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Mathbox for Jeff Madsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ismaxidl | Structured version Visualization version GIF version |
Description: The predicate "is a maximal ideal". (Contributed by Jeff Madsen, 5-Jan-2011.) |
Ref | Expression |
---|---|
ismaxidl.1 | ⊢ 𝐺 = (1st ‘𝑅) |
ismaxidl.2 | ⊢ 𝑋 = ran 𝐺 |
Ref | Expression |
---|---|
ismaxidl | ⊢ (𝑅 ∈ RingOps → (𝑀 ∈ (MaxIdl‘𝑅) ↔ (𝑀 ∈ (Idl‘𝑅) ∧ 𝑀 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝑋))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismaxidl.1 | . . . 4 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | ismaxidl.2 | . . . 4 ⊢ 𝑋 = ran 𝐺 | |
3 | 1, 2 | maxidlval 34796 | . . 3 ⊢ (𝑅 ∈ RingOps → (MaxIdl‘𝑅) = {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝑋)))}) |
4 | 3 | eleq2d 2844 | . 2 ⊢ (𝑅 ∈ RingOps → (𝑀 ∈ (MaxIdl‘𝑅) ↔ 𝑀 ∈ {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝑋)))})) |
5 | neeq1 3022 | . . . . 5 ⊢ (𝑖 = 𝑀 → (𝑖 ≠ 𝑋 ↔ 𝑀 ≠ 𝑋)) | |
6 | sseq1 3875 | . . . . . . 7 ⊢ (𝑖 = 𝑀 → (𝑖 ⊆ 𝑗 ↔ 𝑀 ⊆ 𝑗)) | |
7 | eqeq2 2782 | . . . . . . . 8 ⊢ (𝑖 = 𝑀 → (𝑗 = 𝑖 ↔ 𝑗 = 𝑀)) | |
8 | 7 | orbi1d 901 | . . . . . . 7 ⊢ (𝑖 = 𝑀 → ((𝑗 = 𝑖 ∨ 𝑗 = 𝑋) ↔ (𝑗 = 𝑀 ∨ 𝑗 = 𝑋))) |
9 | 6, 8 | imbi12d 337 | . . . . . 6 ⊢ (𝑖 = 𝑀 → ((𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝑋)) ↔ (𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝑋)))) |
10 | 9 | ralbidv 3140 | . . . . 5 ⊢ (𝑖 = 𝑀 → (∀𝑗 ∈ (Idl‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝑋)) ↔ ∀𝑗 ∈ (Idl‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝑋)))) |
11 | 5, 10 | anbi12d 622 | . . . 4 ⊢ (𝑖 = 𝑀 → ((𝑖 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝑋))) ↔ (𝑀 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝑋))))) |
12 | 11 | elrab 3588 | . . 3 ⊢ (𝑀 ∈ {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝑋)))} ↔ (𝑀 ∈ (Idl‘𝑅) ∧ (𝑀 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝑋))))) |
13 | 3anass 1077 | . . 3 ⊢ ((𝑀 ∈ (Idl‘𝑅) ∧ 𝑀 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝑋))) ↔ (𝑀 ∈ (Idl‘𝑅) ∧ (𝑀 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝑋))))) | |
14 | 12, 13 | bitr4i 270 | . 2 ⊢ (𝑀 ∈ {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝑋)))} ↔ (𝑀 ∈ (Idl‘𝑅) ∧ 𝑀 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝑋)))) |
15 | 4, 14 | syl6bb 279 | 1 ⊢ (𝑅 ∈ RingOps → (𝑀 ∈ (MaxIdl‘𝑅) ↔ (𝑀 ∈ (Idl‘𝑅) ∧ 𝑀 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝑋))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∨ wo 834 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 ≠ wne 2960 ∀wral 3081 {crab 3085 ⊆ wss 3822 ran crn 5404 ‘cfv 6185 1st c1st 7497 RingOpscrngo 34651 Idlcidl 34764 MaxIdlcmaxidl 34766 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pr 5182 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-rab 3090 df-v 3410 df-sbc 3675 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-iota 6149 df-fun 6187 df-fv 6193 df-maxidl 34769 |
This theorem is referenced by: maxidlidl 34798 maxidlnr 34799 maxidlmax 34800 |
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