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Theorem rprmval 33524
Description: The prime elements of a ring 𝑅. (Contributed by Thierry Arnoux, 1-Jul-2024.)
Hypotheses
Ref Expression
rprmval.b 𝐵 = (Base‘𝑅)
rprmval.u 𝑈 = (Unit‘𝑅)
rprmval.1 0 = (0g𝑅)
rprmval.m · = (.r𝑅)
rprmval.d = (∥r𝑅)
Assertion
Ref Expression
rprmval (𝑅𝑉 → (RPrime‘𝑅) = {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥𝐵𝑦𝐵 (𝑝 (𝑥 · 𝑦) → (𝑝 𝑥𝑝 𝑦))})
Distinct variable groups:   0 ,𝑝   𝐵,𝑝   𝑅,𝑝,𝑥,𝑦   𝑈,𝑝
Allowed substitution hints:   𝐵(𝑥,𝑦)   (𝑥,𝑦,𝑝)   · (𝑥,𝑦,𝑝)   𝑈(𝑥,𝑦)   𝑉(𝑥,𝑦,𝑝)   0 (𝑥,𝑦)

Proof of Theorem rprmval
Dummy variables 𝑏 𝑟 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rprm 20450 . 2 RPrime = (𝑟 ∈ V ↦ (Base‘𝑟) / 𝑏{𝑝 ∈ (𝑏 ∖ ((Unit‘𝑟) ∪ {(0g𝑟)})) ∣ ∀𝑥𝑏𝑦𝑏 [(∥r𝑟) / 𝑑](𝑝𝑑(𝑥(.r𝑟)𝑦) → (𝑝𝑑𝑥𝑝𝑑𝑦))})
2 fvexd 6922 . . 3 (𝑟 = 𝑅 → (Base‘𝑟) ∈ V)
3 simpr 484 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → 𝑏 = (Base‘𝑟))
4 fveq2 6907 . . . . . . . 8 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
54adantr 480 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (Base‘𝑟) = (Base‘𝑅))
63, 5eqtrd 2775 . . . . . 6 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → 𝑏 = (Base‘𝑅))
7 rprmval.b . . . . . 6 𝐵 = (Base‘𝑅)
86, 7eqtr4di 2793 . . . . 5 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → 𝑏 = 𝐵)
9 fveq2 6907 . . . . . . . 8 (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
10 rprmval.u . . . . . . . 8 𝑈 = (Unit‘𝑅)
119, 10eqtr4di 2793 . . . . . . 7 (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
12 fveq2 6907 . . . . . . . . 9 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
13 rprmval.1 . . . . . . . . 9 0 = (0g𝑅)
1412, 13eqtr4di 2793 . . . . . . . 8 (𝑟 = 𝑅 → (0g𝑟) = 0 )
1514sneqd 4643 . . . . . . 7 (𝑟 = 𝑅 → {(0g𝑟)} = { 0 })
1611, 15uneq12d 4179 . . . . . 6 (𝑟 = 𝑅 → ((Unit‘𝑟) ∪ {(0g𝑟)}) = (𝑈 ∪ { 0 }))
1716adantr 480 . . . . 5 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → ((Unit‘𝑟) ∪ {(0g𝑟)}) = (𝑈 ∪ { 0 }))
188, 17difeq12d 4137 . . . 4 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (𝑏 ∖ ((Unit‘𝑟) ∪ {(0g𝑟)})) = (𝐵 ∖ (𝑈 ∪ { 0 })))
19 fvexd 6922 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (∥r𝑟) ∈ V)
20 eqidd 2736 . . . . . . . . 9 (((𝑟 = 𝑅𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r𝑟)) → 𝑝 = 𝑝)
21 simpr 484 . . . . . . . . . . 11 (((𝑟 = 𝑅𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r𝑟)) → 𝑑 = (∥r𝑟))
22 fveq2 6907 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (∥r𝑟) = (∥r𝑅))
2322ad2antrr 726 . . . . . . . . . . 11 (((𝑟 = 𝑅𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r𝑟)) → (∥r𝑟) = (∥r𝑅))
2421, 23eqtrd 2775 . . . . . . . . . 10 (((𝑟 = 𝑅𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r𝑟)) → 𝑑 = (∥r𝑅))
25 rprmval.d . . . . . . . . . 10 = (∥r𝑅)
2624, 25eqtr4di 2793 . . . . . . . . 9 (((𝑟 = 𝑅𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r𝑟)) → 𝑑 = )
27 fveq2 6907 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
28 rprmval.m . . . . . . . . . . . 12 · = (.r𝑅)
2927, 28eqtr4di 2793 . . . . . . . . . . 11 (𝑟 = 𝑅 → (.r𝑟) = · )
3029ad2antrr 726 . . . . . . . . . 10 (((𝑟 = 𝑅𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r𝑟)) → (.r𝑟) = · )
3130oveqd 7448 . . . . . . . . 9 (((𝑟 = 𝑅𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r𝑟)) → (𝑥(.r𝑟)𝑦) = (𝑥 · 𝑦))
3220, 26, 31breq123d 5162 . . . . . . . 8 (((𝑟 = 𝑅𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r𝑟)) → (𝑝𝑑(𝑥(.r𝑟)𝑦) ↔ 𝑝 (𝑥 · 𝑦)))
3326breqd 5159 . . . . . . . . 9 (((𝑟 = 𝑅𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r𝑟)) → (𝑝𝑑𝑥𝑝 𝑥))
3426breqd 5159 . . . . . . . . 9 (((𝑟 = 𝑅𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r𝑟)) → (𝑝𝑑𝑦𝑝 𝑦))
3533, 34orbi12d 918 . . . . . . . 8 (((𝑟 = 𝑅𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r𝑟)) → ((𝑝𝑑𝑥𝑝𝑑𝑦) ↔ (𝑝 𝑥𝑝 𝑦)))
3632, 35imbi12d 344 . . . . . . 7 (((𝑟 = 𝑅𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r𝑟)) → ((𝑝𝑑(𝑥(.r𝑟)𝑦) → (𝑝𝑑𝑥𝑝𝑑𝑦)) ↔ (𝑝 (𝑥 · 𝑦) → (𝑝 𝑥𝑝 𝑦))))
3719, 36sbcied 3837 . . . . . 6 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → ([(∥r𝑟) / 𝑑](𝑝𝑑(𝑥(.r𝑟)𝑦) → (𝑝𝑑𝑥𝑝𝑑𝑦)) ↔ (𝑝 (𝑥 · 𝑦) → (𝑝 𝑥𝑝 𝑦))))
388, 37raleqbidv 3344 . . . . 5 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (∀𝑦𝑏 [(∥r𝑟) / 𝑑](𝑝𝑑(𝑥(.r𝑟)𝑦) → (𝑝𝑑𝑥𝑝𝑑𝑦)) ↔ ∀𝑦𝐵 (𝑝 (𝑥 · 𝑦) → (𝑝 𝑥𝑝 𝑦))))
398, 38raleqbidv 3344 . . . 4 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (∀𝑥𝑏𝑦𝑏 [(∥r𝑟) / 𝑑](𝑝𝑑(𝑥(.r𝑟)𝑦) → (𝑝𝑑𝑥𝑝𝑑𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑝 (𝑥 · 𝑦) → (𝑝 𝑥𝑝 𝑦))))
4018, 39rabeqbidv 3452 . . 3 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → {𝑝 ∈ (𝑏 ∖ ((Unit‘𝑟) ∪ {(0g𝑟)})) ∣ ∀𝑥𝑏𝑦𝑏 [(∥r𝑟) / 𝑑](𝑝𝑑(𝑥(.r𝑟)𝑦) → (𝑝𝑑𝑥𝑝𝑑𝑦))} = {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥𝐵𝑦𝐵 (𝑝 (𝑥 · 𝑦) → (𝑝 𝑥𝑝 𝑦))})
412, 40csbied 3946 . 2 (𝑟 = 𝑅(Base‘𝑟) / 𝑏{𝑝 ∈ (𝑏 ∖ ((Unit‘𝑟) ∪ {(0g𝑟)})) ∣ ∀𝑥𝑏𝑦𝑏 [(∥r𝑟) / 𝑑](𝑝𝑑(𝑥(.r𝑟)𝑦) → (𝑝𝑑𝑥𝑝𝑑𝑦))} = {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥𝐵𝑦𝐵 (𝑝 (𝑥 · 𝑦) → (𝑝 𝑥𝑝 𝑦))})
42 elex 3499 . 2 (𝑅𝑉𝑅 ∈ V)
437fvexi 6921 . . . . 5 𝐵 ∈ V
4443difexi 5336 . . . 4 (𝐵 ∖ (𝑈 ∪ { 0 })) ∈ V
4544rabex 5345 . . 3 {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥𝐵𝑦𝐵 (𝑝 (𝑥 · 𝑦) → (𝑝 𝑥𝑝 𝑦))} ∈ V
4645a1i 11 . 2 (𝑅𝑉 → {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥𝐵𝑦𝐵 (𝑝 (𝑥 · 𝑦) → (𝑝 𝑥𝑝 𝑦))} ∈ V)
471, 41, 42, 46fvmptd3 7039 1 (𝑅𝑉 → (RPrime‘𝑅) = {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥𝐵𝑦𝐵 (𝑝 (𝑥 · 𝑦) → (𝑝 𝑥𝑝 𝑦))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847   = wceq 1537  wcel 2106  wral 3059  {crab 3433  Vcvv 3478  [wsbc 3791  csb 3908  cdif 3960  cun 3961  {csn 4631   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  .rcmulr 17299  0gc0g 17486  rcdsr 20371  Unitcui 20372  RPrimecrpm 20449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-rprm 20450
This theorem is referenced by:  isrprm  33525
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