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| Mirrors > Home > MPE Home > Th. List > Mathboxes > idlnegcl | Structured version Visualization version GIF version | ||
| Description: An ideal is closed under negation. (Contributed by Jeff Madsen, 10-Jun-2010.) |
| Ref | Expression |
|---|---|
| idlnegcl.1 | ⊢ 𝐺 = (1st ‘𝑅) |
| idlnegcl.2 | ⊢ 𝑁 = (inv‘𝐺) |
| Ref | Expression |
|---|---|
| idlnegcl | ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → (𝑁‘𝐴) ∈ 𝐼) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | idlnegcl.1 | . . . 4 ⊢ 𝐺 = (1st ‘𝑅) | |
| 2 | eqid 2730 | . . . 4 ⊢ ran 𝐺 = ran 𝐺 | |
| 3 | 1, 2 | idlss 38035 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ ran 𝐺) |
| 4 | ssel2 3927 | . . . . 5 ⊢ ((𝐼 ⊆ ran 𝐺 ∧ 𝐴 ∈ 𝐼) → 𝐴 ∈ ran 𝐺) | |
| 5 | eqid 2730 | . . . . . 6 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅) | |
| 6 | idlnegcl.2 | . . . . . 6 ⊢ 𝑁 = (inv‘𝐺) | |
| 7 | eqid 2730 | . . . . . 6 ⊢ (GId‘(2nd ‘𝑅)) = (GId‘(2nd ‘𝑅)) | |
| 8 | 1, 5, 2, 6, 7 | rngonegmn1l 37960 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ ran 𝐺) → (𝑁‘𝐴) = ((𝑁‘(GId‘(2nd ‘𝑅)))(2nd ‘𝑅)𝐴)) |
| 9 | 4, 8 | sylan2 593 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐼 ⊆ ran 𝐺 ∧ 𝐴 ∈ 𝐼)) → (𝑁‘𝐴) = ((𝑁‘(GId‘(2nd ‘𝑅)))(2nd ‘𝑅)𝐴)) |
| 10 | 9 | anassrs 467 | . . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ⊆ ran 𝐺) ∧ 𝐴 ∈ 𝐼) → (𝑁‘𝐴) = ((𝑁‘(GId‘(2nd ‘𝑅)))(2nd ‘𝑅)𝐴)) |
| 11 | 3, 10 | syldanl 602 | . 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → (𝑁‘𝐴) = ((𝑁‘(GId‘(2nd ‘𝑅)))(2nd ‘𝑅)𝐴)) |
| 12 | 1 | rneqi 5874 | . . . . . 6 ⊢ ran 𝐺 = ran (1st ‘𝑅) |
| 13 | 12, 5, 7 | rngo1cl 37958 | . . . . 5 ⊢ (𝑅 ∈ RingOps → (GId‘(2nd ‘𝑅)) ∈ ran 𝐺) |
| 14 | 1, 2, 6 | rngonegcl 37946 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (GId‘(2nd ‘𝑅)) ∈ ran 𝐺) → (𝑁‘(GId‘(2nd ‘𝑅))) ∈ ran 𝐺) |
| 15 | 13, 14 | mpdan 687 | . . . 4 ⊢ (𝑅 ∈ RingOps → (𝑁‘(GId‘(2nd ‘𝑅))) ∈ ran 𝐺) |
| 16 | 15 | ad2antrr 726 | . . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → (𝑁‘(GId‘(2nd ‘𝑅))) ∈ ran 𝐺) |
| 17 | 1, 5, 2 | idllmulcl 38039 | . . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ (𝑁‘(GId‘(2nd ‘𝑅))) ∈ ran 𝐺)) → ((𝑁‘(GId‘(2nd ‘𝑅)))(2nd ‘𝑅)𝐴) ∈ 𝐼) |
| 18 | 17 | anassrs 467 | . . 3 ⊢ ((((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) ∧ (𝑁‘(GId‘(2nd ‘𝑅))) ∈ ran 𝐺) → ((𝑁‘(GId‘(2nd ‘𝑅)))(2nd ‘𝑅)𝐴) ∈ 𝐼) |
| 19 | 16, 18 | mpdan 687 | . 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → ((𝑁‘(GId‘(2nd ‘𝑅)))(2nd ‘𝑅)𝐴) ∈ 𝐼) |
| 20 | 11, 19 | eqeltrd 2829 | 1 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → (𝑁‘𝐴) ∈ 𝐼) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2110 ⊆ wss 3900 ran crn 5615 ‘cfv 6477 (class class class)co 7341 1st c1st 7914 2nd c2nd 7915 GIdcgi 30460 invcgn 30461 RingOpscrngo 37913 Idlcidl 38026 |
| 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 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rmo 3344 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4858 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-riota 7298 df-ov 7344 df-1st 7916 df-2nd 7917 df-grpo 30463 df-gid 30464 df-ginv 30465 df-ablo 30515 df-ass 37862 df-exid 37864 df-mgmOLD 37868 df-sgrOLD 37880 df-mndo 37886 df-rngo 37914 df-idl 38029 |
| This theorem is referenced by: idlsubcl 38042 |
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