Theorem igenidl 38372

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

Hypotheses

Ref	Expression
igenval.1	⊢ 𝐺 = (1st ‘𝑅)
igenval.2	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
igenidl	⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → (𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅))

Proof of Theorem igenidl

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		igenval.1	. . 3 ⊢ 𝐺 = (1st ‘𝑅)
2		igenval.2	. . 3 ⊢ 𝑋 = ran 𝐺
3	1, 2	igenval 38370	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗})
4	1, 2	rngoidl 38333	. . . . 5 ⊢ (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅))
5		sseq2 3943	. . . . . 6 ⊢ (𝑗 = 𝑋 → (𝑆 ⊆ 𝑗 ↔ 𝑆 ⊆ 𝑋))
6	5	rspcev 3562	. . . . 5 ⊢ ((𝑋 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝑋) → ∃𝑗 ∈ (Idl‘𝑅)𝑆 ⊆ 𝑗)
7	4, 6	sylan 581	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → ∃𝑗 ∈ (Idl‘𝑅)𝑆 ⊆ 𝑗)
8		rabn0 4319	. . . 4 ⊢ ({𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ≠ ∅ ↔ ∃𝑗 ∈ (Idl‘𝑅)𝑆 ⊆ 𝑗)
9	7, 8	sylibr 234	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ≠ ∅)
10		ssrab2 4013	. . . 4 ⊢ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ⊆ (Idl‘𝑅)
11		intidl 38338	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ≠ ∅ ∧ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ⊆ (Idl‘𝑅)) → ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ∈ (Idl‘𝑅))
12	10, 11	mp3an3 1453	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ≠ ∅) → ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ∈ (Idl‘𝑅))
13	9, 12	syldan 592	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ∈ (Idl‘𝑅))
14	3, 13	eqeltrd 2835	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → (𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2930  ∃wrex 3059  {crab 3387   ⊆ wss 3885  ∅c0 4263  ∩ cint 4879  ran crn 5621  ‘cfv 6487  (class class class)co 7356  1^st c1st 7929  RingOpscrngo 38203  Idlcidl 38316   IdlGen cigen 38368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-int 4880  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-fo 6493  df-fv 6495  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-grpo 30552  df-gid 30553  df-ablo 30604  df-rngo 38204  df-idl 38319  df-igen 38369

This theorem is referenced by:  igenval2  38375  isfldidl  38377  ispridlc  38379