Theorem isrprm 31665

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 31664	. . 3 ⊢ (𝑅 ∈ 𝑉 → (RPrime‘𝑅) = {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦))})
7	6	eleq2d 2824	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑃 ∈ (RPrime‘𝑅) ↔ 𝑃 ∈ {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦))}))
8		breq1 5077	. . . . 5 ⊢ (𝑝 = 𝑃 → (𝑝 ∥ (𝑥 · 𝑦) ↔ 𝑃 ∥ (𝑥 · 𝑦)))
9		breq1 5077	. . . . . 6 ⊢ (𝑝 = 𝑃 → (𝑝 ∥ 𝑥 ↔ 𝑃 ∥ 𝑥))
10		breq1 5077	. . . . . 6 ⊢ (𝑝 = 𝑃 → (𝑝 ∥ 𝑦 ↔ 𝑃 ∥ 𝑦))
11	9, 10	orbi12d 916	. . . . 5 ⊢ (𝑝 = 𝑃 → ((𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦) ↔ (𝑃 ∥ 𝑥 ∨ 𝑃 ∥ 𝑦)))
12	8, 11	imbi12d 345	. . . 4 ⊢ (𝑝 = 𝑃 → ((𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦)) ↔ (𝑃 ∥ (𝑥 · 𝑦) → (𝑃 ∥ 𝑥 ∨ 𝑃 ∥ 𝑦))))
13	12	2ralbidv 3129	. . 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 3624	. 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 287	1 ⊢ (𝑅 ∈ 𝑉 → (𝑃 ∈ (RPrime‘𝑅) ↔ (𝑃 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑃 ∥ (𝑥 · 𝑦) → (𝑃 ∥ 𝑥 ∨ 𝑃 ∥ 𝑦)))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {crab 3068   ∖ cdif 3884   ∪ cun 3885  {csn 4561   class class class wbr 5074  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  ._rcmulr 16963  0_gc0g 17150  ∥_rcdsr 19880  Unitcui 19881  RPrimecrpm 19954

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-rprm 19955

This theorem is referenced by: (None)