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Mathbox for Jeff Madsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngoaddneg2 | Structured version Visualization version GIF version |
Description: Adding the negative in a ring gives zero. (Contributed by Jeff Madsen, 10-Jun-2010.) |
Ref | Expression |
---|---|
ringnegcl.1 | ⊢ 𝐺 = (1st ‘𝑅) |
ringnegcl.2 | ⊢ 𝑋 = ran 𝐺 |
ringnegcl.3 | ⊢ 𝑁 = (inv‘𝐺) |
ringaddneg.4 | ⊢ 𝑍 = (GId‘𝐺) |
Ref | Expression |
---|---|
rngoaddneg2 | ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)𝐺𝐴) = 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringnegcl.1 | . . 3 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | 1 | rngogrpo 37559 | . 2 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp) |
3 | ringnegcl.2 | . . 3 ⊢ 𝑋 = ran 𝐺 | |
4 | ringaddneg.4 | . . 3 ⊢ 𝑍 = (GId‘𝐺) | |
5 | ringnegcl.3 | . . 3 ⊢ 𝑁 = (inv‘𝐺) | |
6 | 3, 4, 5 | grpolinv 30451 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)𝐺𝐴) = 𝑍) |
7 | 2, 6 | sylan 578 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)𝐺𝐴) = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ran crn 5682 ‘cfv 6553 (class class class)co 7423 1st c1st 8000 GrpOpcgr 30414 GIdcgi 30415 invcgn 30416 RingOpscrngo 37543 |
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 2166 ax-ext 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pr 5432 ax-un 7745 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4325 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5579 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-1st 8002 df-2nd 8003 df-grpo 30418 df-gid 30419 df-ginv 30420 df-ablo 30470 df-rngo 37544 |
This theorem is referenced by: rngonegmn1r 37591 |
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