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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 35301 | . . . . 5 ⊢ (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅)) |
4 | sseq2 3992 | . . . . . 6 ⊢ (𝑗 = 𝑋 → (𝑆 ⊆ 𝑗 ↔ 𝑆 ⊆ 𝑋)) | |
5 | 4 | rspcev 3622 | . . . . 5 ⊢ ((𝑋 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝑋) → ∃𝑗 ∈ (Idl‘𝑅)𝑆 ⊆ 𝑗) |
6 | 3, 5 | sylan 582 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → ∃𝑗 ∈ (Idl‘𝑅)𝑆 ⊆ 𝑗) |
7 | rabn0 4338 | . . . 4 ⊢ ({𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ≠ ∅ ↔ ∃𝑗 ∈ (Idl‘𝑅)𝑆 ⊆ 𝑗) | |
8 | 6, 7 | sylibr 236 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ≠ ∅) |
9 | intex 5239 | . . 3 ⊢ ({𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ≠ ∅ ↔ ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ∈ V) | |
10 | 8, 9 | sylib 220 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ∈ V) |
11 | 1 | fvexi 6683 | . . . . . 6 ⊢ 𝐺 ∈ V |
12 | 11 | rnex 7616 | . . . . 5 ⊢ ran 𝐺 ∈ V |
13 | 2, 12 | eqeltri 2909 | . . . 4 ⊢ 𝑋 ∈ V |
14 | 13 | elpw2 5247 | . . 3 ⊢ (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋) |
15 | simpl 485 | . . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → 𝑟 = 𝑅) | |
16 | 15 | fveq2d 6673 | . . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (Idl‘𝑟) = (Idl‘𝑅)) |
17 | sseq1 3991 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠 ⊆ 𝑗 ↔ 𝑆 ⊆ 𝑗)) | |
18 | 17 | adantl 484 | . . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑠 ⊆ 𝑗 ↔ 𝑆 ⊆ 𝑗)) |
19 | 16, 18 | rabeqbidv 3485 | . . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠 ⊆ 𝑗} = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗}) |
20 | 19 | inteqd 4880 | . . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ∩ {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠 ⊆ 𝑗} = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗}) |
21 | fveq2 6669 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = (1st ‘𝑅)) | |
22 | 21, 1 | syl6eqr 2874 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = 𝐺) |
23 | 22 | rneqd 5807 | . . . . . 6 ⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = ran 𝐺) |
24 | 23, 2 | syl6eqr 2874 | . . . . 5 ⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = 𝑋) |
25 | 24 | pweqd 4557 | . . . 4 ⊢ (𝑟 = 𝑅 → 𝒫 ran (1st ‘𝑟) = 𝒫 𝑋) |
26 | df-igen 35337 | . . . 4 ⊢ IdlGen = (𝑟 ∈ RingOps, 𝑠 ∈ 𝒫 ran (1st ‘𝑟) ↦ ∩ {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠 ⊆ 𝑗}) | |
27 | 20, 25, 26 | ovmpox 7302 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ 𝒫 𝑋 ∧ ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ∈ V) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗}) |
28 | 14, 27 | syl3an2br 1403 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋 ∧ ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ∈ V) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗}) |
29 | 10, 28 | mpd3an3 1458 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∃wrex 3139 {crab 3142 Vcvv 3494 ⊆ wss 3935 ∅c0 4290 𝒫 cpw 4538 ∩ cint 4875 ran crn 5555 ‘cfv 6354 (class class class)co 7155 1st c1st 7686 RingOpscrngo 35171 Idlcidl 35284 IdlGen cigen 35336 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-fo 6360 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-1st 7688 df-2nd 7689 df-grpo 28269 df-gid 28270 df-ablo 28321 df-rngo 35172 df-idl 35287 df-igen 35337 |
This theorem is referenced by: igenss 35339 igenidl 35340 igenmin 35341 igenidl2 35342 igenval2 35343 |
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