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Theorem zlmval 21544
Description: Augment an abelian group with vector space operations to turn it into a -module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.)
Hypotheses
Ref Expression
zlmval.w 𝑊 = (ℤMod‘𝐺)
zlmval.m · = (.g𝐺)
Assertion
Ref Expression
zlmval (𝐺𝑉𝑊 = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩))

Proof of Theorem zlmval
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 zlmval.w . 2 𝑊 = (ℤMod‘𝐺)
2 elex 3499 . . 3 (𝐺𝑉𝐺 ∈ V)
3 oveq1 7438 . . . . 5 (𝑔 = 𝐺 → (𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) = (𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩))
4 fveq2 6907 . . . . . . 7 (𝑔 = 𝐺 → (.g𝑔) = (.g𝐺))
5 zlmval.m . . . . . . 7 · = (.g𝐺)
64, 5eqtr4di 2793 . . . . . 6 (𝑔 = 𝐺 → (.g𝑔) = · )
76opeq2d 4885 . . . . 5 (𝑔 = 𝐺 → ⟨( ·𝑠 ‘ndx), (.g𝑔)⟩ = ⟨( ·𝑠 ‘ndx), · ⟩)
83, 7oveq12d 7449 . . . 4 (𝑔 = 𝐺 → ((𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g𝑔)⟩) = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩))
9 df-zlm 21533 . . . 4 ℤMod = (𝑔 ∈ V ↦ ((𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g𝑔)⟩))
10 ovex 7464 . . . 4 ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩) ∈ V
118, 9, 10fvmpt 7016 . . 3 (𝐺 ∈ V → (ℤMod‘𝐺) = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩))
122, 11syl 17 . 2 (𝐺𝑉 → (ℤMod‘𝐺) = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩))
131, 12eqtrid 2787 1 (𝐺𝑉𝑊 = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  cop 4637  cfv 6563  (class class class)co 7431   sSet csts 17197  ndxcnx 17227  Scalarcsca 17301   ·𝑠 cvsca 17302  .gcmg 19098  ringczring 21475  ℤModczlm 21529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-zlm 21533
This theorem is referenced by:  zlmlem  21545  zlmlemOLD  21546  zlmsca  21553  zlmvsca  21554  zlmds  33923  zlmdsOLD  33924  zlmtset  33925  zlmtsetOLD  33926
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