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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngonegcl | Structured version Visualization version GIF version |
Description: A ring is closed under negation. (Contributed by Jeff Madsen, 10-Jun-2010.) |
Ref | Expression |
---|---|
ringnegcl.1 | ⊢ 𝐺 = (1st ‘𝑅) |
ringnegcl.2 | ⊢ 𝑋 = ran 𝐺 |
ringnegcl.3 | ⊢ 𝑁 = (inv‘𝐺) |
Ref | Expression |
---|---|
rngonegcl | ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringnegcl.1 | . . 3 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | 1 | rngogrpo 35182 | . 2 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp) |
3 | ringnegcl.2 | . . 3 ⊢ 𝑋 = ran 𝐺 | |
4 | ringnegcl.3 | . . 3 ⊢ 𝑁 = (inv‘𝐺) | |
5 | 3, 4 | grpoinvcl 28295 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋) |
6 | 2, 5 | sylan 582 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ran crn 5550 ‘cfv 6349 1st c1st 7681 GrpOpcgr 28260 invcgn 28262 RingOpscrngo 35166 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-1st 7683 df-2nd 7684 df-grpo 28264 df-gid 28265 df-ginv 28266 df-ablo 28316 df-rngo 35167 |
This theorem is referenced by: rngonegmn1l 35213 rngonegmn1r 35214 rngoneglmul 35215 rngonegrmul 35216 rngosubdi 35217 rngosubdir 35218 idlnegcl 35294 |
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