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Mathbox for Jeff Madsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > maxidlval | Structured version Visualization version GIF version |
Description: The set of maximal ideals of a ring. (Contributed by Jeff Madsen, 5-Jan-2011.) |
Ref | Expression |
---|---|
maxidlval.1 | ⊢ 𝐺 = (1st ‘𝑅) |
maxidlval.2 | ⊢ 𝑋 = ran 𝐺 |
Ref | Expression |
---|---|
maxidlval | ⊢ (𝑅 ∈ RingOps → (MaxIdl‘𝑅) = {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝑋)))}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6885 | . . 3 ⊢ (𝑟 = 𝑅 → (Idl‘𝑟) = (Idl‘𝑅)) | |
2 | fveq2 6885 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = (1st ‘𝑅)) | |
3 | maxidlval.1 | . . . . . . . 8 ⊢ 𝐺 = (1st ‘𝑅) | |
4 | 2, 3 | eqtr4di 2784 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = 𝐺) |
5 | 4 | rneqd 5931 | . . . . . 6 ⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = ran 𝐺) |
6 | maxidlval.2 | . . . . . 6 ⊢ 𝑋 = ran 𝐺 | |
7 | 5, 6 | eqtr4di 2784 | . . . . 5 ⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = 𝑋) |
8 | 7 | neeq2d 2995 | . . . 4 ⊢ (𝑟 = 𝑅 → (𝑖 ≠ ran (1st ‘𝑟) ↔ 𝑖 ≠ 𝑋)) |
9 | 7 | eqeq2d 2737 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑗 = ran (1st ‘𝑟) ↔ 𝑗 = 𝑋)) |
10 | 9 | orbi2d 912 | . . . . . 6 ⊢ (𝑟 = 𝑅 → ((𝑗 = 𝑖 ∨ 𝑗 = ran (1st ‘𝑟)) ↔ (𝑗 = 𝑖 ∨ 𝑗 = 𝑋))) |
11 | 10 | imbi2d 340 | . . . . 5 ⊢ (𝑟 = 𝑅 → ((𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = ran (1st ‘𝑟))) ↔ (𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝑋)))) |
12 | 1, 11 | raleqbidv 3336 | . . . 4 ⊢ (𝑟 = 𝑅 → (∀𝑗 ∈ (Idl‘𝑟)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = ran (1st ‘𝑟))) ↔ ∀𝑗 ∈ (Idl‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝑋)))) |
13 | 8, 12 | anbi12d 630 | . . 3 ⊢ (𝑟 = 𝑅 → ((𝑖 ≠ ran (1st ‘𝑟) ∧ ∀𝑗 ∈ (Idl‘𝑟)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = ran (1st ‘𝑟)))) ↔ (𝑖 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝑋))))) |
14 | 1, 13 | rabeqbidv 3443 | . 2 ⊢ (𝑟 = 𝑅 → {𝑖 ∈ (Idl‘𝑟) ∣ (𝑖 ≠ ran (1st ‘𝑟) ∧ ∀𝑗 ∈ (Idl‘𝑟)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = ran (1st ‘𝑟))))} = {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝑋)))}) |
15 | df-maxidl 37393 | . 2 ⊢ MaxIdl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ (Idl‘𝑟) ∣ (𝑖 ≠ ran (1st ‘𝑟) ∧ ∀𝑗 ∈ (Idl‘𝑟)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = ran (1st ‘𝑟))))}) | |
16 | fvex 6898 | . . 3 ⊢ (Idl‘𝑅) ∈ V | |
17 | 16 | rabex 5325 | . 2 ⊢ {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝑋)))} ∈ V |
18 | 14, 15, 17 | fvmpt 6992 | 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 395 ∨ wo 844 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∀wral 3055 {crab 3426 ⊆ wss 3943 ran crn 5670 ‘cfv 6537 1st c1st 7972 RingOpscrngo 37275 Idlcidl 37388 MaxIdlcmaxidl 37390 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6489 df-fun 6539 df-fv 6545 df-maxidl 37393 |
This theorem is referenced by: ismaxidl 37421 |
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