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Theorem igenmin 35375

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 2820	. . . . 5 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2		eqid 2820	. . . . 5 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
3	1, 2	idlss 35327	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ ran (1^st ‘𝑅))
4		sstr 3968	. . . . . . 7 ⊢ ((𝑆 ⊆ 𝐼 ∧ 𝐼 ⊆ ran (1^st ‘𝑅)) → 𝑆 ⊆ ran (1^st ‘𝑅))
5	4	ancoms 461	. . . . . 6 ⊢ ((𝐼 ⊆ ran (1^st ‘𝑅) ∧ 𝑆 ⊆ 𝐼) → 𝑆 ⊆ ran (1^st ‘𝑅))
6	1, 2	igenval 35372	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ ran (1^st ‘𝑅)) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗})
7	5, 6	sylan2 594	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐼 ⊆ ran (1^st ‘𝑅) ∧ 𝑆 ⊆ 𝐼)) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗})
8	7	anassrs 470	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ⊆ ran (1^st ‘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 1105	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗})
11		sseq2 3986	. . . 4 ⊢ (𝑗 = 𝐼 → (𝑆 ⊆ 𝑗 ↔ 𝑆 ⊆ 𝐼))
12	11	intminss 4895	. . 3 ⊢ ((𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼) → ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ⊆ 𝐼)
13	12	3adant1 1125	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼) → ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ⊆ 𝐼)
14	10, 13	eqsstrd 3998	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 1082 = wceq 1536 ∈ wcel 2113 {crab 3141 ⊆ wss 3929 ∩ cint 4869 ran crn 5549 ‘cfv 6348 (class class class)co 7149 1^st c1st 7680 RingOpscrngo 35205 Idlcidl 35318 IdlGen cigen 35370

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fo 6354 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-1st 7682 df-2nd 7683 df-grpo 28268 df-gid 28269 df-ablo 28320 df-rngo 35206 df-idl 35321 df-igen 35371

This theorem is referenced by: igenval2 35377

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