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Theorem isrprm 32084
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 32083 . . 3 (𝑅𝑉 → (RPrime‘𝑅) = {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥𝐵𝑦𝐵 (𝑝 (𝑥 · 𝑦) → (𝑝 𝑥𝑝 𝑦))})
76eleq2d 2823 . 2 (𝑅𝑉 → (𝑃 ∈ (RPrime‘𝑅) ↔ 𝑃 ∈ {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥𝐵𝑦𝐵 (𝑝 (𝑥 · 𝑦) → (𝑝 𝑥𝑝 𝑦))}))
8 breq1 5106 . . . . 5 (𝑝 = 𝑃 → (𝑝 (𝑥 · 𝑦) ↔ 𝑃 (𝑥 · 𝑦)))
9 breq1 5106 . . . . . 6 (𝑝 = 𝑃 → (𝑝 𝑥𝑃 𝑥))
10 breq1 5106 . . . . . 6 (𝑝 = 𝑃 → (𝑝 𝑦𝑃 𝑦))
119, 10orbi12d 917 . . . . 5 (𝑝 = 𝑃 → ((𝑝 𝑥𝑝 𝑦) ↔ (𝑃 𝑥𝑃 𝑦)))
128, 11imbi12d 344 . . . 4 (𝑝 = 𝑃 → ((𝑝 (𝑥 · 𝑦) → (𝑝 𝑥𝑝 𝑦)) ↔ (𝑃 (𝑥 · 𝑦) → (𝑃 𝑥𝑃 𝑦))))
13122ralbidv 3210 . . 3 (𝑝 = 𝑃 → (∀𝑥𝐵𝑦𝐵 (𝑝 (𝑥 · 𝑦) → (𝑝 𝑥𝑝 𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑃 (𝑥 · 𝑦) → (𝑃 𝑥𝑃 𝑦))))
1413elrab 3643 . 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, 14bitrdi 286 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 205  wa 396  wo 845   = wceq 1541  wcel 2106  wral 3062  {crab 3405  cdif 3905  cun 3906  {csn 4584   class class class wbr 5103  cfv 6493  (class class class)co 7351  Basecbs 17043  .rcmulr 17094  0gc0g 17281  rcdsr 20020  Unitcui 20021  RPrimecrpm 20094
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6445  df-fun 6495  df-fv 6501  df-ov 7354  df-rprm 20095
This theorem is referenced by: (None)
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