Theorem pridlval 37414

Description: The class of prime ideals of a ring 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.)

Hypotheses

Ref	Expression
pridlval.1	⊢ 𝐺 = (1st ‘𝑅)
pridlval.2	⊢ 𝐻 = (2nd ‘𝑅)
pridlval.3	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
pridlval	⊢ (𝑅 ∈ RingOps → (PrIdl‘𝑅) = {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖 ≠ 𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖)))})

Distinct variable groups:   𝑅,𝑖,𝑥,𝑦,𝑎,𝑏   𝑖,𝑋   𝑖,𝐻

Allowed substitution hints:   𝐺(𝑥,𝑦,𝑖,𝑎,𝑏)   𝐻(𝑥,𝑦,𝑎,𝑏)   𝑋(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem pridlval

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6885	. . 3 ⊢ (𝑟 = 𝑅 → (Idl‘𝑟) = (Idl‘𝑅))
2		fveq2 6885	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = (1st ‘𝑅))
3		pridlval.1	. . . . . . . 8 ⊢ 𝐺 = (1st ‘𝑅)
4	2, 3	eqtr4di 2784	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = 𝐺)
5	4	rneqd 5931	. . . . . 6 ⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = ran 𝐺)
6		pridlval.3	. . . . . 6 ⊢ 𝑋 = ran 𝐺
7	5, 6	eqtr4di 2784	. . . . 5 ⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = 𝑋)
8	7	neeq2d 2995	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑖 ≠ ran (1st ‘𝑟) ↔ 𝑖 ≠ 𝑋))
9		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (2nd ‘𝑟) = (2nd ‘𝑅))
10		pridlval.2	. . . . . . . . . . 11 ⊢ 𝐻 = (2nd ‘𝑅)
11	9, 10	eqtr4di 2784	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (2nd ‘𝑟) = 𝐻)
12	11	oveqd 7422	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑥(2nd ‘𝑟)𝑦) = (𝑥𝐻𝑦))
13	12	eleq1d 2812	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((𝑥(2nd ‘𝑟)𝑦) ∈ 𝑖 ↔ (𝑥𝐻𝑦) ∈ 𝑖))
14	13	2ralbidv 3212	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥(2nd ‘𝑟)𝑦) ∈ 𝑖 ↔ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑖))
15	14	imbi1d 341	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥(2nd ‘𝑟)𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖)) ↔ (∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖))))
16	1, 15	raleqbidv 3336	. . . . 5 ⊢ (𝑟 = 𝑅 → (∀𝑏 ∈ (Idl‘𝑟)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥(2nd ‘𝑟)𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖)) ↔ ∀𝑏 ∈ (Idl‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖))))
17	1, 16	raleqbidv 3336	. . . 4 ⊢ (𝑟 = 𝑅 → (∀𝑎 ∈ (Idl‘𝑟)∀𝑏 ∈ (Idl‘𝑟)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥(2nd ‘𝑟)𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖)) ↔ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖))))
18	8, 17	anbi12d 630	. . 3 ⊢ (𝑟 = 𝑅 → ((𝑖 ≠ ran (1st ‘𝑟) ∧ ∀𝑎 ∈ (Idl‘𝑟)∀𝑏 ∈ (Idl‘𝑟)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥(2nd ‘𝑟)𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖))) ↔ (𝑖 ≠ 𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖)))))
19	1, 18	rabeqbidv 3443	. 2 ⊢ (𝑟 = 𝑅 → {𝑖 ∈ (Idl‘𝑟) ∣ (𝑖 ≠ ran (1st ‘𝑟) ∧ ∀𝑎 ∈ (Idl‘𝑟)∀𝑏 ∈ (Idl‘𝑟)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥(2nd ‘𝑟)𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖)))} = {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖 ≠ 𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖)))})
20		df-pridl 37392	. 2 ⊢ PrIdl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ (Idl‘𝑟) ∣ (𝑖 ≠ ran (1st ‘𝑟) ∧ ∀𝑎 ∈ (Idl‘𝑟)∀𝑏 ∈ (Idl‘𝑟)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥(2nd ‘𝑟)𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖)))})
21		fvex 6898	. . 3 ⊢ (Idl‘𝑅) ∈ V
22	21	rabex 5325	. 2 ⊢ {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖 ≠ 𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖)))} ∈ V
23	19, 20, 22	fvmpt 6992	1 ⊢ (𝑅 ∈ RingOps → (PrIdl‘𝑅) = {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖 ≠ 𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖)))})

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 844   = wceq 1533   ∈ wcel 2098   ≠ wne 2934  ∀wral 3055  {crab 3426   ⊆ wss 3943  ran crn 5670  ‘cfv 6537  (class class class)co 7405  1^st c1st 7972  2^nd c2nd 7973  RingOpscrngo 37275  Idlcidl 37388  PrIdlcpridl 37389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6489  df-fun 6539  df-fv 6545  df-ov 7408  df-pridl 37392

This theorem is referenced by:  ispridl  37415