Theorem isrprm 33488

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 33487	. . 3 ⊢ (𝑅 ∈ 𝑉 → (RPrime‘𝑅) = {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦))})
7	6	eleq2d 2814	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑃 ∈ (RPrime‘𝑅) ↔ 𝑃 ∈ {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦))}))
8		breq1 5110	. . . . 5 ⊢ (𝑝 = 𝑃 → (𝑝 ∥ (𝑥 · 𝑦) ↔ 𝑃 ∥ (𝑥 · 𝑦)))
9		breq1 5110	. . . . . 6 ⊢ (𝑝 = 𝑃 → (𝑝 ∥ 𝑥 ↔ 𝑃 ∥ 𝑥))
10		breq1 5110	. . . . . 6 ⊢ (𝑝 = 𝑃 → (𝑝 ∥ 𝑦 ↔ 𝑃 ∥ 𝑦))
11	9, 10	orbi12d 918	. . . . 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 3659	. 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 847   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3405   ∖ cdif 3911   ∪ cun 3912  {csn 4589   class class class wbr 5107  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  ._rcmulr 17221  0_gc0g 17402  ∥_rcdsr 20263  Unitcui 20264  RPrimecrpm 20341

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 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-rprm 20342

This theorem is referenced by:  rprmcl  33489  rprmdvds  33490  rprmnz  33491  rprmnunit  33492  rsprprmprmidlb  33494  rprmirredb  33503