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Theorem zlmval 21284
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 3492 . . 3 (𝐺𝑉𝐺 ∈ V)
3 oveq1 7418 . . . . 5 (𝑔 = 𝐺 → (𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) = (𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩))
4 fveq2 6891 . . . . . . 7 (𝑔 = 𝐺 → (.g𝑔) = (.g𝐺))
5 zlmval.m . . . . . . 7 · = (.g𝐺)
64, 5eqtr4di 2790 . . . . . 6 (𝑔 = 𝐺 → (.g𝑔) = · )
76opeq2d 4880 . . . . 5 (𝑔 = 𝐺 → ⟨( ·𝑠 ‘ndx), (.g𝑔)⟩ = ⟨( ·𝑠 ‘ndx), · ⟩)
83, 7oveq12d 7429 . . . 4 (𝑔 = 𝐺 → ((𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g𝑔)⟩) = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩))
9 df-zlm 21273 . . . 4 ℤMod = (𝑔 ∈ V ↦ ((𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g𝑔)⟩))
10 ovex 7444 . . . 4 ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩) ∈ V
118, 9, 10fvmpt 6998 . . 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 2784 1 (𝐺𝑉𝑊 = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  Vcvv 3474  cop 4634  cfv 6543  (class class class)co 7411   sSet csts 17100  ndxcnx 17130  Scalarcsca 17204   ·𝑠 cvsca 17205  .gcmg 18986  ringczring 21217  ℤModczlm 21269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7414  df-zlm 21273
This theorem is referenced by:  zlmlem  21285  zlmlemOLD  21286  zlmsca  21293  zlmvsca  21294  zlmds  33228  zlmdsOLD  33229  zlmtset  33230  zlmtsetOLD  33231
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