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Mathbox for Jeff Madsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > igenmin | Structured version Visualization version GIF version |
Description: The ideal generated by a set is the minimal ideal containing that set. (Contributed by Jeff Madsen, 10-Jun-2010.) |
Ref | Expression |
---|---|
igenmin | ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼) → (𝑅 IdlGen 𝑆) ⊆ 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2734 | . . . . 5 ⊢ (1st ‘𝑅) = (1st ‘𝑅) | |
2 | eqid 2734 | . . . . 5 ⊢ ran (1st ‘𝑅) = ran (1st ‘𝑅) | |
3 | 1, 2 | idlss 38002 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ ran (1st ‘𝑅)) |
4 | sstr 4003 | . . . . . . 7 ⊢ ((𝑆 ⊆ 𝐼 ∧ 𝐼 ⊆ ran (1st ‘𝑅)) → 𝑆 ⊆ ran (1st ‘𝑅)) | |
5 | 4 | ancoms 458 | . . . . . 6 ⊢ ((𝐼 ⊆ ran (1st ‘𝑅) ∧ 𝑆 ⊆ 𝐼) → 𝑆 ⊆ ran (1st ‘𝑅)) |
6 | 1, 2 | igenval 38047 | . . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ ran (1st ‘𝑅)) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗}) |
7 | 5, 6 | sylan2 593 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐼 ⊆ ran (1st ‘𝑅) ∧ 𝑆 ⊆ 𝐼)) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗}) |
8 | 7 | anassrs 467 | . . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ⊆ ran (1st ‘𝑅)) ∧ 𝑆 ⊆ 𝐼) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗}) |
9 | 3, 8 | syldanl 602 | . . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑆 ⊆ 𝐼) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗}) |
10 | 9 | 3impa 1109 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗}) |
11 | sseq2 4021 | . . . 4 ⊢ (𝑗 = 𝐼 → (𝑆 ⊆ 𝑗 ↔ 𝑆 ⊆ 𝐼)) | |
12 | 11 | intminss 4978 | . . 3 ⊢ ((𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼) → ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ⊆ 𝐼) |
13 | 12 | 3adant1 1129 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼) → ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ⊆ 𝐼) |
14 | 10, 13 | eqsstrd 4033 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼) → (𝑅 IdlGen 𝑆) ⊆ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 {crab 3432 ⊆ wss 3962 ∩ cint 4950 ran crn 5689 ‘cfv 6562 (class class class)co 7430 1st c1st 8010 RingOpscrngo 37880 Idlcidl 37993 IdlGen cigen 38045 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-fo 6568 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-1st 8012 df-2nd 8013 df-grpo 30521 df-gid 30522 df-ablo 30573 df-rngo 37881 df-idl 37996 df-igen 38046 |
This theorem is referenced by: igenval2 38052 |
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