Theorem isrprm 33577

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 33576	. . 3 ⊢ (𝑅 ∈ 𝑉 → (RPrime‘𝑅) = {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦))})
7	6	eleq2d 2822	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑃 ∈ (RPrime‘𝑅) ↔ 𝑃 ∈ {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦))}))
8		breq1 5088	. . . . 5 ⊢ (𝑝 = 𝑃 → (𝑝 ∥ (𝑥 · 𝑦) ↔ 𝑃 ∥ (𝑥 · 𝑦)))
9		breq1 5088	. . . . . 6 ⊢ (𝑝 = 𝑃 → (𝑝 ∥ 𝑥 ↔ 𝑃 ∥ 𝑥))
10		breq1 5088	. . . . . 6 ⊢ (𝑝 = 𝑃 → (𝑝 ∥ 𝑦 ↔ 𝑃 ∥ 𝑦))
11	9, 10	orbi12d 919	. . . . 5 ⊢ (𝑝 = 𝑃 → ((𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦) ↔ (𝑃 ∥ 𝑥 ∨ 𝑃 ∥ 𝑦)))
12	8, 11	imbi12d 344	. . . 4 ⊢ (𝑝 = 𝑃 → ((𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦)) ↔ (𝑃 ∥ (𝑥 · 𝑦) → (𝑃 ∥ 𝑥 ∨ 𝑃 ∥ 𝑦))))
13	12	2ralbidv 3201	. . 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 3634	. 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 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114  ∀wral 3051  {crab 3389   ∖ cdif 3886   ∪ cun 3887  {csn 4567   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  ._rcmulr 17221  0_gc0g 17402  ∥_rcdsr 20334  Unitcui 20335  RPrimecrpm 20412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fv 6506  df-ov 7370  df-rprm 20413

This theorem is referenced by:  rprmcl  33578  rprmdvds  33579  rprmnz  33580  rprmnunit  33581  rsprprmprmidlb  33583  rprmirredb  33592