Theorem rprmval 31170

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 19519	. 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 6666	. . 3 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) ∈ V)
3		simpr 489	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → 𝑏 = (Base‘𝑟))
4		fveq2 6651	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
5	4	adantr 485	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (Base‘𝑟) = (Base‘𝑅))
6	3, 5	eqtrd 2794	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → 𝑏 = (Base‘𝑅))
7		rprmval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
8	6, 7	eqtr4di 2812	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → 𝑏 = 𝐵)
9		fveq2 6651	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
10		rprmval.u	. . . . . . . 8 ⊢ 𝑈 = (Unit‘𝑅)
11	9, 10	eqtr4di 2812	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
12		fveq2 6651	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
13		rprmval.1	. . . . . . . . 9 ⊢ 0 = (0g‘𝑅)
14	12, 13	eqtr4di 2812	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
15	14	sneqd 4527	. . . . . . 7 ⊢ (𝑟 = 𝑅 → {(0g‘𝑟)} = { 0 })
16	11, 15	uneq12d 4065	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((Unit‘𝑟) ∪ {(0g‘𝑟)}) = (𝑈 ∪ { 0 }))
17	16	adantr 485	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → ((Unit‘𝑟) ∪ {(0g‘𝑟)}) = (𝑈 ∪ { 0 }))
18	8, 17	difeq12d 4025	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑏 ∖ ((Unit‘𝑟) ∪ {(0g‘𝑟)})) = (𝐵 ∖ (𝑈 ∪ { 0 })))
19		fvexd 6666	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (∥r‘𝑟) ∈ V)
20		eqidd 2760	. . . . . . . . 9 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r‘𝑟)) → 𝑝 = 𝑝)
21		simpr 489	. . . . . . . . . . 11 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r‘𝑟)) → 𝑑 = (∥r‘𝑟))
22		fveq2 6651	. . . . . . . . . . . 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 2794	. . . . . . . . . 10 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r‘𝑟)) → 𝑑 = (∥r‘𝑅))
25		rprmval.d	. . . . . . . . . 10 ⊢ ∥ = (∥r‘𝑅)
26	24, 25	eqtr4di 2812	. . . . . . . . 9 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r‘𝑟)) → 𝑑 = ∥ )
27		fveq2 6651	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
28		rprmval.m	. . . . . . . . . . . 12 ⊢ · = (.r‘𝑅)
29	27, 28	eqtr4di 2812	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · )
30	29	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r‘𝑟)) → (.r‘𝑟) = · )
31	30	oveqd 7160	. . . . . . . . 9 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r‘𝑟)) → (𝑥(.r‘𝑟)𝑦) = (𝑥 · 𝑦))
32	20, 26, 31	breq123d 5039	. . . . . . . 8 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r‘𝑟)) → (𝑝𝑑(𝑥(.r‘𝑟)𝑦) ↔ 𝑝 ∥ (𝑥 · 𝑦)))
33	26	breqd 5036	. . . . . . . . 9 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r‘𝑟)) → (𝑝𝑑𝑥 ↔ 𝑝 ∥ 𝑥))
34	26	breqd 5036	. . . . . . . . 9 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r‘𝑟)) → (𝑝𝑑𝑦 ↔ 𝑝 ∥ 𝑦))
35	33, 34	orbi12d 917	. . . . . . . 8 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r‘𝑟)) → ((𝑝𝑑𝑥 ∨ 𝑝𝑑𝑦) ↔ (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦)))
36	32, 35	imbi12d 349	. . . . . . 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 3735	. . . . . 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 3317	. . . . 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 3317	. . . 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 3396	. . 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 3837	. 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 3427	. 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
43	7	fvexi 6665	. . . . 5 ⊢ 𝐵 ∈ V
44	43	difexi 5191	. . . 4 ⊢ (𝐵 ∖ (𝑈 ∪ { 0 })) ∈ V
45	44	rabex 5195	. . 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 6775	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 400   ∨ wo 845   = wceq 1539   ∈ wcel 2112  ∀wral 3068  {crab 3072  Vcvv 3407  [wsbc 3693  ⦋csb 3801   ∖ cdif 3851   ∪ cun 3852  {csn 4515   class class class wbr 5025  ‘cfv 6328  (class class class)co 7143  Basecbs 16526  ._rcmulr 16609  0_gc0g 16756  ∥_rcdsr 19444  Unitcui 19445  RPrimecrpm 19518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5162  ax-nul 5169  ax-pr 5291

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2899  df-ral 3073  df-rex 3074  df-rab 3077  df-v 3409  df-sbc 3694  df-csb 3802  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-nul 4222  df-if 4414  df-sn 4516  df-pr 4518  df-op 4522  df-uni 4792  df-br 5026  df-opab 5088  df-mpt 5106  df-id 5423  df-xp 5523  df-rel 5524  df-cnv 5525  df-co 5526  df-dm 5527  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7146  df-rprm 19519

This theorem is referenced by:  isrprm  31171