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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 36478 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ 𝑋) |
4 | 3 | adantr 482 | . . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈 ∈ 𝐼) → 𝐼 ⊆ 𝑋) |
5 | 1idl.2 | . . . . . . . . 9 ⊢ 𝐻 = (2nd ‘𝑅) | |
6 | 1 | rneqi 5893 | . . . . . . . . . 10 ⊢ ran 𝐺 = ran (1st ‘𝑅) |
7 | 2, 6 | eqtri 2765 | . . . . . . . . 9 ⊢ 𝑋 = ran (1st ‘𝑅) |
8 | 1idl.4 | . . . . . . . . 9 ⊢ 𝑈 = (GId‘𝐻) | |
9 | 5, 7, 8 | rngolidm 36399 | . . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → (𝑈𝐻𝑥) = 𝑥) |
10 | 9 | ad2ant2rl 748 | . . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)) → (𝑈𝐻𝑥) = 𝑥) |
11 | 1, 5, 2 | idlrmulcl 36483 | . . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)) → (𝑈𝐻𝑥) ∈ 𝐼) |
12 | 10, 11 | eqeltrrd 2839 | . . . . . 6 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)) → 𝑥 ∈ 𝐼) |
13 | 12 | expr 458 | . . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈 ∈ 𝐼) → (𝑥 ∈ 𝑋 → 𝑥 ∈ 𝐼)) |
14 | 13 | ssrdv 3951 | . . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈 ∈ 𝐼) → 𝑋 ⊆ 𝐼) |
15 | 4, 14 | eqssd 3962 | . . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈 ∈ 𝐼) → 𝐼 = 𝑋) |
16 | 15 | ex 414 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑈 ∈ 𝐼 → 𝐼 = 𝑋)) |
17 | 7, 5, 8 | rngo1cl 36401 | . . . 4 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋) |
18 | 17 | adantr 482 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝑈 ∈ 𝑋) |
19 | eleq2 2827 | . . 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 211 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑈 ∈ 𝐼 ↔ 𝐼 = 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⊆ wss 3911 ran crn 5635 ‘cfv 6497 (class class class)co 7358 1st c1st 7920 2nd c2nd 7921 GIdcgi 29435 RingOpscrngo 36356 Idlcidl 36469 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fo 6503 df-fv 6505 df-riota 7314 df-ov 7361 df-1st 7922 df-2nd 7923 df-grpo 29438 df-gid 29439 df-ablo 29490 df-ass 36305 df-exid 36307 df-mgmOLD 36311 df-sgrOLD 36323 df-mndo 36329 df-rngo 36357 df-idl 36472 |
This theorem is referenced by: 0rngo 36489 divrngidl 36490 maxidln1 36506 |
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