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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 37917 | . 2 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp) | 
| 3 | ringnegcl.2 | . . 3 ⊢ 𝑋 = ran 𝐺 | |
| 4 | ringnegcl.3 | . . 3 ⊢ 𝑁 = (inv‘𝐺) | |
| 5 | 3, 4 | grpoinvcl 30543 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋) | 
| 6 | 2, 5 | sylan 580 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ran crn 5686 ‘cfv 6561 1st c1st 8012 GrpOpcgr 30508 invcgn 30510 RingOpscrngo 37901 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-1st 8014 df-2nd 8015 df-grpo 30512 df-gid 30513 df-ginv 30514 df-ablo 30564 df-rngo 37902 | 
| This theorem is referenced by: rngonegmn1l 37948 rngonegmn1r 37949 rngoneglmul 37950 rngonegrmul 37951 rngosubdi 37952 rngosubdir 37953 idlnegcl 38029 | 
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