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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 2821 | . . . . 5 ⊢ (1st ‘𝑅) = (1st ‘𝑅) | |
2 | eqid 2821 | . . . . 5 ⊢ ran (1st ‘𝑅) = ran (1st ‘𝑅) | |
3 | 1, 2 | idlss 35293 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ ran (1st ‘𝑅)) |
4 | sstr 3974 | . . . . . . 7 ⊢ ((𝑆 ⊆ 𝐼 ∧ 𝐼 ⊆ ran (1st ‘𝑅)) → 𝑆 ⊆ ran (1st ‘𝑅)) | |
5 | 4 | ancoms 461 | . . . . . 6 ⊢ ((𝐼 ⊆ ran (1st ‘𝑅) ∧ 𝑆 ⊆ 𝐼) → 𝑆 ⊆ ran (1st ‘𝑅)) |
6 | 1, 2 | igenval 35338 | . . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ ran (1st ‘𝑅)) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗}) |
7 | 5, 6 | sylan2 594 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐼 ⊆ ran (1st ‘𝑅) ∧ 𝑆 ⊆ 𝐼)) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗}) |
8 | 7 | anassrs 470 | . . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ⊆ ran (1st ‘𝑅)) ∧ 𝑆 ⊆ 𝐼) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗}) |
9 | 3, 8 | syldanl 603 | . . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑆 ⊆ 𝐼) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗}) |
10 | 9 | 3impa 1106 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗}) |
11 | sseq2 3992 | . . . 4 ⊢ (𝑗 = 𝐼 → (𝑆 ⊆ 𝑗 ↔ 𝑆 ⊆ 𝐼)) | |
12 | 11 | intminss 4901 | . . 3 ⊢ ((𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼) → ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ⊆ 𝐼) |
13 | 12 | 3adant1 1126 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼) → ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ⊆ 𝐼) |
14 | 10, 13 | eqsstrd 4004 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼) → (𝑅 IdlGen 𝑆) ⊆ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 {crab 3142 ⊆ wss 3935 ∩ 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: igenval2 35343 |
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