Theorem isrprm 34049

Description: Property for 𝑃 to be a prime element in the ring 𝑅. (Contributed by Thierry Arnoux, 1-Jul-2024.)

Hypotheses

Ref	Expression
isrprm.1	⊢ 𝐵 = (Base‘𝑅)
isrprm.2	⊢ 𝑈 = (Unit‘𝑅)
isrprm.3	⊢ 0 = (0g‘𝑅)
isrprm.4	⊢ ∥ = (∥r‘𝑅)
isrprm.5	⊢ · = (.r‘𝑅)

Assertion

Ref	Expression
isrprm	⊢ (𝑅 ∈ 𝑉 → (𝑃 ∈ (RPrime‘𝑅) ↔ (𝑃 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑃 ∥ (𝑥 · 𝑦) → (𝑃 ∥ 𝑥 ∨ 𝑃 ∥ 𝑦)))))

Distinct variable groups:   𝑥,𝑃,𝑦   𝑥,𝑅,𝑦

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

Proof of Theorem isrprm

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isrprm.1	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
2		isrprm.2	. . . 4 ⊢ 𝑈 = (Unit‘𝑅)
3		isrprm.3	. . . 4 ⊢ 0 = (0g‘𝑅)
4		isrprm.5	. . . 4 ⊢ · = (.r‘𝑅)
5		isrprm.4	. . . 4 ⊢ ∥ = (∥r‘𝑅)
6	1, 2, 3, 4, 5	rprmval 34048	. . 3 ⊢ (𝑅 ∈ 𝑉 → (RPrime‘𝑅) = {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦))})
7	6	eleq2d 2847	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑃 ∈ (RPrime‘𝑅) ↔ 𝑃 ∈ {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦))}))
8		breq1 5106	. . . . 5 ⊢ (𝑝 = 𝑃 → (𝑝 ∥ (𝑥 · 𝑦) ↔ 𝑃 ∥ (𝑥 · 𝑦)))
9		breq1 5106	. . . . . 6 ⊢ (𝑝 = 𝑃 → (𝑝 ∥ 𝑥 ↔ 𝑃 ∥ 𝑥))
10		breq1 5106	. . . . . 6 ⊢ (𝑝 = 𝑃 → (𝑝 ∥ 𝑦 ↔ 𝑃 ∥ 𝑦))
11	9, 10	orbi12d 932	. . . . 5 ⊢ (𝑝 = 𝑃 → ((𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦) ↔ (𝑃 ∥ 𝑥 ∨ 𝑃 ∥ 𝑦)))
12	8, 11	imbi12d 347	. . . 4 ⊢ (𝑝 = 𝑃 → ((𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦)) ↔ (𝑃 ∥ (𝑥 · 𝑦) → (𝑃 ∥ 𝑥 ∨ 𝑃 ∥ 𝑦))))
13	12	2ralbidv 3227	. . 3 ⊢ (𝑝 = 𝑃 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑃 ∥ (𝑥 · 𝑦) → (𝑃 ∥ 𝑥 ∨ 𝑃 ∥ 𝑦))))
14	13	elrab 3645	. 2 ⊢ (𝑃 ∈ {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦))} ↔ (𝑃 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑃 ∥ (𝑥 · 𝑦) → (𝑃 ∥ 𝑥 ∨ 𝑃 ∥ 𝑦))))
15	7, 14	bitrdi 290	1 ⊢ (𝑅 ∈ 𝑉 → (𝑃 ∈ (RPrime‘𝑅) ↔ (𝑃 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑃 ∥ (𝑥 · 𝑦) → (𝑃 ∥ 𝑥 ∨ 𝑃 ∥ 𝑦)))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∀wral 3077  {crab 3413   ∖ cdif 3896   ∪ cun 3897  {csn 4584   class class class wbr 5103  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  ._rcmulr 17429  0_gc0g 17610  ∥_rcdsr 20584  Unitcui 20585  RPrimecrpm 20662

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7423  df-rprm 20663

This theorem is used by:  rprmcl  34050  rprmdvds  34051  rprmnz  34052  rprmnunit  34053  rsprprmprmidlb  34055  rprmirredb  34064