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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 3501 | . . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V) | |
| 3 | oveq1 7438 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑔 sSet 〈(Scalar‘ndx), ℤring〉) = (𝐺 sSet 〈(Scalar‘ndx), ℤring〉)) | |
| 4 | fveq2 6906 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = (.g‘𝐺)) | |
| 5 | zlmval.m | . . . . . . 7 ⊢ · = (.g‘𝐺) | |
| 6 | 4, 5 | eqtr4di 2795 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = · ) | 
| 7 | 6 | opeq2d 4880 | . . . . 5 ⊢ (𝑔 = 𝐺 → 〈( ·𝑠 ‘ndx), (.g‘𝑔)〉 = 〈( ·𝑠 ‘ndx), · 〉) | 
| 8 | 3, 7 | oveq12d 7449 | . . . 4 ⊢ (𝑔 = 𝐺 → ((𝑔 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝑔)〉) = ((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), · 〉)) | 
| 9 | df-zlm 21515 | . . . 4 ⊢ ℤMod = (𝑔 ∈ V ↦ ((𝑔 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝑔)〉)) | |
| 10 | ovex 7464 | . . . 4 ⊢ ((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), · 〉) ∈ V | |
| 11 | 8, 9, 10 | fvmpt 7016 | . . 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 2789 | 1 ⊢ (𝐺 ∈ 𝑉 → 𝑊 = ((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), · 〉)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 ‘cfv 6561 (class class class)co 7431 sSet csts 17200 ndxcnx 17230 Scalarcsca 17300 ·𝑠 cvsca 17301 .gcmg 19085 ℤringczring 21457 ℤModczlm 21511 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-zlm 21515 | 
| This theorem is referenced by: zlmlem 21527 zlmlemOLD 21528 zlmsca 21535 zlmvsca 21536 zlmds 33961 zlmdsOLD 33962 zlmtset 33963 zlmtsetOLD 33964 | 
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