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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 2823 | . . . 4 ⊢ ran 𝐺 = ran 𝐺 | |
3 | 1, 2 | idlss 35296 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ ran 𝐺) |
4 | ssel2 3964 | . . . . 5 ⊢ ((𝐼 ⊆ ran 𝐺 ∧ 𝐴 ∈ 𝐼) → 𝐴 ∈ ran 𝐺) | |
5 | eqid 2823 | . . . . . 6 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅) | |
6 | idlnegcl.2 | . . . . . 6 ⊢ 𝑁 = (inv‘𝐺) | |
7 | eqid 2823 | . . . . . 6 ⊢ (GId‘(2nd ‘𝑅)) = (GId‘(2nd ‘𝑅)) | |
8 | 1, 5, 2, 6, 7 | rngonegmn1l 35221 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ ran 𝐺) → (𝑁‘𝐴) = ((𝑁‘(GId‘(2nd ‘𝑅)))(2nd ‘𝑅)𝐴)) |
9 | 4, 8 | sylan2 594 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐼 ⊆ ran 𝐺 ∧ 𝐴 ∈ 𝐼)) → (𝑁‘𝐴) = ((𝑁‘(GId‘(2nd ‘𝑅)))(2nd ‘𝑅)𝐴)) |
10 | 9 | anassrs 470 | . . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ⊆ ran 𝐺) ∧ 𝐴 ∈ 𝐼) → (𝑁‘𝐴) = ((𝑁‘(GId‘(2nd ‘𝑅)))(2nd ‘𝑅)𝐴)) |
11 | 3, 10 | syldanl 603 | . 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → (𝑁‘𝐴) = ((𝑁‘(GId‘(2nd ‘𝑅)))(2nd ‘𝑅)𝐴)) |
12 | 1 | rneqi 5809 | . . . . . 6 ⊢ ran 𝐺 = ran (1st ‘𝑅) |
13 | 12, 5, 7 | rngo1cl 35219 | . . . . 5 ⊢ (𝑅 ∈ RingOps → (GId‘(2nd ‘𝑅)) ∈ ran 𝐺) |
14 | 1, 2, 6 | rngonegcl 35207 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (GId‘(2nd ‘𝑅)) ∈ ran 𝐺) → (𝑁‘(GId‘(2nd ‘𝑅))) ∈ ran 𝐺) |
15 | 13, 14 | mpdan 685 | . . . 4 ⊢ (𝑅 ∈ RingOps → (𝑁‘(GId‘(2nd ‘𝑅))) ∈ ran 𝐺) |
16 | 15 | ad2antrr 724 | . . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → (𝑁‘(GId‘(2nd ‘𝑅))) ∈ ran 𝐺) |
17 | 1, 5, 2 | idllmulcl 35300 | . . . 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 470 | . . 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 685 | . 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → ((𝑁‘(GId‘(2nd ‘𝑅)))(2nd ‘𝑅)𝐴) ∈ 𝐼) |
20 | 11, 19 | eqeltrd 2915 | 1 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → (𝑁‘𝐴) ∈ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 ran crn 5558 ‘cfv 6357 (class class class)co 7158 1st c1st 7689 2nd c2nd 7690 GIdcgi 28269 invcgn 28270 RingOpscrngo 35174 Idlcidl 35287 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-1st 7691 df-2nd 7692 df-grpo 28272 df-gid 28273 df-ginv 28274 df-ablo 28324 df-ass 35123 df-exid 35125 df-mgmOLD 35129 df-sgrOLD 35141 df-mndo 35147 df-rngo 35175 df-idl 35290 |
This theorem is referenced by: idlsubcl 35303 |
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