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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 5149 | . . 3 ⊢ (𝐺RingOps𝐻 ↔ 〈𝐺, 𝐻〉 ∈ RingOps) | |
2 | relrngo 37080 | . . . . 5 ⊢ Rel RingOps | |
3 | 2 | brrelex12i 5731 | . . . 4 ⊢ (𝐺RingOps𝐻 → (𝐺 ∈ V ∧ 𝐻 ∈ V)) |
4 | op1stg 7991 | . . . 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 2731 | . . 3 ⊢ (1st ‘〈𝐺, 𝐻〉) = (1st ‘〈𝐺, 𝐻〉) | |
8 | 7 | rngoablo 37092 | . 2 ⊢ (〈𝐺, 𝐻〉 ∈ RingOps → (1st ‘〈𝐺, 𝐻〉) ∈ AbelOp) |
9 | 6, 8 | eqeltrrd 2833 | 1 ⊢ (〈𝐺, 𝐻〉 ∈ RingOps → 𝐺 ∈ AbelOp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 Vcvv 3473 〈cop 4634 class class class wbr 5148 ‘cfv 6543 1st c1st 7977 AbelOpcablo 30079 RingOpscrngo 37078 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 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-rn 5687 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7415 df-1st 7979 df-2nd 7980 df-rngo 37079 |
This theorem is referenced by: isdivrngo 37134 |
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