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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 35351 | . . . . . 6 ⊢ (𝑅 ∈ RingOps → (𝑀 ∈ (MaxIdl‘𝑅) ↔ (𝑀 ∈ (Idl‘𝑅) ∧ 𝑀 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝑋))))) |
4 | 3 | biimpa 479 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → (𝑀 ∈ (Idl‘𝑅) ∧ 𝑀 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝑋)))) |
5 | 4 | simp3d 1139 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ∀𝑗 ∈ (Idl‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝑋))) |
6 | sseq2 3986 | . . . . . 6 ⊢ (𝑗 = 𝐼 → (𝑀 ⊆ 𝑗 ↔ 𝑀 ⊆ 𝐼)) | |
7 | eqeq1 2824 | . . . . . . 7 ⊢ (𝑗 = 𝐼 → (𝑗 = 𝑀 ↔ 𝐼 = 𝑀)) | |
8 | eqeq1 2824 | . . . . . . 7 ⊢ (𝑗 = 𝐼 → (𝑗 = 𝑋 ↔ 𝐼 = 𝑋)) | |
9 | 7, 8 | orbi12d 915 | . . . . . 6 ⊢ (𝑗 = 𝐼 → ((𝑗 = 𝑀 ∨ 𝑗 = 𝑋) ↔ (𝐼 = 𝑀 ∨ 𝐼 = 𝑋))) |
10 | 6, 9 | imbi12d 347 | . . . . 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 594 | . . 3 ⊢ ((𝐼 ∈ (Idl‘𝑅) ∧ (𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅))) → (𝑀 ⊆ 𝐼 → (𝐼 = 𝑀 ∨ 𝐼 = 𝑋))) |
13 | 12 | ancoms 461 | . 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑀 ⊆ 𝐼 → (𝐼 = 𝑀 ∨ 𝐼 = 𝑋))) |
14 | 13 | impr 457 | 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 398 ∨ wo 843 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 ≠ wne 3015 ∀wral 3137 ⊆ wss 3929 ran crn 5549 ‘cfv 6348 1st c1st 7680 RingOpscrngo 35205 Idlcidl 35318 MaxIdlcmaxidl 35320 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 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 35323 |
This theorem is referenced by: (None) |
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