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Theorem sralem 20790
Description: Lemma for srabase 20792 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 2996 . . . . 5 (πΈβ€˜ndx) β‰  (Scalarβ€˜ndx)
41, 3setsnid 17142 . . . 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 2996 . . . . 5 (πΈβ€˜ndx) β‰  ( ·𝑠 β€˜ndx)
71, 6setsnid 17142 . . . 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 2996 . . . . 5 (πΈβ€˜ndx) β‰  (Β·π‘–β€˜ndx)
101, 9setsnid 17142 . . . 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 2765 . . 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 20789 . . . . . 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 2773 . . . 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 6896 . . 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 2792 . 2 ((π‘Š ∈ V ∧ πœ‘) β†’ (πΈβ€˜π‘Š) = (πΈβ€˜π΄))
201str0 17122 . . 3 βˆ… = (πΈβ€˜βˆ…)
21 fvprc 6884 . . . 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 6937 . . . . 5 (Β¬ π‘Š ∈ V β†’ ((subringAlg β€˜π‘Š)β€˜π‘†) = βˆ…)
2412, 23sylan9eqr 2795 . . . 4 ((Β¬ π‘Š ∈ V ∧ πœ‘) β†’ 𝐴 = βˆ…)
2524fveq2d 6896 . . 3 ((Β¬ π‘Š ∈ V ∧ πœ‘) β†’ (πΈβ€˜π΄) = (πΈβ€˜βˆ…))
2620, 22, 253eqtr4a 2799 . 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 2941  Vcvv 3475   βŠ† wss 3949  βˆ…c0 4323  βŸ¨cop 4635  β€˜cfv 6544  (class class class)co 7409   sSet csts 17096  Slot cslot 17114  ndxcnx 17126  Basecbs 17144   β†Ύs cress 17173  .rcmulr 17198  Scalarcsca 17200   ·𝑠 cvsca 17201  Β·π‘–cip 17202  subringAlg csra 20781
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-sets 17097  df-slot 17115  df-sra 20785
This theorem is referenced by:  srabase  20792  sraaddg  20794  sramulr  20796  sratset  20803  srads  20806  cchhllem  28144
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