Theorem rprmval 33536

Description: The prime elements of a ring 𝑅. (Contributed by Thierry Arnoux, 1-Jul-2024.)

Hypotheses

Ref	Expression
rprmval.b	⊢ 𝐵 = (Base‘𝑅)
rprmval.u	⊢ 𝑈 = (Unit‘𝑅)
rprmval.1	⊢ 0 = (0g‘𝑅)
rprmval.m	⊢ · = (.r‘𝑅)
rprmval.d	⊢ ∥ = (∥r‘𝑅)

Assertion

Ref	Expression
rprmval	⊢ (𝑅 ∈ 𝑉 → (RPrime‘𝑅) = {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦))})

Distinct variable groups:   0 ,𝑝   𝐵,𝑝   𝑅,𝑝,𝑥,𝑦   𝑈,𝑝

Allowed substitution hints:   𝐵(𝑥,𝑦)   ∥ (𝑥,𝑦,𝑝)   · (𝑥,𝑦,𝑝)   𝑈(𝑥,𝑦)   𝑉(𝑥,𝑦,𝑝)   0 (𝑥,𝑦)

Proof of Theorem rprmval

Dummy variables 𝑏 𝑟 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rprm 20398	. 2 ⊢ RPrime = (𝑟 ∈ V ↦ ⦋(Base‘𝑟) / 𝑏⦌{𝑝 ∈ (𝑏 ∖ ((Unit‘𝑟) ∪ {(0g‘𝑟)})) ∣ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 [(∥r‘𝑟) / 𝑑](𝑝𝑑(𝑥(.r‘𝑟)𝑦) → (𝑝𝑑𝑥 ∨ 𝑝𝑑𝑦))})
2		fvexd 6896	. . 3 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) ∈ V)
3		simpr 484	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → 𝑏 = (Base‘𝑟))
4		fveq2 6881	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
5	4	adantr 480	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (Base‘𝑟) = (Base‘𝑅))
6	3, 5	eqtrd 2771	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → 𝑏 = (Base‘𝑅))
7		rprmval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
8	6, 7	eqtr4di 2789	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → 𝑏 = 𝐵)
9		fveq2 6881	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
10		rprmval.u	. . . . . . . 8 ⊢ 𝑈 = (Unit‘𝑅)
11	9, 10	eqtr4di 2789	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
12		fveq2 6881	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
13		rprmval.1	. . . . . . . . 9 ⊢ 0 = (0g‘𝑅)
14	12, 13	eqtr4di 2789	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
15	14	sneqd 4618	. . . . . . 7 ⊢ (𝑟 = 𝑅 → {(0g‘𝑟)} = { 0 })
16	11, 15	uneq12d 4149	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((Unit‘𝑟) ∪ {(0g‘𝑟)}) = (𝑈 ∪ { 0 }))
17	16	adantr 480	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → ((Unit‘𝑟) ∪ {(0g‘𝑟)}) = (𝑈 ∪ { 0 }))
18	8, 17	difeq12d 4107	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑏 ∖ ((Unit‘𝑟) ∪ {(0g‘𝑟)})) = (𝐵 ∖ (𝑈 ∪ { 0 })))
19		fvexd 6896	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (∥r‘𝑟) ∈ V)
20		eqidd 2737	. . . . . . . . 9 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r‘𝑟)) → 𝑝 = 𝑝)
21		simpr 484	. . . . . . . . . . 11 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r‘𝑟)) → 𝑑 = (∥r‘𝑟))
22		fveq2 6881	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (∥r‘𝑟) = (∥r‘𝑅))
23	22	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r‘𝑟)) → (∥r‘𝑟) = (∥r‘𝑅))
24	21, 23	eqtrd 2771	. . . . . . . . . 10 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r‘𝑟)) → 𝑑 = (∥r‘𝑅))
25		rprmval.d	. . . . . . . . . 10 ⊢ ∥ = (∥r‘𝑅)
26	24, 25	eqtr4di 2789	. . . . . . . . 9 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r‘𝑟)) → 𝑑 = ∥ )
27		fveq2 6881	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
28		rprmval.m	. . . . . . . . . . . 12 ⊢ · = (.r‘𝑅)
29	27, 28	eqtr4di 2789	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · )
30	29	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r‘𝑟)) → (.r‘𝑟) = · )
31	30	oveqd 7427	. . . . . . . . 9 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r‘𝑟)) → (𝑥(.r‘𝑟)𝑦) = (𝑥 · 𝑦))
32	20, 26, 31	breq123d 5138	. . . . . . . 8 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r‘𝑟)) → (𝑝𝑑(𝑥(.r‘𝑟)𝑦) ↔ 𝑝 ∥ (𝑥 · 𝑦)))
33	26	breqd 5135	. . . . . . . . 9 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r‘𝑟)) → (𝑝𝑑𝑥 ↔ 𝑝 ∥ 𝑥))
34	26	breqd 5135	. . . . . . . . 9 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r‘𝑟)) → (𝑝𝑑𝑦 ↔ 𝑝 ∥ 𝑦))
35	33, 34	orbi12d 918	. . . . . . . 8 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r‘𝑟)) → ((𝑝𝑑𝑥 ∨ 𝑝𝑑𝑦) ↔ (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦)))
36	32, 35	imbi12d 344	. . . . . . 7 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r‘𝑟)) → ((𝑝𝑑(𝑥(.r‘𝑟)𝑦) → (𝑝𝑑𝑥 ∨ 𝑝𝑑𝑦)) ↔ (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦))))
37	19, 36	sbcied 3814	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → ([(∥r‘𝑟) / 𝑑](𝑝𝑑(𝑥(.r‘𝑟)𝑦) → (𝑝𝑑𝑥 ∨ 𝑝𝑑𝑦)) ↔ (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦))))
38	8, 37	raleqbidv 3329	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (∀𝑦 ∈ 𝑏 [(∥r‘𝑟) / 𝑑](𝑝𝑑(𝑥(.r‘𝑟)𝑦) → (𝑝𝑑𝑥 ∨ 𝑝𝑑𝑦)) ↔ ∀𝑦 ∈ 𝐵 (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦))))
39	8, 38	raleqbidv 3329	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 [(∥r‘𝑟) / 𝑑](𝑝𝑑(𝑥(.r‘𝑟)𝑦) → (𝑝𝑑𝑥 ∨ 𝑝𝑑𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦))))
40	18, 39	rabeqbidv 3439	. . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → {𝑝 ∈ (𝑏 ∖ ((Unit‘𝑟) ∪ {(0g‘𝑟)})) ∣ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 [(∥r‘𝑟) / 𝑑](𝑝𝑑(𝑥(.r‘𝑟)𝑦) → (𝑝𝑑𝑥 ∨ 𝑝𝑑𝑦))} = {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦))})
41	2, 40	csbied 3915	. 2 ⊢ (𝑟 = 𝑅 → ⦋(Base‘𝑟) / 𝑏⦌{𝑝 ∈ (𝑏 ∖ ((Unit‘𝑟) ∪ {(0g‘𝑟)})) ∣ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 [(∥r‘𝑟) / 𝑑](𝑝𝑑(𝑥(.r‘𝑟)𝑦) → (𝑝𝑑𝑥 ∨ 𝑝𝑑𝑦))} = {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦))})
42		elex 3485	. 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
43	7	fvexi 6895	. . . . 5 ⊢ 𝐵 ∈ V
44	43	difexi 5305	. . . 4 ⊢ (𝐵 ∖ (𝑈 ∪ { 0 })) ∈ V
45	44	rabex 5314	. . 3 ⊢ {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦))} ∈ V
46	45	a1i 11	. 2 ⊢ (𝑅 ∈ 𝑉 → {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦))} ∈ V)
47	1, 41, 42, 46	fvmptd3 7014	1 ⊢ (𝑅 ∈ 𝑉 → (RPrime‘𝑅) = {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦))})

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∀wral 3052  {crab 3420  Vcvv 3464  [wsbc 3770  ⦋csb 3879   ∖ cdif 3928   ∪ cun 3929  {csn 4606   class class class wbr 5124  ‘cfv 6536  (class class class)co 7410  Basecbs 17233  ._rcmulr 17277  0_gc0g 17458  ∥_rcdsr 20319  Unitcui 20320  RPrimecrpm 20397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6489  df-fun 6538  df-fv 6544  df-ov 7413  df-rprm 20398

This theorem is referenced by:  isrprm  33537