MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  zlmval Structured version   Visualization version   GIF version

Theorem zlmval 20066
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 3352 . . 3 (𝐺𝑉𝐺 ∈ V)
3 oveq1 6820 . . . . 5 (𝑔 = 𝐺 → (𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) = (𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩))
4 fveq2 6352 . . . . . . 7 (𝑔 = 𝐺 → (.g𝑔) = (.g𝐺))
5 zlmval.m . . . . . . 7 · = (.g𝐺)
64, 5syl6eqr 2812 . . . . . 6 (𝑔 = 𝐺 → (.g𝑔) = · )
76opeq2d 4560 . . . . 5 (𝑔 = 𝐺 → ⟨( ·𝑠 ‘ndx), (.g𝑔)⟩ = ⟨( ·𝑠 ‘ndx), · ⟩)
83, 7oveq12d 6831 . . . 4 (𝑔 = 𝐺 → ((𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g𝑔)⟩) = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩))
9 df-zlm 20055 . . . 4 ℤMod = (𝑔 ∈ V ↦ ((𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g𝑔)⟩))
10 ovex 6841 . . . 4 ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩) ∈ V
118, 9, 10fvmpt 6444 . . 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, 12syl5eq 2806 1 (𝐺𝑉𝑊 = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  Vcvv 3340  cop 4327  cfv 6049  (class class class)co 6813  ndxcnx 16056   sSet csts 16057  Scalarcsca 16146   ·𝑠 cvsca 16147  .gcmg 17741  ringzring 20020  ℤModczlm 20051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-zlm 20055
This theorem is referenced by:  zlmlem  20067  zlmsca  20071  zlmvsca  20072  zlmds  30317  zlmtset  30318
  Copyright terms: Public domain W3C validator