igenmin - Mathbox for Jeff Madsen

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Theorem igenmin 38523

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 2761	. . . . 5 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2		eqid 2761	. . . . 5 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
3	1, 2	idlss 38475	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ ran (1^st ‘𝑅))
4		sstr 3942	. . . . . . 7 ⊢ ((𝑆 ⊆ 𝐼 ∧ 𝐼 ⊆ ran (1^st ‘𝑅)) → 𝑆 ⊆ ran (1^st ‘𝑅))
5	4	ancoms 462	. . . . . 6 ⊢ ((𝐼 ⊆ ran (1^st ‘𝑅) ∧ 𝑆 ⊆ 𝐼) → 𝑆 ⊆ ran (1^st ‘𝑅))
6	1, 2	igenval 38520	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ ran (1^st ‘𝑅)) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗})
7	5, 6	sylan2 602	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐼 ⊆ ran (1^st ‘𝑅) ∧ 𝑆 ⊆ 𝐼)) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗})
8	7	anassrs 471	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ⊆ ran (1^st ‘𝑅)) ∧ 𝑆 ⊆ 𝐼) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗})
9	3, 8	syldanl 611	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑆 ⊆ 𝐼) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗})
10	9	3impa 1121	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗})
11		sseq2 3960	. . . 4 ⊢ (𝑗 = 𝐼 → (𝑆 ⊆ 𝑗 ↔ 𝑆 ⊆ 𝐼))
12	11	intminss 4929	. . 3 ⊢ ((𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼) → ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ⊆ 𝐼)
13	12	3adant1 1142	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼) → ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ⊆ 𝐼)
14	10, 13	eqsstrd 3968	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 {crab 3413 ⊆ wss 3902 ∩ cint 4902 ran crn 5644 ‘cfv 6515 (class class class)co 7390 1^st c1st 7962 RingOpscrngo 38353 Idlcidl 38466 IdlGen cigen 38518

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7712

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3076 df-rex 3086 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-id 5538 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-fo 6521 df-fv 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-1st 7964 df-2nd 7965 df-grpo 30652 df-gid 30653 df-ablo 30704 df-rngo 38354 df-idl 38469 df-igen 38519

This theorem is referenced by: igenval2 38525

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