Theorem zlmval 21452

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 3457	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
3		oveq1 7353	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) = (𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩))
4		fveq2 6822	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = (.g‘𝐺))
5		zlmval.m	. . . . . . 7 ⊢ · = (.g‘𝐺)
6	4, 5	eqtr4di 2784	. . . . . 6 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = · )
7	6	opeq2d 4829	. . . . 5 ⊢ (𝑔 = 𝐺 → ⟨( ·𝑠 ‘ndx), (.g‘𝑔)⟩ = ⟨( ·𝑠 ‘ndx), · ⟩)
8	3, 7	oveq12d 7364	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g‘𝑔)⟩) = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩))
9		df-zlm 21441	. . . 4 ⊢ ℤMod = (𝑔 ∈ V ↦ ((𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g‘𝑔)⟩))
10		ovex 7379	. . . 4 ⊢ ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩) ∈ V
11	8, 9, 10	fvmpt 6929	. . 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 2778	1 ⊢ (𝐺 ∈ 𝑉 → 𝑊 = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  Vcvv 3436  ⟨cop 4579  ‘cfv 6481  (class class class)co 7346   sSet csts 17074  ndxcnx 17104  Scalarcsca 17164   ·_𝑠 cvsca 17165  ._gcmg 18980  ℤ_ringczring 21383  ℤModczlm 21437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6437  df-fun 6483  df-fv 6489  df-ov 7349  df-zlm 21441

This theorem is referenced by:  zlmlem  21453  zlmsca  21457  zlmvsca  21458  zlmds  33975  zlmtset  33976