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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 36892 | . . . . 5 ⊢ (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅)) |
4 | sseq2 4009 | . . . . . 6 ⊢ (𝑗 = 𝑋 → (𝑆 ⊆ 𝑗 ↔ 𝑆 ⊆ 𝑋)) | |
5 | 4 | rspcev 3613 | . . . . 5 ⊢ ((𝑋 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝑋) → ∃𝑗 ∈ (Idl‘𝑅)𝑆 ⊆ 𝑗) |
6 | 3, 5 | sylan 581 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → ∃𝑗 ∈ (Idl‘𝑅)𝑆 ⊆ 𝑗) |
7 | rabn0 4386 | . . . 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 5338 | . . 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 6906 | . . . . . 6 ⊢ 𝐺 ∈ V |
12 | 11 | rnex 7903 | . . . . 5 ⊢ ran 𝐺 ∈ V |
13 | 2, 12 | eqeltri 2830 | . . . 4 ⊢ 𝑋 ∈ V |
14 | 13 | elpw2 5346 | . . 3 ⊢ (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋) |
15 | simpl 484 | . . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → 𝑟 = 𝑅) | |
16 | 15 | fveq2d 6896 | . . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (Idl‘𝑟) = (Idl‘𝑅)) |
17 | sseq1 4008 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠 ⊆ 𝑗 ↔ 𝑆 ⊆ 𝑗)) | |
18 | 17 | adantl 483 | . . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑠 ⊆ 𝑗 ↔ 𝑆 ⊆ 𝑗)) |
19 | 16, 18 | rabeqbidv 3450 | . . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠 ⊆ 𝑗} = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗}) |
20 | 19 | inteqd 4956 | . . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ∩ {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠 ⊆ 𝑗} = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗}) |
21 | fveq2 6892 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = (1st ‘𝑅)) | |
22 | 21, 1 | eqtr4di 2791 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = 𝐺) |
23 | 22 | rneqd 5938 | . . . . . 6 ⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = ran 𝐺) |
24 | 23, 2 | eqtr4di 2791 | . . . . 5 ⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = 𝑋) |
25 | 24 | pweqd 4620 | . . . 4 ⊢ (𝑟 = 𝑅 → 𝒫 ran (1st ‘𝑟) = 𝒫 𝑋) |
26 | df-igen 36928 | . . . 4 ⊢ IdlGen = (𝑟 ∈ RingOps, 𝑠 ∈ 𝒫 ran (1st ‘𝑟) ↦ ∩ {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠 ⊆ 𝑗}) | |
27 | 20, 25, 26 | ovmpox 7561 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ 𝒫 𝑋 ∧ ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ∈ V) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗}) |
28 | 14, 27 | syl3an2br 1408 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋 ∧ ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ∈ V) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗}) |
29 | 10, 28 | mpd3an3 1463 | 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 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∃wrex 3071 {crab 3433 Vcvv 3475 ⊆ wss 3949 ∅c0 4323 𝒫 cpw 4603 ∩ cint 4951 ran crn 5678 ‘cfv 6544 (class class class)co 7409 1st c1st 7973 RingOpscrngo 36762 Idlcidl 36875 IdlGen cigen 36927 |
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-pow 5364 ax-pr 5428 ax-un 7725 |
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-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 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-fn 6547 df-f 6548 df-fo 6550 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-grpo 29746 df-gid 29747 df-ablo 29798 df-rngo 36763 df-idl 36878 df-igen 36928 |
This theorem is referenced by: igenss 36930 igenidl 36931 igenmin 36932 igenidl2 36933 igenval2 36934 |
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