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Theorem sralem 20654
Description: Lemma for srabase 20656 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 2999 . . . . 5 (πΈβ€˜ndx) β‰  (Scalarβ€˜ndx)
41, 3setsnid 17088 . . . 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 2999 . . . . 5 (πΈβ€˜ndx) β‰  ( ·𝑠 β€˜ndx)
71, 6setsnid 17088 . . . 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 2999 . . . . 5 (πΈβ€˜ndx) β‰  (Β·π‘–β€˜ndx)
101, 9setsnid 17088 . . . 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 2769 . . 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 483 . . . . 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 20653 . . . . . 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 594 . . . . 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 2777 . . . 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 6851 . . 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 2796 . 2 ((π‘Š ∈ V ∧ πœ‘) β†’ (πΈβ€˜π‘Š) = (πΈβ€˜π΄))
201str0 17068 . . 3 βˆ… = (πΈβ€˜βˆ…)
21 fvprc 6839 . . . 4 (Β¬ π‘Š ∈ V β†’ (πΈβ€˜π‘Š) = βˆ…)
2221adantr 482 . . 3 ((Β¬ π‘Š ∈ V ∧ πœ‘) β†’ (πΈβ€˜π‘Š) = βˆ…)
23 fv2prc 6892 . . . . 5 (Β¬ π‘Š ∈ V β†’ ((subringAlg β€˜π‘Š)β€˜π‘†) = βˆ…)
2412, 23sylan9eqr 2799 . . . 4 ((Β¬ π‘Š ∈ V ∧ πœ‘) β†’ 𝐴 = βˆ…)
2524fveq2d 6851 . . 3 ((Β¬ π‘Š ∈ V ∧ πœ‘) β†’ (πΈβ€˜π΄) = (πΈβ€˜βˆ…))
2620, 22, 253eqtr4a 2803 . 2 ((Β¬ π‘Š ∈ V ∧ πœ‘) β†’ (πΈβ€˜π‘Š) = (πΈβ€˜π΄))
2719, 26pm2.61ian 811 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 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  Vcvv 3448   βŠ† wss 3915  βˆ…c0 4287  βŸ¨cop 4597  β€˜cfv 6501  (class class class)co 7362   sSet csts 17042  Slot cslot 17060  ndxcnx 17072  Basecbs 17090   β†Ύs cress 17119  .rcmulr 17141  Scalarcsca 17143   ·𝑠 cvsca 17144  Β·π‘–cip 17145  subringAlg csra 20645
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-sets 17043  df-slot 17061  df-sra 20649
This theorem is referenced by:  srabase  20656  sraaddg  20658  sramulr  20660  sratset  20667  srads  20670  cchhllem  27877
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