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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 35203 | . 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 28303 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)𝐺𝐴) = 𝑍) |
7 | 2, 6 | sylan 582 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)𝐺𝐴) = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ran crn 5556 ‘cfv 6355 (class class class)co 7156 1st c1st 7687 GrpOpcgr 28266 GIdcgi 28267 invcgn 28268 RingOpscrngo 35187 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-1st 7689 df-2nd 7690 df-grpo 28270 df-gid 28271 df-ginv 28272 df-ablo 28322 df-rngo 35188 |
This theorem is referenced by: rngonegmn1r 35235 |
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