maxidlidl - Mathbox for Jeff Madsen

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Theorem maxidlidl 36199

Description: A maximal ideal is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.)

**Assertion**
Ref	Expression
maxidlidl	⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ∈ (Idl‘𝑅))

Proof of Theorem maxidlidl Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2738	. . . 4 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2		eqid 2738	. . . 4 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
3	1, 2	ismaxidl 36198	. . 3 ⊢ (𝑅 ∈ RingOps → (𝑀 ∈ (MaxIdl‘𝑅) ↔ (𝑀 ∈ (Idl‘𝑅) ∧ 𝑀 ≠ ran (1^st ‘𝑅) ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = ran (1^st ‘𝑅))))))
4		3anass 1094	. . 3 ⊢ ((𝑀 ∈ (Idl‘𝑅) ∧ 𝑀 ≠ ran (1^st ‘𝑅) ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = ran (1^st ‘𝑅)))) ↔ (𝑀 ∈ (Idl‘𝑅) ∧ (𝑀 ≠ ran (1^st ‘𝑅) ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = ran (1^st ‘𝑅))))))
5	3, 4	bitrdi 287	. 2 ⊢ (𝑅 ∈ RingOps → (𝑀 ∈ (MaxIdl‘𝑅) ↔ (𝑀 ∈ (Idl‘𝑅) ∧ (𝑀 ≠ ran (1^st ‘𝑅) ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = ran (1^st ‘𝑅)))))))
6	5	simprbda 499	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ∈ (Idl‘𝑅))

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 3887 ran crn 5590 ‘cfv 6433 1^st c1st 7829 RingOpscrngo 36052 Idlcidl 36165 MaxIdlcmaxidl 36167

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 5223 ax-nul 5230 ax-pr 5352

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 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-iota 6391 df-fun 6435 df-fv 6441 df-maxidl 36170

This theorem is referenced by: maxidln1 36202 maxidln0 36203

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