Theorem igenval 38690

Description: Obsolete theorem, use rspvalint 21350 instead. The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof shortened by Mario Carneiro, 20-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
igenval	⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗})

Distinct variable groups:   𝑅,𝑗   𝑆,𝑗   𝑗,𝑋

Allowed substitution hint:   𝐺(𝑗)

Proof of Theorem igenval

Dummy variables 𝑟 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		igenval.1	. . . . . 6 ⊢ 𝐺 = (1st ‘𝑅)
2		igenval.2	. . . . . 6 ⊢ 𝑋 = ran 𝐺
3	1, 2	rngoidl 38653	. . . . 5 ⊢ (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅))
4		sseq2 3964	. . . . . 6 ⊢ (𝑗 = 𝑋 → (𝑆 ⊆ 𝑗 ↔ 𝑆 ⊆ 𝑋))
5	4	rspcev 3582	. . . . 5 ⊢ ((𝑋 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝑋) → ∃𝑗 ∈ (Idl‘𝑅)𝑆 ⊆ 𝑗)
6	3, 5	sylan 591	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → ∃𝑗 ∈ (Idl‘𝑅)𝑆 ⊆ 𝑗)
7		rabn0 4347	. . . 4 ⊢ ({𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ≠ ∅ ↔ ∃𝑗 ∈ (Idl‘𝑅)𝑆 ⊆ 𝑗)
8	6, 7	sylibr 237	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ≠ ∅)
9		intex 5316	. . 3 ⊢ ({𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ≠ ∅ ↔ ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ∈ V)
10	8, 9	sylib 221	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ∈ V)
11	1	fvexi 6897	. . . . . 6 ⊢ 𝐺 ∈ V
12	11	rnex 7908	. . . . 5 ⊢ ran 𝐺 ∈ V
13	2, 12	eqeltri 2859	. . . 4 ⊢ 𝑋 ∈ V
14	13	elpw2 5306	. . 3 ⊢ (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋)
15		simpl 487	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → 𝑟 = 𝑅)
16	15	fveq2d 6887	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (Idl‘𝑟) = (Idl‘𝑅))
17		sseq1 3963	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠 ⊆ 𝑗 ↔ 𝑆 ⊆ 𝑗))
18	17	adantl 486	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑠 ⊆ 𝑗 ↔ 𝑆 ⊆ 𝑗))
19	16, 18	rabeqbidv 3434	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠 ⊆ 𝑗} = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗})
20	19	inteqd 4918	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ∩ {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠 ⊆ 𝑗} = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗})
21		fveq2 6883	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = (1st ‘𝑅))
22	21, 1	eqtr4di 2816	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = 𝐺)
23	22	rneqd 5930	. . . . . 6 ⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = ran 𝐺)
24	23, 2	eqtr4di 2816	. . . . 5 ⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = 𝑋)
25	24	pweqd 4580	. . . 4 ⊢ (𝑟 = 𝑅 → 𝒫 ran (1st ‘𝑟) = 𝒫 𝑋)
26		df-igen 38689	. . . 4 ⊢ IdlGen = (𝑟 ∈ RingOps, 𝑠 ∈ 𝒫 ran (1st ‘𝑟) ↦ ∩ {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠 ⊆ 𝑗})
27	20, 25, 26	ovmpox 7565	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ 𝒫 𝑋 ∧ ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ∈ V) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗})
28	14, 27	syl3an2br 1434	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋 ∧ ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ∈ V) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗})
29	10, 28	mpd3an3 1491	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∃wrex 3089  {crab 3416  Vcvv 3455   ⊆ wss 3906  ∅c0 4287  𝒫 cpw 4563  ∩ cint 4913  ran crn 5664  ‘cfv 6538  (class class class)co 7412  1^st c1st 7985  RingOpscrngo 38523  Idlcidl 38636   IdlGen cigen 38688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988  df-grpo 30823  df-gid 30824  df-ablo 30875  df-rngo 38524  df-idl 38639  df-igen 38689

This theorem is referenced by:  igenss  38691  igenidl  38692  igenmin  38693  igenidl2  38694  igenval2  38695