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 35175 | . . . . 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 5800 | . . . . . . . . . 10 ⊢ ran 𝐺 = ran (1st ‘𝑅) |
7 | 2, 6 | eqtri 2841 | . . . . . . . . 9 ⊢ 𝑋 = ran (1st ‘𝑅) |
8 | 1idl.4 | . . . . . . . . 9 ⊢ 𝑈 = (GId‘𝐻) | |
9 | 5, 7, 8 | rngolidm 35096 | . . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → (𝑈𝐻𝑥) = 𝑥) |
10 | 9 | ad2ant2rl 745 | . . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)) → (𝑈𝐻𝑥) = 𝑥) |
11 | 1, 5, 2 | idlrmulcl 35180 | . . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)) → (𝑈𝐻𝑥) ∈ 𝐼) |
12 | 10, 11 | eqeltrrd 2911 | . . . . . 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 3970 | . . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈 ∈ 𝐼) → 𝑋 ⊆ 𝐼) |
15 | 4, 14 | eqssd 3981 | . . 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 35098 | . . . 4 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋) |
18 | 17 | adantr 481 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝑈 ∈ 𝑋) |
19 | eleq2 2898 | . . 3 ⊢ (𝐼 = 𝑋 → (𝑈 ∈ 𝐼 ↔ 𝑈 ∈ 𝑋)) | |
20 | 18, 19 | syl5ibrcom 248 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝐼 = 𝑋 → 𝑈 ∈ 𝐼)) |
21 | 16, 20 | impbid 213 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑈 ∈ 𝐼 ↔ 𝐼 = 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 ran crn 5549 ‘cfv 6348 (class class class)co 7145 1st c1st 7676 2nd c2nd 7677 GIdcgi 28194 RingOpscrngo 35053 Idlcidl 35166 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fo 6354 df-fv 6356 df-riota 7103 df-ov 7148 df-1st 7678 df-2nd 7679 df-grpo 28197 df-gid 28198 df-ablo 28249 df-ass 35002 df-exid 35004 df-mgmOLD 35008 df-sgrOLD 35020 df-mndo 35026 df-rngo 35054 df-idl 35169 |
This theorem is referenced by: 0rngo 35186 divrngidl 35187 maxidln1 35203 |
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