Theorem zlmval 21065

Description: Augment an abelian group with vector space operations to turn it into a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.)

Hypotheses

Ref	Expression
zlmval.w	⊢ 𝑊 = (ℤMod‘𝐺)
zlmval.m	⊢ · = (.g‘𝐺)

Assertion

Ref	Expression
zlmval	⊢ (𝐺 ∈ 𝑉 → 𝑊 = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩))

Proof of Theorem zlmval

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		zlmval.w	. 2 ⊢ 𝑊 = (ℤMod‘𝐺)
2		elex 3493	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
3		oveq1 7416	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) = (𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩))
4		fveq2 6892	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = (.g‘𝐺))
5		zlmval.m	. . . . . . 7 ⊢ · = (.g‘𝐺)
6	4, 5	eqtr4di 2791	. . . . . 6 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = · )
7	6	opeq2d 4881	. . . . 5 ⊢ (𝑔 = 𝐺 → ⟨( ·𝑠 ‘ndx), (.g‘𝑔)⟩ = ⟨( ·𝑠 ‘ndx), · ⟩)
8	3, 7	oveq12d 7427	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g‘𝑔)⟩) = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩))
9		df-zlm 21054	. . . 4 ⊢ ℤMod = (𝑔 ∈ V ↦ ((𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g‘𝑔)⟩))
10		ovex 7442	. . . 4 ⊢ ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩) ∈ V
11	8, 9, 10	fvmpt 6999	. . 3 ⊢ (𝐺 ∈ V → (ℤMod‘𝐺) = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩))
12	2, 11	syl 17	. 2 ⊢ (𝐺 ∈ 𝑉 → (ℤMod‘𝐺) = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩))
13	1, 12	eqtrid 2785	1 ⊢ (𝐺 ∈ 𝑉 → 𝑊 = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩))

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  ℤ_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