maxidlval - Mathbox for Jeff Madsen

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Theorem maxidlval 39049

Description: Obsolete theorem, use mxidlval 34029 instead. The set of maximal ideals of a ring. (Contributed by Jeff Madsen, 5-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
maxidlval.1	⊢ 𝐺 = (1^st ‘𝑅)
maxidlval.2	⊢ 𝑋 = ran 𝐺

**Assertion**
Ref	Expression
maxidlval	⊢ (𝑅 ∈ RingOps → (MaxIdl‘𝑅) = {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝑋)))})

Distinct variable group: 𝑅,𝑖,𝑗 Allowed substitution hints: 𝐺(𝑖, 𝑗) 𝑋(𝑖, 𝑗)

Proof of Theorem maxidlval Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6885	. . 3 ⊢ (𝑟 = 𝑅 → (Idl‘𝑟) = (Idl‘𝑅))
2		fveq2 6885	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1^st ‘𝑟) = (1^st ‘𝑅))
3		maxidlval.1	. . . . . . . 8 ⊢ 𝐺 = (1^st ‘𝑅)
4	2, 3	eqtr4di 2813	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (1^st ‘𝑟) = 𝐺)
5	4	rneqd 5919	. . . . . 6 ⊢ (𝑟 = 𝑅 → ran (1^st ‘𝑟) = ran 𝐺)
6		maxidlval.2	. . . . . 6 ⊢ 𝑋 = ran 𝐺
7	5, 6	eqtr4di 2813	. . . . 5 ⊢ (𝑟 = 𝑅 → ran (1^st ‘𝑟) = 𝑋)
8	7	neeq2d 3015	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑖 ≠ ran (1^st ‘𝑟) ↔ 𝑖 ≠ 𝑋))
9	7	eqeq2d 2771	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑗 = ran (1^st ‘𝑟) ↔ 𝑗 = 𝑋))
10	9	orbi2d 929	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((𝑗 = 𝑖 ∨ 𝑗 = ran (1^st ‘𝑟)) ↔ (𝑗 = 𝑖 ∨ 𝑗 = 𝑋)))
11	10	imbi2d 343	. . . . 5 ⊢ (𝑟 = 𝑅 → ((𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = ran (1^st ‘𝑟))) ↔ (𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝑋))))
12	1, 11	raleqbidv 3334	. . . 4 ⊢ (𝑟 = 𝑅 → (∀𝑗 ∈ (Idl‘𝑟)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = ran (1^st ‘𝑟))) ↔ ∀𝑗 ∈ (Idl‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝑋))))
13	8, 12	anbi12d 644	. . 3 ⊢ (𝑟 = 𝑅 → ((𝑖 ≠ ran (1^st ‘𝑟) ∧ ∀𝑗 ∈ (Idl‘𝑟)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = ran (1^st ‘𝑟)))) ↔ (𝑖 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝑋)))))
14	1, 13	rabeqbidv 3429	. 2 ⊢ (𝑟 = 𝑅 → {𝑖 ∈ (Idl‘𝑟) ∣ (𝑖 ≠ ran (1^st ‘𝑟) ∧ ∀𝑗 ∈ (Idl‘𝑟)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = ran (1^st ‘𝑟))))} = {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝑋)))})
15		df-maxidl 39022	. 2 ⊢ MaxIdl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ (Idl‘𝑟) ∣ (𝑖 ≠ ran (1^st ‘𝑟) ∧ ∀𝑗 ∈ (Idl‘𝑟)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = ran (1^st ‘𝑟))))})
16		fvex 6898	. . 3 ⊢ (Idl‘𝑅) ∈ V
17	16	rabex 5299	. 2 ⊢ {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝑋)))} ∈ V
18	14, 15, 17	fvmpt 6993	1 ⊢ (𝑅 ∈ RingOps → (MaxIdl‘𝑅) = {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝑋)))})

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∨ wo 861 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ∀wral 3076 {crab 3412 ⊆ wss 3895 ran crn 5651 ‘cfv 6538 1^st c1st 7999 RingOpscrngo 38904 Idlcidl 39017 MaxIdlcmaxidl 39019

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pr 5390

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-iota 6494 df-fun 6540 df-fv 6546 df-maxidl 39022

This theorem is used by: ismaxidl 39050

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