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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rngoablo2 | Structured version Visualization version GIF version | ||
| Description: In a unital ring the addition is an abelian group. (Contributed by FL, 31-Aug-2009.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| rngoablo2 | ⊢ (〈𝐺, 𝐻〉 ∈ RingOps → 𝐺 ∈ AbelOp) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-br 5143 | . . 3 ⊢ (𝐺RingOps𝐻 ↔ 〈𝐺, 𝐻〉 ∈ RingOps) | |
| 2 | relrngo 37904 | . . . . 5 ⊢ Rel RingOps | |
| 3 | 2 | brrelex12i 5739 | . . . 4 ⊢ (𝐺RingOps𝐻 → (𝐺 ∈ V ∧ 𝐻 ∈ V)) | 
| 4 | op1stg 8027 | . . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ V) → (1st ‘〈𝐺, 𝐻〉) = 𝐺) | |
| 5 | 3, 4 | syl 17 | . . 3 ⊢ (𝐺RingOps𝐻 → (1st ‘〈𝐺, 𝐻〉) = 𝐺) | 
| 6 | 1, 5 | sylbir 235 | . 2 ⊢ (〈𝐺, 𝐻〉 ∈ RingOps → (1st ‘〈𝐺, 𝐻〉) = 𝐺) | 
| 7 | eqid 2736 | . . 3 ⊢ (1st ‘〈𝐺, 𝐻〉) = (1st ‘〈𝐺, 𝐻〉) | |
| 8 | 7 | rngoablo 37916 | . 2 ⊢ (〈𝐺, 𝐻〉 ∈ RingOps → (1st ‘〈𝐺, 𝐻〉) ∈ AbelOp) | 
| 9 | 6, 8 | eqeltrrd 2841 | 1 ⊢ (〈𝐺, 𝐻〉 ∈ RingOps → 𝐺 ∈ AbelOp) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 Vcvv 3479 〈cop 4631 class class class wbr 5142 ‘cfv 6560 1st c1st 8013 AbelOpcablo 30564 RingOpscrngo 37902 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-ov 7435 df-1st 8015 df-2nd 8016 df-rngo 37903 | 
| This theorem is referenced by: isdivrngo 37958 | 
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