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Theorem rprmval 32907
Description: The prime elements of a ring 𝑅. (Contributed by Thierry Arnoux, 1-Jul-2024.)
Hypotheses
Ref Expression
rprmval.b 𝐡 = (Baseβ€˜π‘…)
rprmval.u π‘ˆ = (Unitβ€˜π‘…)
rprmval.1 0 = (0gβ€˜π‘…)
rprmval.m Β· = (.rβ€˜π‘…)
rprmval.d βˆ₯ = (βˆ₯rβ€˜π‘…)
Assertion
Ref Expression
rprmval (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))})
Distinct variable groups:   0 ,𝑝   𝐡,𝑝   𝑅,𝑝,π‘₯,𝑦   π‘ˆ,𝑝
Allowed substitution hints:   𝐡(π‘₯,𝑦)   βˆ₯ (π‘₯,𝑦,𝑝)   Β· (π‘₯,𝑦,𝑝)   π‘ˆ(π‘₯,𝑦)   𝑉(π‘₯,𝑦,𝑝)   0 (π‘₯,𝑦)
Proof of Theorem rprmval
Dummy variables 𝑏 π‘Ÿ 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rprm 20324 . 2 RPrime = (π‘Ÿ ∈ V ↦ ⦋(Baseβ€˜π‘Ÿ) / π‘β¦Œ{𝑝 ∈ (𝑏 βˆ– ((Unitβ€˜π‘Ÿ) βˆͺ {(0gβ€˜π‘Ÿ)})) ∣ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 [(βˆ₯rβ€˜π‘Ÿ) / 𝑑](𝑝𝑑(π‘₯(.rβ€˜π‘Ÿ)𝑦) β†’ (𝑝𝑑π‘₯ ∨ 𝑝𝑑𝑦))})
2 fvexd 6905 . . 3 (π‘Ÿ = 𝑅 β†’ (Baseβ€˜π‘Ÿ) ∈ V)
3 simpr 483 . . . . . . 7 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (Baseβ€˜π‘Ÿ)) β†’ 𝑏 = (Baseβ€˜π‘Ÿ))
4 fveq2 6890 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (Baseβ€˜π‘Ÿ) = (Baseβ€˜π‘…))
54adantr 479 . . . . . . 7 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (Baseβ€˜π‘Ÿ)) β†’ (Baseβ€˜π‘Ÿ) = (Baseβ€˜π‘…))
63, 5eqtrd 2770 . . . . . 6 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (Baseβ€˜π‘Ÿ)) β†’ 𝑏 = (Baseβ€˜π‘…))
7 rprmval.b . . . . . 6 𝐡 = (Baseβ€˜π‘…)
86, 7eqtr4di 2788 . . . . 5 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (Baseβ€˜π‘Ÿ)) β†’ 𝑏 = 𝐡)
9 fveq2 6890 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (Unitβ€˜π‘Ÿ) = (Unitβ€˜π‘…))
10 rprmval.u . . . . . . . 8 π‘ˆ = (Unitβ€˜π‘…)
119, 10eqtr4di 2788 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (Unitβ€˜π‘Ÿ) = π‘ˆ)
12 fveq2 6890 . . . . . . . . 9 (π‘Ÿ = 𝑅 β†’ (0gβ€˜π‘Ÿ) = (0gβ€˜π‘…))
13 rprmval.1 . . . . . . . . 9 0 = (0gβ€˜π‘…)
1412, 13eqtr4di 2788 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (0gβ€˜π‘Ÿ) = 0 )
1514sneqd 4639 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ {(0gβ€˜π‘Ÿ)} = { 0 })
1611, 15uneq12d 4163 . . . . . 6 (π‘Ÿ = 𝑅 β†’ ((Unitβ€˜π‘Ÿ) βˆͺ {(0gβ€˜π‘Ÿ)}) = (π‘ˆ βˆͺ { 0 }))
1716adantr 479 . . . . 5 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (Baseβ€˜π‘Ÿ)) β†’ ((Unitβ€˜π‘Ÿ) βˆͺ {(0gβ€˜π‘Ÿ)}) = (π‘ˆ βˆͺ { 0 }))
188, 17difeq12d 4122 . . . 4 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (Baseβ€˜π‘Ÿ)) β†’ (𝑏 βˆ– ((Unitβ€˜π‘Ÿ) βˆͺ {(0gβ€˜π‘Ÿ)})) = (𝐡 βˆ– (π‘ˆ βˆͺ { 0 })))
19 fvexd 6905 . . . . . . 7 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (Baseβ€˜π‘Ÿ)) β†’ (βˆ₯rβ€˜π‘Ÿ) ∈ V)
20 eqidd 2731 . . . . . . . . 9 (((π‘Ÿ = 𝑅 ∧ 𝑏 = (Baseβ€˜π‘Ÿ)) ∧ 𝑑 = (βˆ₯rβ€˜π‘Ÿ)) β†’ 𝑝 = 𝑝)
21 simpr 483 . . . . . . . . . . 11 (((π‘Ÿ = 𝑅 ∧ 𝑏 = (Baseβ€˜π‘Ÿ)) ∧ 𝑑 = (βˆ₯rβ€˜π‘Ÿ)) β†’ 𝑑 = (βˆ₯rβ€˜π‘Ÿ))
22 fveq2 6890 . . . . . . . . . . . 12 (π‘Ÿ = 𝑅 β†’ (βˆ₯rβ€˜π‘Ÿ) = (βˆ₯rβ€˜π‘…))
2322ad2antrr 722 . . . . . . . . . . 11 (((π‘Ÿ = 𝑅 ∧ 𝑏 = (Baseβ€˜π‘Ÿ)) ∧ 𝑑 = (βˆ₯rβ€˜π‘Ÿ)) β†’ (βˆ₯rβ€˜π‘Ÿ) = (βˆ₯rβ€˜π‘…))
2421, 23eqtrd 2770 . . . . . . . . . 10 (((π‘Ÿ = 𝑅 ∧ 𝑏 = (Baseβ€˜π‘Ÿ)) ∧ 𝑑 = (βˆ₯rβ€˜π‘Ÿ)) β†’ 𝑑 = (βˆ₯rβ€˜π‘…))
25 rprmval.d . . . . . . . . . 10 βˆ₯ = (βˆ₯rβ€˜π‘…)
2624, 25eqtr4di 2788 . . . . . . . . 9 (((π‘Ÿ = 𝑅 ∧ 𝑏 = (Baseβ€˜π‘Ÿ)) ∧ 𝑑 = (βˆ₯rβ€˜π‘Ÿ)) β†’ 𝑑 = βˆ₯ )
27 fveq2 6890 . . . . . . . . . . . 12 (π‘Ÿ = 𝑅 β†’ (.rβ€˜π‘Ÿ) = (.rβ€˜π‘…))
28 rprmval.m . . . . . . . . . . . 12 Β· = (.rβ€˜π‘…)
2927, 28eqtr4di 2788 . . . . . . . . . . 11 (π‘Ÿ = 𝑅 β†’ (.rβ€˜π‘Ÿ) = Β· )
3029ad2antrr 722 . . . . . . . . . 10 (((π‘Ÿ = 𝑅 ∧ 𝑏 = (Baseβ€˜π‘Ÿ)) ∧ 𝑑 = (βˆ₯rβ€˜π‘Ÿ)) β†’ (.rβ€˜π‘Ÿ) = Β· )
3130oveqd 7428 . . . . . . . . 9 (((π‘Ÿ = 𝑅 ∧ 𝑏 = (Baseβ€˜π‘Ÿ)) ∧ 𝑑 = (βˆ₯rβ€˜π‘Ÿ)) β†’ (π‘₯(.rβ€˜π‘Ÿ)𝑦) = (π‘₯ Β· 𝑦))
3220, 26, 31breq123d 5161 . . . . . . . 8 (((π‘Ÿ = 𝑅 ∧ 𝑏 = (Baseβ€˜π‘Ÿ)) ∧ 𝑑 = (βˆ₯rβ€˜π‘Ÿ)) β†’ (𝑝𝑑(π‘₯(.rβ€˜π‘Ÿ)𝑦) ↔ 𝑝 βˆ₯ (π‘₯ Β· 𝑦)))
3326breqd 5158 . . . . . . . . 9 (((π‘Ÿ = 𝑅 ∧ 𝑏 = (Baseβ€˜π‘Ÿ)) ∧ 𝑑 = (βˆ₯rβ€˜π‘Ÿ)) β†’ (𝑝𝑑π‘₯ ↔ 𝑝 βˆ₯ π‘₯))
3426breqd 5158 . . . . . . . . 9 (((π‘Ÿ = 𝑅 ∧ 𝑏 = (Baseβ€˜π‘Ÿ)) ∧ 𝑑 = (βˆ₯rβ€˜π‘Ÿ)) β†’ (𝑝𝑑𝑦 ↔ 𝑝 βˆ₯ 𝑦))
3533, 34orbi12d 915 . . . . . . . 8 (((π‘Ÿ = 𝑅 ∧ 𝑏 = (Baseβ€˜π‘Ÿ)) ∧ 𝑑 = (βˆ₯rβ€˜π‘Ÿ)) β†’ ((𝑝𝑑π‘₯ ∨ 𝑝𝑑𝑦) ↔ (𝑝 βˆ₯ π‘₯ ∨ 𝑝 βˆ₯ 𝑦)))
3632, 35imbi12d 343 . . . . . . 7 (((π‘Ÿ = 𝑅 ∧ 𝑏 = (Baseβ€˜π‘Ÿ)) ∧ 𝑑 = (βˆ₯rβ€˜π‘Ÿ)) β†’ ((𝑝𝑑(π‘₯(.rβ€˜π‘Ÿ)𝑦) β†’ (𝑝𝑑π‘₯ ∨ 𝑝𝑑𝑦)) ↔ (𝑝 βˆ₯ (π‘₯ Β· 𝑦) β†’ (𝑝 βˆ₯ π‘₯ ∨ 𝑝 βˆ₯ 𝑦))))
3719, 36sbcied 3821 . . . . . 6 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (Baseβ€˜π‘Ÿ)) β†’ ([(βˆ₯rβ€˜π‘Ÿ) / 𝑑](𝑝𝑑(π‘₯(.rβ€˜π‘Ÿ)𝑦) β†’ (𝑝𝑑π‘₯ ∨ 𝑝𝑑𝑦)) ↔ (𝑝 βˆ₯ (π‘₯ Β· 𝑦) β†’ (𝑝 βˆ₯ π‘₯ ∨ 𝑝 βˆ₯ 𝑦))))
388, 37raleqbidv 3340 . . . . 5 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (Baseβ€˜π‘Ÿ)) β†’ (βˆ€π‘¦ ∈ 𝑏 [(βˆ₯rβ€˜π‘Ÿ) / 𝑑](𝑝𝑑(π‘₯(.rβ€˜π‘Ÿ)𝑦) β†’ (𝑝𝑑π‘₯ ∨ 𝑝𝑑𝑦)) ↔ βˆ€π‘¦ ∈ 𝐡 (𝑝 βˆ₯ (π‘₯ Β· 𝑦) β†’ (𝑝 βˆ₯ π‘₯ ∨ 𝑝 βˆ₯ 𝑦))))
398, 38raleqbidv 3340 . . . 4 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (Baseβ€˜π‘Ÿ)) β†’ (βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 [(βˆ₯rβ€˜π‘Ÿ) / 𝑑](𝑝𝑑(π‘₯(.rβ€˜π‘Ÿ)𝑦) β†’ (𝑝𝑑π‘₯ ∨ 𝑝𝑑𝑦)) ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (𝑝 βˆ₯ (π‘₯ Β· 𝑦) β†’ (𝑝 βˆ₯ π‘₯ ∨ 𝑝 βˆ₯ 𝑦))))
4018, 39rabeqbidv 3447 . . 3 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (Baseβ€˜π‘Ÿ)) β†’ {𝑝 ∈ (𝑏 βˆ– ((Unitβ€˜π‘Ÿ) βˆͺ {(0gβ€˜π‘Ÿ)})) ∣ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 [(βˆ₯rβ€˜π‘Ÿ) / 𝑑](𝑝𝑑(π‘₯(.rβ€˜π‘Ÿ)𝑦) β†’ (𝑝𝑑π‘₯ ∨ 𝑝𝑑𝑦))} = {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))})
412, 40csbied 3930 . 2 (π‘Ÿ = 𝑅 β†’ ⦋(Baseβ€˜π‘Ÿ) / π‘β¦Œ{𝑝 ∈ (𝑏 βˆ– ((Unitβ€˜π‘Ÿ) βˆͺ {(0gβ€˜π‘Ÿ)})) ∣ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 [(βˆ₯rβ€˜π‘Ÿ) / 𝑑](𝑝𝑑(π‘₯(.rβ€˜π‘Ÿ)𝑦) β†’ (𝑝𝑑π‘₯ ∨ 𝑝𝑑𝑦))} = {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))})
42 elex 3491 . 2 (𝑅 ∈ 𝑉 β†’ 𝑅 ∈ V)
437fvexi 6904 . . . . 5 𝐡 ∈ V
4443difexi 5327 . . . 4 (𝐡 βˆ– (π‘ˆ βˆͺ { 0 })) ∈ V
4544rabex 5331 . . 3 {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))} ∈ V
4645a1i 11 . 2 (𝑅 ∈ 𝑉 β†’ {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))} ∈ V)
471, 41, 42, 46fvmptd3 7020 1 (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))})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∨ wo 843   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  {crab 3430  Vcvv 3472  [wsbc 3776  β¦‹csb 3892   βˆ– cdif 3944   βˆͺ cun 3945  {csn 4627   class class class wbr 5147  β€˜cfv 6542  (class class class)co 7411  Basecbs 17148  .rcmulr 17202  0gc0g 17389  βˆ₯rcdsr 20245  Unitcui 20246  RPrimecrpm 20323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7414  df-rprm 20324
This theorem is referenced by:  isrprm  32908
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