![]() |
Mathbox for Jeff Madsen |
< Previous
Next >
Nearby theorems |
|
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 36872 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ 𝑋) |
4 | 3 | adantr 481 | . . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈 ∈ 𝐼) → 𝐼 ⊆ 𝑋) |
5 | 1idl.2 | . . . . . . . . 9 ⊢ 𝐻 = (2nd ‘𝑅) | |
6 | 1 | rneqi 5934 | . . . . . . . . . 10 ⊢ ran 𝐺 = ran (1st ‘𝑅) |
7 | 2, 6 | eqtri 2760 | . . . . . . . . 9 ⊢ 𝑋 = ran (1st ‘𝑅) |
8 | 1idl.4 | . . . . . . . . 9 ⊢ 𝑈 = (GId‘𝐻) | |
9 | 5, 7, 8 | rngolidm 36793 | . . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → (𝑈𝐻𝑥) = 𝑥) |
10 | 9 | ad2ant2rl 747 | . . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)) → (𝑈𝐻𝑥) = 𝑥) |
11 | 1, 5, 2 | idlrmulcl 36877 | . . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)) → (𝑈𝐻𝑥) ∈ 𝐼) |
12 | 10, 11 | eqeltrrd 2834 | . . . . . 6 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)) → 𝑥 ∈ 𝐼) |
13 | 12 | expr 457 | . . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈 ∈ 𝐼) → (𝑥 ∈ 𝑋 → 𝑥 ∈ 𝐼)) |
14 | 13 | ssrdv 3987 | . . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈 ∈ 𝐼) → 𝑋 ⊆ 𝐼) |
15 | 4, 14 | eqssd 3998 | . . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈 ∈ 𝐼) → 𝐼 = 𝑋) |
16 | 15 | ex 413 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑈 ∈ 𝐼 → 𝐼 = 𝑋)) |
17 | 7, 5, 8 | rngo1cl 36795 | . . . 4 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋) |
18 | 17 | adantr 481 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝑈 ∈ 𝑋) |
19 | eleq2 2822 | . . 3 ⊢ (𝐼 = 𝑋 → (𝑈 ∈ 𝐼 ↔ 𝑈 ∈ 𝑋)) | |
20 | 18, 19 | syl5ibrcom 246 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝐼 = 𝑋 → 𝑈 ∈ 𝐼)) |
21 | 16, 20 | impbid 211 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑈 ∈ 𝐼 ↔ 𝐼 = 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ⊆ wss 3947 ran crn 5676 ‘cfv 6540 (class class class)co 7405 1st c1st 7969 2nd c2nd 7970 GIdcgi 29730 RingOpscrngo 36750 Idlcidl 36863 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-fo 6546 df-fv 6548 df-riota 7361 df-ov 7408 df-1st 7971 df-2nd 7972 df-grpo 29733 df-gid 29734 df-ablo 29785 df-ass 36699 df-exid 36701 df-mgmOLD 36705 df-sgrOLD 36717 df-mndo 36723 df-rngo 36751 df-idl 36866 |
This theorem is referenced by: 0rngo 36883 divrngidl 36884 maxidln1 36900 |
Copyright terms: Public domain | W3C validator |