| 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 38023 | . . . . 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 5948 | . . . . . . . . . 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 37944 | . . . . . . . 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 38028 | . . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)) → (𝑈𝐻𝑥) ∈ 𝐼) |
| 12 | 10, 11 | eqeltrrd 2842 | . . . . . 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 3989 | . . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈 ∈ 𝐼) → 𝑋 ⊆ 𝐼) |
| 15 | 4, 14 | eqssd 4001 | . . 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 37946 | . . . 4 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋) |
| 18 | 17 | adantr 480 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝑈 ∈ 𝑋) |
| 19 | eleq2 2830 | . . 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 1540 ∈ wcel 2108 ⊆ wss 3951 ran crn 5686 ‘cfv 6561 (class class class)co 7431 1st c1st 8012 2nd c2nd 8013 GIdcgi 30509 RingOpscrngo 37901 Idlcidl 38014 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-riota 7388 df-ov 7434 df-1st 8014 df-2nd 8015 df-grpo 30512 df-gid 30513 df-ablo 30564 df-ass 37850 df-exid 37852 df-mgmOLD 37856 df-sgrOLD 37868 df-mndo 37874 df-rngo 37902 df-idl 38017 |
| This theorem is referenced by: 0rngo 38034 divrngidl 38035 maxidln1 38051 |
| Copyright terms: Public domain | W3C validator |