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Theorem rlmval2 21092
Description: Value of the ring module extended. (Contributed by AV, 2-Dec-2018.) (Revised by Thierry Arnoux, 16-Jun-2019.)
Assertion
Ref Expression
rlmval2 (π‘Š ∈ 𝑋 β†’ (ringLModβ€˜π‘Š) = (((π‘Š sSet ⟨(Scalarβ€˜ndx), π‘ŠβŸ©) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘Š)⟩))

Proof of Theorem rlmval2
StepHypRef Expression
1 rlmval 21091 . . 3 (ringLModβ€˜π‘Š) = ((subringAlg β€˜π‘Š)β€˜(Baseβ€˜π‘Š))
21a1i 11 . 2 (π‘Š ∈ 𝑋 β†’ (ringLModβ€˜π‘Š) = ((subringAlg β€˜π‘Š)β€˜(Baseβ€˜π‘Š)))
3 ssid 4004 . . 3 (Baseβ€˜π‘Š) βŠ† (Baseβ€˜π‘Š)
4 sraval 21067 . . 3 ((π‘Š ∈ 𝑋 ∧ (Baseβ€˜π‘Š) βŠ† (Baseβ€˜π‘Š)) β†’ ((subringAlg β€˜π‘Š)β€˜(Baseβ€˜π‘Š)) = (((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs (Baseβ€˜π‘Š))⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘Š)⟩))
53, 4mpan2 689 . 2 (π‘Š ∈ 𝑋 β†’ ((subringAlg β€˜π‘Š)β€˜(Baseβ€˜π‘Š)) = (((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs (Baseβ€˜π‘Š))⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘Š)⟩))
6 eqid 2728 . . . . . . 7 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
76ressid 17232 . . . . . 6 (π‘Š ∈ 𝑋 β†’ (π‘Š β†Ύs (Baseβ€˜π‘Š)) = π‘Š)
87opeq2d 4885 . . . . 5 (π‘Š ∈ 𝑋 β†’ ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs (Baseβ€˜π‘Š))⟩ = ⟨(Scalarβ€˜ndx), π‘ŠβŸ©)
98oveq2d 7442 . . . 4 (π‘Š ∈ 𝑋 β†’ (π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs (Baseβ€˜π‘Š))⟩) = (π‘Š sSet ⟨(Scalarβ€˜ndx), π‘ŠβŸ©))
109oveq1d 7441 . . 3 (π‘Š ∈ 𝑋 β†’ ((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs (Baseβ€˜π‘Š))⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) = ((π‘Š sSet ⟨(Scalarβ€˜ndx), π‘ŠβŸ©) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩))
1110oveq1d 7441 . 2 (π‘Š ∈ 𝑋 β†’ (((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs (Baseβ€˜π‘Š))⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘Š)⟩) = (((π‘Š sSet ⟨(Scalarβ€˜ndx), π‘ŠβŸ©) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘Š)⟩))
122, 5, 113eqtrd 2772 1 (π‘Š ∈ 𝑋 β†’ (ringLModβ€˜π‘Š) = (((π‘Š sSet ⟨(Scalarβ€˜ndx), π‘ŠβŸ©) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘Š)⟩))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098   βŠ† wss 3949  βŸ¨cop 4638  β€˜cfv 6553  (class class class)co 7426   sSet csts 17139  ndxcnx 17169  Basecbs 17187   β†Ύs cress 17216  .rcmulr 17241  Scalarcsca 17243   ·𝑠 cvsca 17244  Β·π‘–cip 17245  subringAlg csra 21063  ringLModcrglmod 21064
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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-ress 17217  df-sra 21065  df-rgmod 21066
This theorem is referenced by: (None)
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