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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 5142 | . . 3 ⊢ (𝐺RingOps𝐻 ↔ ⟨𝐺, 𝐻⟩ ∈ RingOps) | |
2 | relrngo 37277 | . . . . 5 ⊢ Rel RingOps | |
3 | 2 | brrelex12i 5724 | . . . 4 ⊢ (𝐺RingOps𝐻 → (𝐺 ∈ V ∧ 𝐻 ∈ V)) |
4 | op1stg 7986 | . . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ V) → (1st ‘⟨𝐺, 𝐻⟩) = 𝐺) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝐺RingOps𝐻 → (1st ‘⟨𝐺, 𝐻⟩) = 𝐺) |
6 | 1, 5 | sylbir 234 | . 2 ⊢ (⟨𝐺, 𝐻⟩ ∈ RingOps → (1st ‘⟨𝐺, 𝐻⟩) = 𝐺) |
7 | eqid 2726 | . . 3 ⊢ (1st ‘⟨𝐺, 𝐻⟩) = (1st ‘⟨𝐺, 𝐻⟩) | |
8 | 7 | rngoablo 37289 | . 2 ⊢ (⟨𝐺, 𝐻⟩ ∈ RingOps → (1st ‘⟨𝐺, 𝐻⟩) ∈ AbelOp) |
9 | 6, 8 | eqeltrrd 2828 | 1 ⊢ (⟨𝐺, 𝐻⟩ ∈ RingOps → 𝐺 ∈ AbelOp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ⟨cop 4629 class class class wbr 5141 ‘cfv 6537 1st c1st 7972 AbelOpcablo 30306 RingOpscrngo 37275 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fv 6545 df-ov 7408 df-1st 7974 df-2nd 7975 df-rngo 37276 |
This theorem is referenced by: isdivrngo 37331 |
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