Mathbox for Jeff Madsen |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > maxidlmax | Structured version Visualization version GIF version |
Description: A maximal ideal is a maximal proper ideal. (Contributed by Jeff Madsen, 16-Jun-2011.) |
Ref | Expression |
---|---|
maxidlnr.1 | ⊢ 𝐺 = (1st ‘𝑅) |
maxidlnr.2 | ⊢ 𝑋 = ran 𝐺 |
Ref | Expression |
---|---|
maxidlmax | ⊢ (((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑀 ⊆ 𝐼)) → (𝐼 = 𝑀 ∨ 𝐼 = 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | maxidlnr.1 | . . . . . . 7 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | maxidlnr.2 | . . . . . . 7 ⊢ 𝑋 = ran 𝐺 | |
3 | 1, 2 | ismaxidl 35199 | . . . . . 6 ⊢ (𝑅 ∈ RingOps → (𝑀 ∈ (MaxIdl‘𝑅) ↔ (𝑀 ∈ (Idl‘𝑅) ∧ 𝑀 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝑋))))) |
4 | 3 | biimpa 477 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → (𝑀 ∈ (Idl‘𝑅) ∧ 𝑀 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝑋)))) |
5 | 4 | simp3d 1136 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ∀𝑗 ∈ (Idl‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝑋))) |
6 | sseq2 3990 | . . . . . 6 ⊢ (𝑗 = 𝐼 → (𝑀 ⊆ 𝑗 ↔ 𝑀 ⊆ 𝐼)) | |
7 | eqeq1 2822 | . . . . . . 7 ⊢ (𝑗 = 𝐼 → (𝑗 = 𝑀 ↔ 𝐼 = 𝑀)) | |
8 | eqeq1 2822 | . . . . . . 7 ⊢ (𝑗 = 𝐼 → (𝑗 = 𝑋 ↔ 𝐼 = 𝑋)) | |
9 | 7, 8 | orbi12d 912 | . . . . . 6 ⊢ (𝑗 = 𝐼 → ((𝑗 = 𝑀 ∨ 𝑗 = 𝑋) ↔ (𝐼 = 𝑀 ∨ 𝐼 = 𝑋))) |
10 | 6, 9 | imbi12d 346 | . . . . 5 ⊢ (𝑗 = 𝐼 → ((𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝑋)) ↔ (𝑀 ⊆ 𝐼 → (𝐼 = 𝑀 ∨ 𝐼 = 𝑋)))) |
11 | 10 | rspcva 3618 | . . . 4 ⊢ ((𝐼 ∈ (Idl‘𝑅) ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝑋))) → (𝑀 ⊆ 𝐼 → (𝐼 = 𝑀 ∨ 𝐼 = 𝑋))) |
12 | 5, 11 | sylan2 592 | . . 3 ⊢ ((𝐼 ∈ (Idl‘𝑅) ∧ (𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅))) → (𝑀 ⊆ 𝐼 → (𝐼 = 𝑀 ∨ 𝐼 = 𝑋))) |
13 | 12 | ancoms 459 | . 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑀 ⊆ 𝐼 → (𝐼 = 𝑀 ∨ 𝐼 = 𝑋))) |
14 | 13 | impr 455 | 1 ⊢ (((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑀 ⊆ 𝐼)) → (𝐼 = 𝑀 ∨ 𝐼 = 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∨ wo 841 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 ⊆ wss 3933 ran crn 5549 ‘cfv 6348 1st c1st 7676 RingOpscrngo 35053 Idlcidl 35166 MaxIdlcmaxidl 35168 |
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-pr 5320 |
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-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 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-fv 6356 df-maxidl 35171 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |