Theorem pridlval 38628

Description: Obsolete theorem, use prmidlval 21441 instead. The class of prime ideals of a ring 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

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 6881	. . 3 ⊢ (𝑟 = 𝑅 → (Idl‘𝑟) = (Idl‘𝑅))
2		fveq2 6881	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = (1st ‘𝑅))
3		pridlval.1	. . . . . . . 8 ⊢ 𝐺 = (1st ‘𝑅)
4	2, 3	eqtr4di 2814	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = 𝐺)
5	4	rneqd 5928	. . . . . 6 ⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = ran 𝐺)
6		pridlval.3	. . . . . 6 ⊢ 𝑋 = ran 𝐺
7	5, 6	eqtr4di 2814	. . . . 5 ⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = 𝑋)
8	7	neeq2d 3016	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑖 ≠ ran (1st ‘𝑟) ↔ 𝑖 ≠ 𝑋))
9		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (2nd ‘𝑟) = (2nd ‘𝑅))
10		pridlval.2	. . . . . . . . . . 11 ⊢ 𝐻 = (2nd ‘𝑅)
11	9, 10	eqtr4di 2814	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (2nd ‘𝑟) = 𝐻)
12	11	oveqd 7427	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑥(2nd ‘𝑟)𝑦) = (𝑥𝐻𝑦))
13	12	eleq1d 2846	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((𝑥(2nd ‘𝑟)𝑦) ∈ 𝑖 ↔ (𝑥𝐻𝑦) ∈ 𝑖))
14	13	2ralbidv 3227	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥(2nd ‘𝑟)𝑦) ∈ 𝑖 ↔ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑖))
15	14	imbi1d 344	. . . . . 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 643	. . 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 3432	. 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 38606	. 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 6894	. . 3 ⊢ (Idl‘𝑅) ∈ V
22	21	rabex 5309	. 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 6989	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 400   ∨ wo 860   = wceq 1568   ∈ wcel 2141   ≠ wne 2956  ∀wral 3077  {crab 3414   ⊆ wss 3904  ran crn 5662  ‘cfv 6536  (class class class)co 7410  1^st c1st 7983  2^nd c2nd 7984  RingOpscrngo 38489  Idlcidl 38602  PrIdlcpridl 38603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-pridl 38606

This theorem is referenced by:  ispridl  38629