maxidlval - Mathbox for Jeff Madsen

	Mathbox for Jeff Madsen		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > maxidlval		Structured version Visualization version GIF version

Theorem maxidlval 35332

Description: The set of maximal ideals of a ring. (Contributed by Jeff Madsen, 5-Jan-2011.)

**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 6670	. . 3 ⊢ (𝑟 = 𝑅 → (Idl‘𝑟) = (Idl‘𝑅))
2		fveq2 6670	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1^st ‘𝑟) = (1^st ‘𝑅))
3		maxidlval.1	. . . . . . . 8 ⊢ 𝐺 = (1^st ‘𝑅)
4	2, 3	syl6eqr 2874	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (1^st ‘𝑟) = 𝐺)
5	4	rneqd 5808	. . . . . 6 ⊢ (𝑟 = 𝑅 → ran (1^st ‘𝑟) = ran 𝐺)
6		maxidlval.2	. . . . . 6 ⊢ 𝑋 = ran 𝐺
7	5, 6	syl6eqr 2874	. . . . 5 ⊢ (𝑟 = 𝑅 → ran (1^st ‘𝑟) = 𝑋)
8	7	neeq2d 3076	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑖 ≠ ran (1^st ‘𝑟) ↔ 𝑖 ≠ 𝑋))
9	7	eqeq2d 2832	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑗 = ran (1^st ‘𝑟) ↔ 𝑗 = 𝑋))
10	9	orbi2d 912	. . . . . 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 3401	. . . 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 632	. . 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 3485	. 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 35305	. 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 6683	. . 3 ⊢ (Idl‘𝑅) ∈ V
17	16	rabex 5235	. 2 ⊢ {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝑋)))} ∈ V
18	14, 15, 17	fvmpt 6768	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

Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 {crab 3142 ⊆ wss 3936 ran crn 5556 ‘cfv 6355 1^st c1st 7687 RingOpscrngo 35187 Idlcidl 35300 MaxIdlcmaxidl 35302

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fv 6363 df-maxidl 35305

This theorem is referenced by: ismaxidl 35333

W3C validator