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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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