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Theorem rlmval2 19015
Description: Value of the ring module extended. (Contributed by AV, 2-Dec-2018.) (Revised by Thierry Arnoux, 16-Jun-2019.)
Assertion
Ref Expression
rlmval2 (𝑊𝑋 → (ringLMod‘𝑊) = (((𝑊 sSet ⟨(Scalar‘ndx), 𝑊⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩))

Proof of Theorem rlmval2
StepHypRef Expression
1 rlmval 19012 . . 3 (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊))
21a1i 11 . 2 (𝑊𝑋 → (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊)))
3 ssid 3587 . . 3 (Base‘𝑊) ⊆ (Base‘𝑊)
4 sraval 18997 . . 3 ((𝑊𝑋 ∧ (Base‘𝑊) ⊆ (Base‘𝑊)) → ((subringAlg ‘𝑊)‘(Base‘𝑊)) = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s (Base‘𝑊))⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩))
53, 4mpan2 703 . 2 (𝑊𝑋 → ((subringAlg ‘𝑊)‘(Base‘𝑊)) = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s (Base‘𝑊))⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩))
6 eqid 2610 . . . . . . 7 (Base‘𝑊) = (Base‘𝑊)
76ressid 15762 . . . . . 6 (𝑊𝑋 → (𝑊s (Base‘𝑊)) = 𝑊)
87opeq2d 4347 . . . . 5 (𝑊𝑋 → ⟨(Scalar‘ndx), (𝑊s (Base‘𝑊))⟩ = ⟨(Scalar‘ndx), 𝑊⟩)
98oveq2d 6565 . . . 4 (𝑊𝑋 → (𝑊 sSet ⟨(Scalar‘ndx), (𝑊s (Base‘𝑊))⟩) = (𝑊 sSet ⟨(Scalar‘ndx), 𝑊⟩))
109oveq1d 6564 . . 3 (𝑊𝑋 → ((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s (Base‘𝑊))⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) = ((𝑊 sSet ⟨(Scalar‘ndx), 𝑊⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩))
1110oveq1d 6564 . 2 (𝑊𝑋 → (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s (Base‘𝑊))⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩) = (((𝑊 sSet ⟨(Scalar‘ndx), 𝑊⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩))
122, 5, 113eqtrd 2648 1 (𝑊𝑋 → (ringLMod‘𝑊) = (((𝑊 sSet ⟨(Scalar‘ndx), 𝑊⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  wss 3540  cop 4131  cfv 5804  (class class class)co 6549  ndxcnx 15692   sSet csts 15693  Basecbs 15695  s cress 15696  .rcmulr 15769  Scalarcsca 15771   ·𝑠 cvsca 15772  ·𝑖cip 15773  subringAlg csra 18989  ringLModcrglmod 18990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-ress 15702  df-sra 18993  df-rgmod 18994
This theorem is referenced by: (None)
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