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Mirrors > Home > MPE Home > Th. List > Mathboxes > maxidlnr | Structured version Visualization version GIF version |
Description: A maximal ideal is proper. (Contributed by Jeff Madsen, 16-Jun-2011.) |
Ref | Expression |
---|---|
maxidlnr.1 | ⊢ 𝐺 = (1st ‘𝑅) |
maxidlnr.2 | ⊢ 𝑋 = ran 𝐺 |
Ref | Expression |
---|---|
maxidlnr | ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ≠ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | maxidlnr.1 | . . . 4 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | maxidlnr.2 | . . . 4 ⊢ 𝑋 = ran 𝐺 | |
3 | 1, 2 | ismaxidl 36195 | . . 3 ⊢ (𝑅 ∈ RingOps → (𝑀 ∈ (MaxIdl‘𝑅) ↔ (𝑀 ∈ (Idl‘𝑅) ∧ 𝑀 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝑋))))) |
4 | 3 | biimpa 477 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → (𝑀 ∈ (Idl‘𝑅) ∧ 𝑀 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝑋)))) |
5 | 4 | simp2d 1142 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ≠ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∨ wo 844 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ⊆ wss 3888 ran crn 5592 ‘cfv 6435 1st c1st 7829 RingOpscrngo 36049 Idlcidl 36162 MaxIdlcmaxidl 36164 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5225 ax-nul 5232 ax-pr 5354 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3433 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4259 df-if 4462 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-br 5077 df-opab 5139 df-mpt 5160 df-id 5491 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-iota 6393 df-fun 6437 df-fv 6443 df-maxidl 36167 |
This theorem is referenced by: maxidln1 36199 |
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