Theorem isrprm 33547

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 33546	. . 3 ⊢ (𝑅 ∈ 𝑉 → (RPrime‘𝑅) = {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦))})
7	6	eleq2d 2820	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑃 ∈ (RPrime‘𝑅) ↔ 𝑃 ∈ {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦))}))
8		breq1 5099	. . . . 5 ⊢ (𝑝 = 𝑃 → (𝑝 ∥ (𝑥 · 𝑦) ↔ 𝑃 ∥ (𝑥 · 𝑦)))
9		breq1 5099	. . . . . 6 ⊢ (𝑝 = 𝑃 → (𝑝 ∥ 𝑥 ↔ 𝑃 ∥ 𝑥))
10		breq1 5099	. . . . . 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 3198	. . 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 3644	. 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 1541   ∈ wcel 2113  ∀wral 3049  {crab 3397   ∖ cdif 3896   ∪ cun 3897  {csn 4578   class class class wbr 5096  ‘cfv 6490  (class class class)co 7356  Basecbs 17134  ._rcmulr 17176  0_gc0g 17357  ∥_rcdsr 20288  Unitcui 20289  RPrimecrpm 20366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-iota 6446  df-fun 6492  df-fv 6498  df-ov 7359  df-rprm 20367

This theorem is referenced by:  rprmcl  33548  rprmdvds  33549  rprmnz  33550  rprmnunit  33551  rsprprmprmidlb  33553  rprmirredb  33562