Theorem ismaxidl 38669

Description: Obsolete theorem, use ismxidl 33723 instead. The predicate "is a maximal ideal". (Contributed by Jeff Madsen, 5-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ismaxidl.1	⊢ 𝐺 = (1st ‘𝑅)
ismaxidl.2	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
ismaxidl	⊢ (𝑅 ∈ RingOps → (𝑀 ∈ (MaxIdl‘𝑅) ↔ (𝑀 ∈ (Idl‘𝑅) ∧ 𝑀 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝑋)))))

Distinct variable groups:   𝑅,𝑗   𝑗,𝑀

Allowed substitution hints:   𝐺(𝑗)   𝑋(𝑗)

Proof of Theorem ismaxidl

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ismaxidl.1	. . . 4 ⊢ 𝐺 = (1st ‘𝑅)
2		ismaxidl.2	. . . 4 ⊢ 𝑋 = ran 𝐺
3	1, 2	maxidlval 38668	. . 3 ⊢ (𝑅 ∈ RingOps → (MaxIdl‘𝑅) = {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝑋)))})
4	3	eleq2d 2849	. 2 ⊢ (𝑅 ∈ RingOps → (𝑀 ∈ (MaxIdl‘𝑅) ↔ 𝑀 ∈ {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝑋)))}))
5		neeq1 3020	. . . . 5 ⊢ (𝑖 = 𝑀 → (𝑖 ≠ 𝑋 ↔ 𝑀 ≠ 𝑋))
6		sseq1 3963	. . . . . . 7 ⊢ (𝑖 = 𝑀 → (𝑖 ⊆ 𝑗 ↔ 𝑀 ⊆ 𝑗))
7		eqeq2 2775	. . . . . . . 8 ⊢ (𝑖 = 𝑀 → (𝑗 = 𝑖 ↔ 𝑗 = 𝑀))
8	7	orbi1d 929	. . . . . . 7 ⊢ (𝑖 = 𝑀 → ((𝑗 = 𝑖 ∨ 𝑗 = 𝑋) ↔ (𝑗 = 𝑀 ∨ 𝑗 = 𝑋)))
9	6, 8	imbi12d 347	. . . . . 6 ⊢ (𝑖 = 𝑀 → ((𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝑋)) ↔ (𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝑋))))
10	9	ralbidv 3188	. . . . 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 643	. . . 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 3651	. . 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 1111	. . 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 281	. 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	bitrdi 290	1 ⊢ (𝑅 ∈ RingOps → (𝑀 ∈ (MaxIdl‘𝑅) ↔ (𝑀 ∈ (Idl‘𝑅) ∧ 𝑀 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝑋)))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  {crab 3416   ⊆ wss 3906  ran crn 5664  ‘cfv 6538  1^st c1st 7985  RingOpscrngo 38523  Idlcidl 38636  MaxIdlcmaxidl 38638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6494  df-fun 6540  df-fv 6546  df-maxidl 38641

This theorem is referenced by:  maxidlidl  38670  maxidlnr  38671  maxidlmax  38672