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Mirrors > Home > MPE Home > Th. List > zlmval | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
zlmval.w | ⊢ 𝑊 = (ℤMod‘𝐺) |
zlmval.m | ⊢ · = (.g‘𝐺) |
Ref | Expression |
---|---|
zlmval | ⊢ (𝐺 ∈ 𝑉 → 𝑊 = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zlmval.w | . 2 ⊢ 𝑊 = (ℤMod‘𝐺) | |
2 | elex 3492 | . . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V) | |
3 | oveq1 7418 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) = (𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩)) | |
4 | fveq2 6891 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = (.g‘𝐺)) | |
5 | zlmval.m | . . . . . . 7 ⊢ · = (.g‘𝐺) | |
6 | 4, 5 | eqtr4di 2790 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = · ) |
7 | 6 | opeq2d 4880 | . . . . 5 ⊢ (𝑔 = 𝐺 → ⟨( ·𝑠 ‘ndx), (.g‘𝑔)⟩ = ⟨( ·𝑠 ‘ndx), · ⟩) |
8 | 3, 7 | oveq12d 7429 | . . . 4 ⊢ (𝑔 = 𝐺 → ((𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g‘𝑔)⟩) = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩)) |
9 | df-zlm 21273 | . . . 4 ⊢ ℤMod = (𝑔 ∈ V ↦ ((𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g‘𝑔)⟩)) | |
10 | ovex 7444 | . . . 4 ⊢ ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩) ∈ V | |
11 | 8, 9, 10 | fvmpt 6998 | . . 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 2784 | 1 ⊢ (𝐺 ∈ 𝑉 → 𝑊 = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ⟨cop 4634 ‘cfv 6543 (class class class)co 7411 sSet csts 17100 ndxcnx 17130 Scalarcsca 17204 ·𝑠 cvsca 17205 .gcmg 18986 ℤringczring 21217 ℤModczlm 21269 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7414 df-zlm 21273 |
This theorem is referenced by: zlmlem 21285 zlmlemOLD 21286 zlmsca 21293 zlmvsca 21294 zlmds 33228 zlmdsOLD 33229 zlmtset 33230 zlmtsetOLD 33231 |
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