Theorem igenval 38435

Description: The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof shortened by Mario Carneiro, 20-Dec-2013.)

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 38398	. . . . 5 ⊢ (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅))
4		sseq2 3948	. . . . . 6 ⊢ (𝑗 = 𝑋 → (𝑆 ⊆ 𝑗 ↔ 𝑆 ⊆ 𝑋))
5	4	rspcev 3567	. . . . 5 ⊢ ((𝑋 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝑋) → ∃𝑗 ∈ (Idl‘𝑅)𝑆 ⊆ 𝑗)
6	3, 5	sylan 586	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → ∃𝑗 ∈ (Idl‘𝑅)𝑆 ⊆ 𝑗)
7		rabn0 4324	. . . 4 ⊢ ({𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ≠ ∅ ↔ ∃𝑗 ∈ (Idl‘𝑅)𝑆 ⊆ 𝑗)
8	6, 7	sylibr 235	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ≠ ∅)
9		intex 5279	. . 3 ⊢ ({𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ≠ ∅ ↔ ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ∈ V)
10	8, 9	sylib 219	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ∈ V)
11	1	fvexi 6848	. . . . . 6 ⊢ 𝐺 ∈ V
12	11	rnex 7857	. . . . 5 ⊢ ran 𝐺 ∈ V
13	2, 12	eqeltri 2836	. . . 4 ⊢ 𝑋 ∈ V
14	13	elpw2 5269	. . 3 ⊢ (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋)
15		simpl 483	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → 𝑟 = 𝑅)
16	15	fveq2d 6838	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (Idl‘𝑟) = (Idl‘𝑅))
17		sseq1 3947	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠 ⊆ 𝑗 ↔ 𝑆 ⊆ 𝑗))
18	17	adantl 482	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑠 ⊆ 𝑗 ↔ 𝑆 ⊆ 𝑗))
19	16, 18	rabeqbidv 3410	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠 ⊆ 𝑗} = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗})
20	19	inteqd 4889	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ∩ {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠 ⊆ 𝑗} = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗})
21		fveq2 6834	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = (1st ‘𝑅))
22	21, 1	eqtr4di 2793	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = 𝐺)
23	22	rneqd 5887	. . . . . 6 ⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = ran 𝐺)
24	23, 2	eqtr4di 2793	. . . . 5 ⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = 𝑋)
25	24	pweqd 4553	. . . 4 ⊢ (𝑟 = 𝑅 → 𝒫 ran (1st ‘𝑟) = 𝒫 𝑋)
26		df-igen 38434	. . . 4 ⊢ IdlGen = (𝑟 ∈ RingOps, 𝑠 ∈ 𝒫 ran (1st ‘𝑟) ↦ ∩ {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠 ⊆ 𝑗})
27	20, 25, 26	ovmpox 7516	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ 𝒫 𝑋 ∧ ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ∈ V) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗})
28	14, 27	syl3an2br 1415	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋 ∧ ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗} ∈ V) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗})
29	10, 28	mpd3an3 1470	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗})

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2935  ∃wrex 3064  {crab 3392  Vcvv 3432   ⊆ wss 3890  ∅c0 4268  𝒫 cpw 4536  ∩ cint 4884  ran crn 5626  ‘cfv 6492  (class class class)co 7363  1^st c1st 7936  RingOpscrngo 38268  Idlcidl 38381   IdlGen cigen 38433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fo 6498  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-grpo 30589  df-gid 30590  df-ablo 30641  df-rngo 38269  df-idl 38384  df-igen 38434

This theorem is referenced by:  igenss  38436  igenidl  38437  igenmin  38438  igenidl2  38439  igenval2  38440