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Mirrors > Home > MPE Home > Th. List > Mathboxes > igenval | Structured version Visualization version GIF version |
Description: The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof shortened by Mario Carneiro, 20-Dec-2013.) |
Ref | Expression |
---|---|
igenval.1 | ⊢ 𝐺 = (1st ‘𝑅) |
igenval.2 | ⊢ 𝑋 = ran 𝐺 |
Ref | Expression |
---|---|
igenval | ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | igenval.1 | . . . . . 6 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | igenval.2 | . . . . . 6 ⊢ 𝑋 = ran 𝐺 | |
3 | 1, 2 | rngoidl 36887 | . . . . 5 ⊢ (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅)) |
4 | sseq2 4008 | . . . . . 6 ⊢ (𝑗 = 𝑋 → (𝑆 ⊆ 𝑗 ↔ 𝑆 ⊆ 𝑋)) | |
5 | 4 | rspcev 3612 | . . . . 5 ⊢ ((𝑋 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝑋) → ∃𝑗 ∈ (Idl‘𝑅)𝑆 ⊆ 𝑗) |
6 | 3, 5 | sylan 580 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → ∃𝑗 ∈ (Idl‘𝑅)𝑆 ⊆ 𝑗) |
7 | rabn0 4385 | . . . 4 ⊢ ({𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ≠ ∅ ↔ ∃𝑗 ∈ (Idl‘𝑅)𝑆 ⊆ 𝑗) | |
8 | 6, 7 | sylibr 233 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ≠ ∅) |
9 | intex 5337 | . . 3 ⊢ ({𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ≠ ∅ ↔ ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ∈ V) | |
10 | 8, 9 | sylib 217 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ∈ V) |
11 | 1 | fvexi 6905 | . . . . . 6 ⊢ 𝐺 ∈ V |
12 | 11 | rnex 7902 | . . . . 5 ⊢ ran 𝐺 ∈ V |
13 | 2, 12 | eqeltri 2829 | . . . 4 ⊢ 𝑋 ∈ V |
14 | 13 | elpw2 5345 | . . 3 ⊢ (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋) |
15 | simpl 483 | . . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → 𝑟 = 𝑅) | |
16 | 15 | fveq2d 6895 | . . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (Idl‘𝑟) = (Idl‘𝑅)) |
17 | sseq1 4007 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠 ⊆ 𝑗 ↔ 𝑆 ⊆ 𝑗)) | |
18 | 17 | adantl 482 | . . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑠 ⊆ 𝑗 ↔ 𝑆 ⊆ 𝑗)) |
19 | 16, 18 | rabeqbidv 3449 | . . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠 ⊆ 𝑗} = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗}) |
20 | 19 | inteqd 4955 | . . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ∩ {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠 ⊆ 𝑗} = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗}) |
21 | fveq2 6891 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = (1st ‘𝑅)) | |
22 | 21, 1 | eqtr4di 2790 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = 𝐺) |
23 | 22 | rneqd 5937 | . . . . . 6 ⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = ran 𝐺) |
24 | 23, 2 | eqtr4di 2790 | . . . . 5 ⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = 𝑋) |
25 | 24 | pweqd 4619 | . . . 4 ⊢ (𝑟 = 𝑅 → 𝒫 ran (1st ‘𝑟) = 𝒫 𝑋) |
26 | df-igen 36923 | . . . 4 ⊢ IdlGen = (𝑟 ∈ RingOps, 𝑠 ∈ 𝒫 ran (1st ‘𝑟) ↦ ∩ {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠 ⊆ 𝑗}) | |
27 | 20, 25, 26 | ovmpox 7560 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ 𝒫 𝑋 ∧ ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ∈ V) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗}) |
28 | 14, 27 | syl3an2br 1407 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋 ∧ ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ∈ V) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗}) |
29 | 10, 28 | mpd3an3 1462 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∃wrex 3070 {crab 3432 Vcvv 3474 ⊆ wss 3948 ∅c0 4322 𝒫 cpw 4602 ∩ cint 4950 ran crn 5677 ‘cfv 6543 (class class class)co 7408 1st c1st 7972 RingOpscrngo 36757 Idlcidl 36870 IdlGen cigen 36922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fo 6549 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7974 df-2nd 7975 df-grpo 29741 df-gid 29742 df-ablo 29793 df-rngo 36758 df-idl 36873 df-igen 36923 |
This theorem is referenced by: igenss 36925 igenidl 36926 igenmin 36927 igenidl2 36928 igenval2 36929 |
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