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Theorem isrprm 32317
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 𝐡 = (Baseβ€˜π‘…)
isrprm.2 π‘ˆ = (Unitβ€˜π‘…)
isrprm.3 0 = (0gβ€˜π‘…)
isrprm.4 βˆ₯ = (βˆ₯rβ€˜π‘…)
isrprm.5 Β· = (.rβ€˜π‘…)
Assertion
Ref Expression
isrprm (𝑅 ∈ 𝑉 β†’ (𝑃 ∈ (RPrimeβ€˜π‘…) ↔ (𝑃 ∈ (𝐡 βˆ– (π‘ˆ βˆͺ { 0 })) ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (𝑃 βˆ₯ (π‘₯ Β· 𝑦) β†’ (𝑃 βˆ₯ π‘₯ ∨ 𝑃 βˆ₯ 𝑦)))))
Distinct variable groups:   π‘₯,𝑃,𝑦   π‘₯,𝑅,𝑦
Allowed substitution hints:   𝐡(π‘₯,𝑦)   βˆ₯ (π‘₯,𝑦)   Β· (π‘₯,𝑦)   π‘ˆ(π‘₯,𝑦)   𝑉(π‘₯,𝑦)   0 (π‘₯,𝑦)

Proof of Theorem isrprm
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 isrprm.1 . . . 4 𝐡 = (Baseβ€˜π‘…)
2 isrprm.2 . . . 4 π‘ˆ = (Unitβ€˜π‘…)
3 isrprm.3 . . . 4 0 = (0gβ€˜π‘…)
4 isrprm.5 . . . 4 Β· = (.rβ€˜π‘…)
5 isrprm.4 . . . 4 βˆ₯ = (βˆ₯rβ€˜π‘…)
61, 2, 3, 4, 5rprmval 32316 . . 3 (𝑅 ∈ 𝑉 β†’ (RPrimeβ€˜π‘…) = {𝑝 ∈ (𝐡 βˆ– (π‘ˆ βˆͺ { 0 })) ∣ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (𝑝 βˆ₯ (π‘₯ Β· 𝑦) β†’ (𝑝 βˆ₯ π‘₯ ∨ 𝑝 βˆ₯ 𝑦))})
76eleq2d 2820 . 2 (𝑅 ∈ 𝑉 β†’ (𝑃 ∈ (RPrimeβ€˜π‘…) ↔ 𝑃 ∈ {𝑝 ∈ (𝐡 βˆ– (π‘ˆ βˆͺ { 0 })) ∣ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (𝑝 βˆ₯ (π‘₯ Β· 𝑦) β†’ (𝑝 βˆ₯ π‘₯ ∨ 𝑝 βˆ₯ 𝑦))}))
8 breq1 5112 . . . . 5 (𝑝 = 𝑃 β†’ (𝑝 βˆ₯ (π‘₯ Β· 𝑦) ↔ 𝑃 βˆ₯ (π‘₯ Β· 𝑦)))
9 breq1 5112 . . . . . 6 (𝑝 = 𝑃 β†’ (𝑝 βˆ₯ π‘₯ ↔ 𝑃 βˆ₯ π‘₯))
10 breq1 5112 . . . . . 6 (𝑝 = 𝑃 β†’ (𝑝 βˆ₯ 𝑦 ↔ 𝑃 βˆ₯ 𝑦))
119, 10orbi12d 918 . . . . 5 (𝑝 = 𝑃 β†’ ((𝑝 βˆ₯ π‘₯ ∨ 𝑝 βˆ₯ 𝑦) ↔ (𝑃 βˆ₯ π‘₯ ∨ 𝑃 βˆ₯ 𝑦)))
128, 11imbi12d 345 . . . 4 (𝑝 = 𝑃 β†’ ((𝑝 βˆ₯ (π‘₯ Β· 𝑦) β†’ (𝑝 βˆ₯ π‘₯ ∨ 𝑝 βˆ₯ 𝑦)) ↔ (𝑃 βˆ₯ (π‘₯ Β· 𝑦) β†’ (𝑃 βˆ₯ π‘₯ ∨ 𝑃 βˆ₯ 𝑦))))
13122ralbidv 3209 . . 3 (𝑝 = 𝑃 β†’ (βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (𝑝 βˆ₯ (π‘₯ Β· 𝑦) β†’ (𝑝 βˆ₯ π‘₯ ∨ 𝑝 βˆ₯ 𝑦)) ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (𝑃 βˆ₯ (π‘₯ Β· 𝑦) β†’ (𝑃 βˆ₯ π‘₯ ∨ 𝑃 βˆ₯ 𝑦))))
1413elrab 3649 . 2 (𝑃 ∈ {𝑝 ∈ (𝐡 βˆ– (π‘ˆ βˆͺ { 0 })) ∣ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (𝑝 βˆ₯ (π‘₯ Β· 𝑦) β†’ (𝑝 βˆ₯ π‘₯ ∨ 𝑝 βˆ₯ 𝑦))} ↔ (𝑃 ∈ (𝐡 βˆ– (π‘ˆ βˆͺ { 0 })) ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (𝑃 βˆ₯ (π‘₯ Β· 𝑦) β†’ (𝑃 βˆ₯ π‘₯ ∨ 𝑃 βˆ₯ 𝑦))))
157, 14bitrdi 287 1 (𝑅 ∈ 𝑉 β†’ (𝑃 ∈ (RPrimeβ€˜π‘…) ↔ (𝑃 ∈ (𝐡 βˆ– (π‘ˆ βˆͺ { 0 })) ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (𝑃 βˆ₯ (π‘₯ Β· 𝑦) β†’ (𝑃 βˆ₯ π‘₯ ∨ 𝑃 βˆ₯ 𝑦)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  {crab 3406   βˆ– cdif 3911   βˆͺ cun 3912  {csn 4590   class class class wbr 5109  β€˜cfv 6500  (class class class)co 7361  Basecbs 17091  .rcmulr 17142  0gc0g 17329  βˆ₯rcdsr 20075  Unitcui 20076  RPrimecrpm 20151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-ov 7364  df-rprm 20152
This theorem is referenced by: (None)
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