Theorem zlmval 19683

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 3185	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
3		oveq1 6556	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) = (𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩))
4		fveq2 6103	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = (.g‘𝐺))
5		zlmval.m	. . . . . . 7 ⊢ · = (.g‘𝐺)
6	4, 5	syl6eqr 2662	. . . . . 6 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = · )
7	6	opeq2d 4347	. . . . 5 ⊢ (𝑔 = 𝐺 → ⟨( ·𝑠 ‘ndx), (.g‘𝑔)⟩ = ⟨( ·𝑠 ‘ndx), · ⟩)
8	3, 7	oveq12d 6567	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g‘𝑔)⟩) = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩))
9		df-zlm 19672	. . . 4 ⊢ ℤMod = (𝑔 ∈ V ↦ ((𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g‘𝑔)⟩))
10		ovex 6577	. . . 4 ⊢ ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩) ∈ V
11	8, 9, 10	fvmpt 6191	. . 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	syl5eq 2656	1 ⊢ (𝐺 ∈ 𝑉 → 𝑊 = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩))

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131  ‘cfv 5804  (class class class)co 6549  ndxcnx 15692   sSet csts 15693  Scalarcsca 15771   ·_𝑠 cvsca 15772  ._gcmg 17363  ℤ_ringzring 19637  ℤModczlm 19668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-zlm 19672

This theorem is referenced by:  zlmlem  19684  zlmsca  19688  zlmvsca  19689  zlmds  29336  zlmtset  29337