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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 6892 | . . 3 ⊢ (𝑟 = 𝑅 → (Idl‘𝑟) = (Idl‘𝑅)) | |
2 | fveq2 6892 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = (1st ‘𝑅)) | |
3 | maxidlval.1 | . . . . . . . 8 ⊢ 𝐺 = (1st ‘𝑅) | |
4 | 2, 3 | eqtr4di 2791 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = 𝐺) |
5 | 4 | rneqd 5938 | . . . . . 6 ⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = ran 𝐺) |
6 | maxidlval.2 | . . . . . 6 ⊢ 𝑋 = ran 𝐺 | |
7 | 5, 6 | eqtr4di 2791 | . . . . 5 ⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = 𝑋) |
8 | 7 | neeq2d 3002 | . . . 4 ⊢ (𝑟 = 𝑅 → (𝑖 ≠ ran (1st ‘𝑟) ↔ 𝑖 ≠ 𝑋)) |
9 | 7 | eqeq2d 2744 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑗 = ran (1st ‘𝑟) ↔ 𝑗 = 𝑋)) |
10 | 9 | orbi2d 915 | . . . . . 6 ⊢ (𝑟 = 𝑅 → ((𝑗 = 𝑖 ∨ 𝑗 = ran (1st ‘𝑟)) ↔ (𝑗 = 𝑖 ∨ 𝑗 = 𝑋))) |
11 | 10 | imbi2d 341 | . . . . 5 ⊢ (𝑟 = 𝑅 → ((𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = ran (1st ‘𝑟))) ↔ (𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝑋)))) |
12 | 1, 11 | raleqbidv 3343 | . . . 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 632 | . . 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 3450 | . 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 36880 | . 2 ⊢ MaxIdl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ (Idl‘𝑟) ∣ (𝑖 ≠ ran (1st ‘𝑟) ∧ ∀𝑗 ∈ (Idl‘𝑟)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = ran (1st ‘𝑟))))}) | |
16 | fvex 6905 | . . 3 ⊢ (Idl‘𝑅) ∈ V | |
17 | 16 | rabex 5333 | . 2 ⊢ {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝑋)))} ∈ V |
18 | 14, 15, 17 | fvmpt 6999 | 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 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 {crab 3433 ⊆ wss 3949 ran crn 5678 ‘cfv 6544 1st c1st 7973 RingOpscrngo 36762 Idlcidl 36875 MaxIdlcmaxidl 36877 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-iota 6496 df-fun 6546 df-fv 6552 df-maxidl 36880 |
This theorem is referenced by: ismaxidl 36908 |
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