Theorem zlmval 21545

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 3474	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
3		oveq1 7397	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) = (𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩))
4		fveq2 6861	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = (.g‘𝐺))
5		zlmval.m	. . . . . . 7 ⊢ · = (.g‘𝐺)
6	4, 5	eqtr4di 2814	. . . . . 6 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = · )
7	6	opeq2d 4837	. . . . 5 ⊢ (𝑔 = 𝐺 → ⟨( ·𝑠 ‘ndx), (.g‘𝑔)⟩ = ⟨( ·𝑠 ‘ndx), · ⟩)
8	3, 7	oveq12d 7408	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g‘𝑔)⟩) = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩))
9		df-zlm 21534	. . . 4 ⊢ ℤMod = (𝑔 ∈ V ↦ ((𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g‘𝑔)⟩))
10		ovex 7423	. . . 4 ⊢ ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩) ∈ V
11	8, 9, 10	fvmpt 6969	. . 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 2808	1 ⊢ (𝐺 ∈ 𝑉 → 𝑊 = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩))

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  Vcvv 3453  ⟨cop 4587  ‘cfv 6515  (class class class)co 7390   sSet csts 17180  ndxcnx 17210  Scalarcsca 17270   ·_𝑠 cvsca 17271  ._gcmg 19090  ℤ_ringczring 21476  ℤModczlm 21530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-iota 6471  df-fun 6517  df-fv 6523  df-ov 7393  df-zlm 21534

This theorem is referenced by:  zlmlem  21546  zlmsca  21550  zlmvsca  21551  zlmds  34218  zlmtset  34219