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Theorem rlmval2 19175
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 19172 . . 3 (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊))
21a1i 11 . 2 (𝑊𝑋 → (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊)))
3 ssid 3616 . . 3 (Base‘𝑊) ⊆ (Base‘𝑊)
4 sraval 19157 . . 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 706 . 2 (𝑊𝑋 → ((subringAlg ‘𝑊)‘(Base‘𝑊)) = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s (Base‘𝑊))⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩))
6 eqid 2620 . . . . . . 7 (Base‘𝑊) = (Base‘𝑊)
76ressid 15916 . . . . . 6 (𝑊𝑋 → (𝑊s (Base‘𝑊)) = 𝑊)
87opeq2d 4400 . . . . 5 (𝑊𝑋 → ⟨(Scalar‘ndx), (𝑊s (Base‘𝑊))⟩ = ⟨(Scalar‘ndx), 𝑊⟩)
98oveq2d 6651 . . . 4 (𝑊𝑋 → (𝑊 sSet ⟨(Scalar‘ndx), (𝑊s (Base‘𝑊))⟩) = (𝑊 sSet ⟨(Scalar‘ndx), 𝑊⟩))
109oveq1d 6650 . . 3 (𝑊𝑋 → ((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s (Base‘𝑊))⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) = ((𝑊 sSet ⟨(Scalar‘ndx), 𝑊⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩))
1110oveq1d 6650 . 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 2658 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 1481  wcel 1988  wss 3567  cop 4174  cfv 5876  (class class class)co 6635  ndxcnx 15835   sSet csts 15836  Basecbs 15838  s cress 15839  .rcmulr 15923  Scalarcsca 15925   ·𝑠 cvsca 15926  ·𝑖cip 15927  subringAlg csra 19149  ringLModcrglmod 19150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-ress 15846  df-sra 19153  df-rgmod 19154
This theorem is referenced by: (None)
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