Theorem zlmval 20663

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 3512	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
3		oveq1 7163	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) = (𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩))
4		fveq2 6670	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = (.g‘𝐺))
5		zlmval.m	. . . . . . 7 ⊢ · = (.g‘𝐺)
6	4, 5	syl6eqr 2874	. . . . . 6 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = · )
7	6	opeq2d 4810	. . . . 5 ⊢ (𝑔 = 𝐺 → ⟨( ·𝑠 ‘ndx), (.g‘𝑔)⟩ = ⟨( ·𝑠 ‘ndx), · ⟩)
8	3, 7	oveq12d 7174	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g‘𝑔)⟩) = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩))
9		df-zlm 20652	. . . 4 ⊢ ℤMod = (𝑔 ∈ V ↦ ((𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g‘𝑔)⟩))
10		ovex 7189	. . . 4 ⊢ ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩) ∈ V
11	8, 9, 10	fvmpt 6768	. . 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 2868	1 ⊢ (𝐺 ∈ 𝑉 → 𝑊 = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  Vcvv 3494  ⟨cop 4573  ‘cfv 6355  (class class class)co 7156  ndxcnx 16480   sSet csts 16481  Scalarcsca 16568   ·_𝑠 cvsca 16569  ._gcmg 18224  ℤ_ringzring 20617  ℤModczlm 20648

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-iota 6314  df-fun 6357  df-fv 6363  df-ov 7159  df-zlm 20652

This theorem is referenced by:  zlmlem  20664  zlmsca  20668  zlmvsca  20669  zlmds  31205  zlmtset  31206