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

Theorem zlmval 21065
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 3493 . . 3 (𝐺𝑉𝐺 ∈ V)
3 oveq1 7416 . . . . 5 (𝑔 = 𝐺 → (𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) = (𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩))
4 fveq2 6892 . . . . . . 7 (𝑔 = 𝐺 → (.g𝑔) = (.g𝐺))
5 zlmval.m . . . . . . 7 · = (.g𝐺)
64, 5eqtr4di 2791 . . . . . 6 (𝑔 = 𝐺 → (.g𝑔) = · )
76opeq2d 4881 . . . . 5 (𝑔 = 𝐺 → ⟨( ·𝑠 ‘ndx), (.g𝑔)⟩ = ⟨( ·𝑠 ‘ndx), · ⟩)
83, 7oveq12d 7427 . . . 4 (𝑔 = 𝐺 → ((𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g𝑔)⟩) = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩))
9 df-zlm 21054 . . . 4 ℤMod = (𝑔 ∈ V ↦ ((𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g𝑔)⟩))
10 ovex 7442 . . . 4 ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩) ∈ V
118, 9, 10fvmpt 6999 . . 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 2785 1 (𝐺𝑉𝑊 = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  Vcvv 3475  cop 4635  cfv 6544  (class class class)co 7409   sSet csts 17096  ndxcnx 17126  Scalarcsca 17200   ·𝑠 cvsca 17201  .gcmg 18950  ringczring 21017  ℤModczlm 21050
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-zlm 21054
This theorem is referenced by:  zlmlem  21066  zlmlemOLD  21067  zlmsca  21074  zlmvsca  21075  zlmds  32973  zlmdsOLD  32974  zlmtset  32975  zlmtsetOLD  32976
  Copyright terms: Public domain W3C validator