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Theorem sralem 21022
Description: Lemma for srabase 21024 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 (πœ‘ β†’ 𝐴 = ((subringAlg β€˜π‘Š)β€˜π‘†))
srapart.s (πœ‘ β†’ 𝑆 βŠ† (Baseβ€˜π‘Š))
sralem.1 𝐸 = Slot (πΈβ€˜ndx)
sralem.2 (Scalarβ€˜ndx) β‰  (πΈβ€˜ndx)
sralem.3 ( ·𝑠 β€˜ndx) β‰  (πΈβ€˜ndx)
sralem.4 (Β·π‘–β€˜ndx) β‰  (πΈβ€˜ndx)
Assertion
Ref Expression
sralem (πœ‘ β†’ (πΈβ€˜π‘Š) = (πΈβ€˜π΄))

Proof of Theorem sralem
StepHypRef Expression
1 sralem.1 . . . . 5 𝐸 = Slot (πΈβ€˜ndx)
2 sralem.2 . . . . . 6 (Scalarβ€˜ndx) β‰  (πΈβ€˜ndx)
32necomi 2989 . . . . 5 (πΈβ€˜ndx) β‰  (Scalarβ€˜ndx)
41, 3setsnid 17149 . . . 4 (πΈβ€˜π‘Š) = (πΈβ€˜(π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩))
5 sralem.3 . . . . . 6 ( ·𝑠 β€˜ndx) β‰  (πΈβ€˜ndx)
65necomi 2989 . . . . 5 (πΈβ€˜ndx) β‰  ( ·𝑠 β€˜ndx)
71, 6setsnid 17149 . . . 4 (πΈβ€˜(π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩)) = (πΈβ€˜((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩))
8 sralem.4 . . . . . 6 (Β·π‘–β€˜ndx) β‰  (πΈβ€˜ndx)
98necomi 2989 . . . . 5 (πΈβ€˜ndx) β‰  (Β·π‘–β€˜ndx)
101, 9setsnid 17149 . . . 4 (πΈβ€˜((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩)) = (πΈβ€˜(((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘Š)⟩))
114, 7, 103eqtri 2758 . . 3 (πΈβ€˜π‘Š) = (πΈβ€˜(((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘Š)⟩))
12 srapart.a . . . . . 6 (πœ‘ β†’ 𝐴 = ((subringAlg β€˜π‘Š)β€˜π‘†))
1312adantl 481 . . . . 5 ((π‘Š ∈ V ∧ πœ‘) β†’ 𝐴 = ((subringAlg β€˜π‘Š)β€˜π‘†))
14 srapart.s . . . . . 6 (πœ‘ β†’ 𝑆 βŠ† (Baseβ€˜π‘Š))
15 sraval 21021 . . . . . 6 ((π‘Š ∈ V ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) β†’ ((subringAlg β€˜π‘Š)β€˜π‘†) = (((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘Š)⟩))
1614, 15sylan2 592 . . . . 5 ((π‘Š ∈ V ∧ πœ‘) β†’ ((subringAlg β€˜π‘Š)β€˜π‘†) = (((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘Š)⟩))
1713, 16eqtrd 2766 . . . 4 ((π‘Š ∈ V ∧ πœ‘) β†’ 𝐴 = (((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘Š)⟩))
1817fveq2d 6888 . . 3 ((π‘Š ∈ V ∧ πœ‘) β†’ (πΈβ€˜π΄) = (πΈβ€˜(((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘Š)⟩)))
1911, 18eqtr4id 2785 . 2 ((π‘Š ∈ V ∧ πœ‘) β†’ (πΈβ€˜π‘Š) = (πΈβ€˜π΄))
201str0 17129 . . 3 βˆ… = (πΈβ€˜βˆ…)
21 fvprc 6876 . . . 4 (Β¬ π‘Š ∈ V β†’ (πΈβ€˜π‘Š) = βˆ…)
2221adantr 480 . . 3 ((Β¬ π‘Š ∈ V ∧ πœ‘) β†’ (πΈβ€˜π‘Š) = βˆ…)
23 fv2prc 6929 . . . . 5 (Β¬ π‘Š ∈ V β†’ ((subringAlg β€˜π‘Š)β€˜π‘†) = βˆ…)
2412, 23sylan9eqr 2788 . . . 4 ((Β¬ π‘Š ∈ V ∧ πœ‘) β†’ 𝐴 = βˆ…)
2524fveq2d 6888 . . 3 ((Β¬ π‘Š ∈ V ∧ πœ‘) β†’ (πΈβ€˜π΄) = (πΈβ€˜βˆ…))
2620, 22, 253eqtr4a 2792 . 2 ((Β¬ π‘Š ∈ V ∧ πœ‘) β†’ (πΈβ€˜π‘Š) = (πΈβ€˜π΄))
2719, 26pm2.61ian 809 1 (πœ‘ β†’ (πΈβ€˜π‘Š) = (πΈβ€˜π΄))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  Vcvv 3468   βŠ† wss 3943  βˆ…c0 4317  βŸ¨cop 4629  β€˜cfv 6536  (class class class)co 7404   sSet csts 17103  Slot cslot 17121  ndxcnx 17133  Basecbs 17151   β†Ύs cress 17180  .rcmulr 17205  Scalarcsca 17207   ·𝑠 cvsca 17208  Β·π‘–cip 17209  subringAlg csra 21017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-sets 17104  df-slot 17122  df-sra 21019
This theorem is referenced by:  srabase  21024  sraaddg  21026  sramulr  21028  sratset  21035  srads  21038  cchhllem  28648
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