 ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝐸𝑊) = ∅)
23 fv2prc 6951 . . . . 5 𝑊 ∈ V → ((subringAlg ‘𝑊)‘𝑆) = ∅)
2412, 23sylan9eqr 2799 . . . 4 ((¬ 𝑊 ∈ V ∧ 𝜑) → 𝐴 = ∅)
2524fveq2d 6910 . . 3 ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝐸𝐴) = (𝐸‘∅))
2620, 22, 253eqtr4a 2803 . 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 1540  wcel 2108  wne 2940  Vcvv 3480  wss 3951  c0 4333  cop 4632  cfv 6561  (class class class)co 7431   sSet csts 17200  Slot cslot 17218  ndxcnx 17230  Basecbs 17247  s cress 17274  .rcmulr 17298  Scalarcsca 17300   ·𝑠 cvsca 17301  ·𝑖cip 17302  subringAlg csra 21170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-sets 17201  df-slot 17219  df-sra 21172
This theorem is referenced by:  srabase  21177  sraaddg  21179  sramulr  21181  sratset  21188  srads  21191  cchhllem  28901
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