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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 38041 | . . . . . 6 ⊢ (𝑅 ∈ RingOps → (𝑀 ∈ (MaxIdl‘𝑅) ↔ (𝑀 ∈ (Idl‘𝑅) ∧ 𝑀 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝑋))))) |
4 | 3 | biimpa 476 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → (𝑀 ∈ (Idl‘𝑅) ∧ 𝑀 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝑋)))) |
5 | 4 | simp3d 1145 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ∀𝑗 ∈ (Idl‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝑋))) |
6 | sseq2 4025 | . . . . . 6 ⊢ (𝑗 = 𝐼 → (𝑀 ⊆ 𝑗 ↔ 𝑀 ⊆ 𝐼)) | |
7 | eqeq1 2741 | . . . . . . 7 ⊢ (𝑗 = 𝐼 → (𝑗 = 𝑀 ↔ 𝐼 = 𝑀)) | |
8 | eqeq1 2741 | . . . . . . 7 ⊢ (𝑗 = 𝐼 → (𝑗 = 𝑋 ↔ 𝐼 = 𝑋)) | |
9 | 7, 8 | orbi12d 919 | . . . . . 6 ⊢ (𝑗 = 𝐼 → ((𝑗 = 𝑀 ∨ 𝑗 = 𝑋) ↔ (𝐼 = 𝑀 ∨ 𝐼 = 𝑋))) |
10 | 6, 9 | imbi12d 344 | . . . . 5 ⊢ (𝑗 = 𝐼 → ((𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝑋)) ↔ (𝑀 ⊆ 𝐼 → (𝐼 = 𝑀 ∨ 𝐼 = 𝑋)))) |
11 | 10 | rspcva 3623 | . . . 4 ⊢ ((𝐼 ∈ (Idl‘𝑅) ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝑋))) → (𝑀 ⊆ 𝐼 → (𝐼 = 𝑀 ∨ 𝐼 = 𝑋))) |
12 | 5, 11 | sylan2 593 | . . 3 ⊢ ((𝐼 ∈ (Idl‘𝑅) ∧ (𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅))) → (𝑀 ⊆ 𝐼 → (𝐼 = 𝑀 ∨ 𝐼 = 𝑋))) |
13 | 12 | ancoms 458 | . 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑀 ⊆ 𝐼 → (𝐼 = 𝑀 ∨ 𝐼 = 𝑋))) |
14 | 13 | impr 454 | 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 395 ∨ wo 848 ∧ w3a 1087 = wceq 1539 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ⊆ wss 3966 ran crn 5694 ‘cfv 6569 1st c1st 8020 RingOpscrngo 37895 Idlcidl 38008 MaxIdlcmaxidl 38010 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 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 5305 ax-nul 5315 ax-pr 5441 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 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-rab 3437 df-v 3483 df-dif 3969 df-un 3971 df-in 3973 df-ss 3983 df-nul 4343 df-if 4535 df-pw 4610 df-sn 4635 df-pr 4637 df-op 4641 df-uni 4916 df-br 5152 df-opab 5214 df-mpt 5235 df-id 5587 df-xp 5699 df-rel 5700 df-cnv 5701 df-co 5702 df-dm 5703 df-rn 5704 df-iota 6522 df-fun 6571 df-fv 6577 df-maxidl 38013 |
This theorem is referenced by: (None) |
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