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Theorem rlmval2 20464
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 20461 . . 3 (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊))
21a1i 11 . 2 (𝑊𝑋 → (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊)))
3 ssid 3943 . . 3 (Base‘𝑊) ⊆ (Base‘𝑊)
4 sraval 20438 . . 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 688 . 2 (𝑊𝑋 → ((subringAlg ‘𝑊)‘(Base‘𝑊)) = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s (Base‘𝑊))⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩))
6 eqid 2738 . . . . . . 7 (Base‘𝑊) = (Base‘𝑊)
76ressid 16954 . . . . . 6 (𝑊𝑋 → (𝑊s (Base‘𝑊)) = 𝑊)
87opeq2d 4811 . . . . 5 (𝑊𝑋 → ⟨(Scalar‘ndx), (𝑊s (Base‘𝑊))⟩ = ⟨(Scalar‘ndx), 𝑊⟩)
98oveq2d 7291 . . . 4 (𝑊𝑋 → (𝑊 sSet ⟨(Scalar‘ndx), (𝑊s (Base‘𝑊))⟩) = (𝑊 sSet ⟨(Scalar‘ndx), 𝑊⟩))
109oveq1d 7290 . . 3 (𝑊𝑋 → ((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s (Base‘𝑊))⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) = ((𝑊 sSet ⟨(Scalar‘ndx), 𝑊⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩))
1110oveq1d 7290 . 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 2782 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 1539  wcel 2106  wss 3887  cop 4567  cfv 6433  (class class class)co 7275   sSet csts 16864  ndxcnx 16894  Basecbs 16912  s cress 16941  .rcmulr 16963  Scalarcsca 16965   ·𝑠 cvsca 16966  ·𝑖cip 16967  subringAlg csra 20430  ringLModcrglmod 20431
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-ress 16942  df-sra 20434  df-rgmod 20435
This theorem is referenced by: (None)
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