igenmin - Mathbox for Jeff Madsen

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Theorem igenmin 36149

Description: The ideal generated by a set is the minimal ideal containing that set. (Contributed by Jeff Madsen, 10-Jun-2010.)

**Assertion**
Ref	Expression
igenmin	⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)

Proof of Theorem igenmin Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2738	. . . . 5 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2		eqid 2738	. . . . 5 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
3	1, 2	idlss 36101	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ ran (1^st ‘𝑅))
4		sstr 3925	. . . . . . 7 ⊢ ((𝑆 ⊆ 𝐼 ∧ 𝐼 ⊆ ran (1^st ‘𝑅)) → 𝑆 ⊆ ran (1^st ‘𝑅))
5	4	ancoms 458	. . . . . 6 ⊢ ((𝐼 ⊆ ran (1^st ‘𝑅) ∧ 𝑆 ⊆ 𝐼) → 𝑆 ⊆ ran (1^st ‘𝑅))
6	1, 2	igenval 36146	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ ran (1^st ‘𝑅)) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗})
7	5, 6	sylan2 592	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐼 ⊆ ran (1^st ‘𝑅) ∧ 𝑆 ⊆ 𝐼)) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗})
8	7	anassrs 467	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ⊆ ran (1^st ‘𝑅)) ∧ 𝑆 ⊆ 𝐼) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗})
9	3, 8	syldanl 601	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑆 ⊆ 𝐼) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗})
10	9	3impa 1108	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗})
11		sseq2 3943	. . . 4 ⊢ (𝑗 = 𝐼 → (𝑆 ⊆ 𝑗 ↔ 𝑆 ⊆ 𝐼))
12	11	intminss 4902	. . 3 ⊢ ((𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼) → ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ⊆ 𝐼)
13	12	3adant1 1128	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼) → ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ⊆ 𝐼)
14	10, 13	eqsstrd 3955	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 1085 = wceq 1539 ∈ wcel 2108 {crab 3067 ⊆ wss 3883 ∩ cint 4876 ran crn 5581 ‘cfv 6418 (class class class)co 7255 1^st c1st 7802 RingOpscrngo 35979 Idlcidl 36092 IdlGen cigen 36144

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-fo 6424 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-1st 7804 df-2nd 7805 df-grpo 28756 df-gid 28757 df-ablo 28808 df-rngo 35980 df-idl 36095 df-igen 36145

This theorem is referenced by: igenval2 36151

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