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Mathbox for Jeff Madsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 1idl | Structured version Visualization version GIF version |
Description: Two ways of expressing the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) |
Ref | Expression |
---|---|
1idl.1 | ⊢ 𝐺 = (1st ‘𝑅) |
1idl.2 | ⊢ 𝐻 = (2nd ‘𝑅) |
1idl.3 | ⊢ 𝑋 = ran 𝐺 |
1idl.4 | ⊢ 𝑈 = (GId‘𝐻) |
Ref | Expression |
---|---|
1idl | ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑈 ∈ 𝐼 ↔ 𝐼 = 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1idl.1 | . . . . . 6 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | 1idl.3 | . . . . . 6 ⊢ 𝑋 = ran 𝐺 | |
3 | 1, 2 | idlss 38003 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ 𝑋) |
4 | 3 | adantr 480 | . . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈 ∈ 𝐼) → 𝐼 ⊆ 𝑋) |
5 | 1idl.2 | . . . . . . . . 9 ⊢ 𝐻 = (2nd ‘𝑅) | |
6 | 1 | rneqi 5951 | . . . . . . . . . 10 ⊢ ran 𝐺 = ran (1st ‘𝑅) |
7 | 2, 6 | eqtri 2763 | . . . . . . . . 9 ⊢ 𝑋 = ran (1st ‘𝑅) |
8 | 1idl.4 | . . . . . . . . 9 ⊢ 𝑈 = (GId‘𝐻) | |
9 | 5, 7, 8 | rngolidm 37924 | . . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → (𝑈𝐻𝑥) = 𝑥) |
10 | 9 | ad2ant2rl 749 | . . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)) → (𝑈𝐻𝑥) = 𝑥) |
11 | 1, 5, 2 | idlrmulcl 38008 | . . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)) → (𝑈𝐻𝑥) ∈ 𝐼) |
12 | 10, 11 | eqeltrrd 2840 | . . . . . 6 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)) → 𝑥 ∈ 𝐼) |
13 | 12 | expr 456 | . . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈 ∈ 𝐼) → (𝑥 ∈ 𝑋 → 𝑥 ∈ 𝐼)) |
14 | 13 | ssrdv 4001 | . . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈 ∈ 𝐼) → 𝑋 ⊆ 𝐼) |
15 | 4, 14 | eqssd 4013 | . . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈 ∈ 𝐼) → 𝐼 = 𝑋) |
16 | 15 | ex 412 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑈 ∈ 𝐼 → 𝐼 = 𝑋)) |
17 | 7, 5, 8 | rngo1cl 37926 | . . . 4 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋) |
18 | 17 | adantr 480 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝑈 ∈ 𝑋) |
19 | eleq2 2828 | . . 3 ⊢ (𝐼 = 𝑋 → (𝑈 ∈ 𝐼 ↔ 𝑈 ∈ 𝑋)) | |
20 | 18, 19 | syl5ibrcom 247 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝐼 = 𝑋 → 𝑈 ∈ 𝐼)) |
21 | 16, 20 | impbid 212 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑈 ∈ 𝐼 ↔ 𝐼 = 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 ran crn 5690 ‘cfv 6563 (class class class)co 7431 1st c1st 8011 2nd c2nd 8012 GIdcgi 30519 RingOpscrngo 37881 Idlcidl 37994 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fo 6569 df-fv 6571 df-riota 7388 df-ov 7434 df-1st 8013 df-2nd 8014 df-grpo 30522 df-gid 30523 df-ablo 30574 df-ass 37830 df-exid 37832 df-mgmOLD 37836 df-sgrOLD 37848 df-mndo 37854 df-rngo 37882 df-idl 37997 |
This theorem is referenced by: 0rngo 38014 divrngidl 38015 maxidln1 38031 |
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