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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 5031 | . . 3 ⊢ (𝐺RingOps𝐻 ↔ 〈𝐺, 𝐻〉 ∈ RingOps) | |
2 | relrngo 35334 | . . . . 5 ⊢ Rel RingOps | |
3 | 2 | brrelex12i 5571 | . . . 4 ⊢ (𝐺RingOps𝐻 → (𝐺 ∈ V ∧ 𝐻 ∈ V)) |
4 | op1stg 7683 | . . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ V) → (1st ‘〈𝐺, 𝐻〉) = 𝐺) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝐺RingOps𝐻 → (1st ‘〈𝐺, 𝐻〉) = 𝐺) |
6 | 1, 5 | sylbir 238 | . 2 ⊢ (〈𝐺, 𝐻〉 ∈ RingOps → (1st ‘〈𝐺, 𝐻〉) = 𝐺) |
7 | eqid 2798 | . . 3 ⊢ (1st ‘〈𝐺, 𝐻〉) = (1st ‘〈𝐺, 𝐻〉) | |
8 | 7 | rngoablo 35346 | . 2 ⊢ (〈𝐺, 𝐻〉 ∈ RingOps → (1st ‘〈𝐺, 𝐻〉) ∈ AbelOp) |
9 | 6, 8 | eqeltrrd 2891 | 1 ⊢ (〈𝐺, 𝐻〉 ∈ RingOps → 𝐺 ∈ AbelOp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 〈cop 4531 class class class wbr 5030 ‘cfv 6324 1st c1st 7669 AbelOpcablo 28327 RingOpscrngo 35332 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-1st 7671 df-2nd 7672 df-rngo 35333 |
This theorem is referenced by: isdivrngo 35388 |
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