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Theorem isrprm 31073
 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 = (0g𝑅)
isrprm.4 = (∥r𝑅)
isrprm.5 · = (.r𝑅)
Assertion
Ref Expression
isrprm (𝑅𝑉 → (𝑃 ∈ (RPrime‘𝑅) ↔ (𝑃 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∧ ∀𝑥𝐵𝑦𝐵 (𝑃 (𝑥 · 𝑦) → (𝑃 𝑥𝑃 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.
StepHypRef Expression
1 isrprm.1 . . . 4 𝐵 = (Base‘𝑅)
2 isrprm.2 . . . 4 𝑈 = (Unit‘𝑅)
3 isrprm.3 . . . 4 0 = (0g𝑅)
4 isrprm.5 . . . 4 · = (.r𝑅)
5 isrprm.4 . . . 4 = (∥r𝑅)
61, 2, 3, 4, 5rprmval 31072 . . 3 (𝑅𝑉 → (RPrime‘𝑅) = {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥𝐵𝑦𝐵 (𝑝 (𝑥 · 𝑦) → (𝑝 𝑥𝑝 𝑦))})
76eleq2d 2878 . 2 (𝑅𝑉 → (𝑃 ∈ (RPrime‘𝑅) ↔ 𝑃 ∈ {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥𝐵𝑦𝐵 (𝑝 (𝑥 · 𝑦) → (𝑝 𝑥𝑝 𝑦))}))
8 breq1 5036 . . . . 5 (𝑝 = 𝑃 → (𝑝 (𝑥 · 𝑦) ↔ 𝑃 (𝑥 · 𝑦)))
9 breq1 5036 . . . . . 6 (𝑝 = 𝑃 → (𝑝 𝑥𝑃 𝑥))
10 breq1 5036 . . . . . 6 (𝑝 = 𝑃 → (𝑝 𝑦𝑃 𝑦))
119, 10orbi12d 916 . . . . 5 (𝑝 = 𝑃 → ((𝑝 𝑥𝑝 𝑦) ↔ (𝑃 𝑥𝑃 𝑦)))
128, 11imbi12d 348 . . . 4 (𝑝 = 𝑃 → ((𝑝 (𝑥 · 𝑦) → (𝑝 𝑥𝑝 𝑦)) ↔ (𝑃 (𝑥 · 𝑦) → (𝑃 𝑥𝑃 𝑦))))
13122ralbidv 3167 . . 3 (𝑝 = 𝑃 → (∀𝑥𝐵𝑦𝐵 (𝑝 (𝑥 · 𝑦) → (𝑝 𝑥𝑝 𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑃 (𝑥 · 𝑦) → (𝑃 𝑥𝑃 𝑦))))
1413elrab 3631 . 2 (𝑃 ∈ {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥𝐵𝑦𝐵 (𝑝 (𝑥 · 𝑦) → (𝑝 𝑥𝑝 𝑦))} ↔ (𝑃 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∧ ∀𝑥𝐵𝑦𝐵 (𝑃 (𝑥 · 𝑦) → (𝑃 𝑥𝑃 𝑦))))
157, 14syl6bb 290 1 (𝑅𝑉 → (𝑃 ∈ (RPrime‘𝑅) ↔ (𝑃 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∧ ∀𝑥𝐵𝑦𝐵 (𝑃 (𝑥 · 𝑦) → (𝑃 𝑥𝑃 𝑦)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2112  ∀wral 3109  {crab 3113   ∖ cdif 3881   ∪ cun 3882  {csn 4528   class class class wbr 5033  ‘cfv 6328  (class class class)co 7139  Basecbs 16478  .rcmulr 16561  0gc0g 16708  ∥rcdsr 19387  Unitcui 19388  RPrimecrpm 19461 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-rprm 19462 This theorem is referenced by: (None)
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