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Theorem rprmval 33544
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 20433 . 2 RPrime = (𝑟 ∈ V ↦ (Base‘𝑟) / 𝑏{𝑝 ∈ (𝑏 ∖ ((Unit‘𝑟) ∪ {(0g𝑟)})) ∣ ∀𝑥𝑏𝑦𝑏 [(∥r𝑟) / 𝑑](𝑝𝑑(𝑥(.r𝑟)𝑦) → (𝑝𝑑𝑥𝑝𝑑𝑦))})
2 fvexd 6921 . . 3 (𝑟 = 𝑅 → (Base‘𝑟) ∈ V)
3 simpr 484 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → 𝑏 = (Base‘𝑟))
4 fveq2 6906 . . . . . . . 8 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
54adantr 480 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (Base‘𝑟) = (Base‘𝑅))
63, 5eqtrd 2777 . . . . . 6 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → 𝑏 = (Base‘𝑅))
7 rprmval.b . . . . . 6 𝐵 = (Base‘𝑅)
86, 7eqtr4di 2795 . . . . 5 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → 𝑏 = 𝐵)
9 fveq2 6906 . . . . . . . 8 (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
10 rprmval.u . . . . . . . 8 𝑈 = (Unit‘𝑅)
119, 10eqtr4di 2795 . . . . . . 7 (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
12 fveq2 6906 . . . . . . . . 9 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
13 rprmval.1 . . . . . . . . 9 0 = (0g𝑅)
1412, 13eqtr4di 2795 . . . . . . . 8 (𝑟 = 𝑅 → (0g𝑟) = 0 )
1514sneqd 4638 . . . . . . 7 (𝑟 = 𝑅 → {(0g𝑟)} = { 0 })
1611, 15uneq12d 4169 . . . . . 6 (𝑟 = 𝑅 → ((Unit‘𝑟) ∪ {(0g𝑟)}) = (𝑈 ∪ { 0 }))
1716adantr 480 . . . . 5 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → ((Unit‘𝑟) ∪ {(0g𝑟)}) = (𝑈 ∪ { 0 }))
188, 17difeq12d 4127 . . . 4 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (𝑏 ∖ ((Unit‘𝑟) ∪ {(0g𝑟)})) = (𝐵 ∖ (𝑈 ∪ { 0 })))
19 fvexd 6921 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (∥r𝑟) ∈ V)
20 eqidd 2738 . . . . . . . . 9 (((𝑟 = 𝑅𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r𝑟)) → 𝑝 = 𝑝)
21 simpr 484 . . . . . . . . . . 11 (((𝑟 = 𝑅𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r𝑟)) → 𝑑 = (∥r𝑟))
22 fveq2 6906 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (∥r𝑟) = (∥r𝑅))
2322ad2antrr 726 . . . . . . . . . . 11 (((𝑟 = 𝑅𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r𝑟)) → (∥r𝑟) = (∥r𝑅))
2421, 23eqtrd 2777 . . . . . . . . . 10 (((𝑟 = 𝑅𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r𝑟)) → 𝑑 = (∥r𝑅))
25 rprmval.d . . . . . . . . . 10 = (∥r𝑅)
2624, 25eqtr4di 2795 . . . . . . . . 9 (((𝑟 = 𝑅𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r𝑟)) → 𝑑 = )
27 fveq2 6906 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
28 rprmval.m . . . . . . . . . . . 12 · = (.r𝑅)
2927, 28eqtr4di 2795 . . . . . . . . . . 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 5157 . . . . . . . 8 (((𝑟 = 𝑅𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r𝑟)) → (𝑝𝑑(𝑥(.r𝑟)𝑦) ↔ 𝑝 (𝑥 · 𝑦)))
3326breqd 5154 . . . . . . . . 9 (((𝑟 = 𝑅𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r𝑟)) → (𝑝𝑑𝑥𝑝 𝑥))
3426breqd 5154 . . . . . . . . 9 (((𝑟 = 𝑅𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r𝑟)) → (𝑝𝑑𝑦𝑝 𝑦))
3533, 34orbi12d 919 . . . . . . . 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 3832 . . . . . 6 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → ([(∥r𝑟) / 𝑑](𝑝𝑑(𝑥(.r𝑟)𝑦) → (𝑝𝑑𝑥𝑝𝑑𝑦)) ↔ (𝑝 (𝑥 · 𝑦) → (𝑝 𝑥𝑝 𝑦))))
388, 37raleqbidv 3346 . . . . 5 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (∀𝑦𝑏 [(∥r𝑟) / 𝑑](𝑝𝑑(𝑥(.r𝑟)𝑦) → (𝑝𝑑𝑥𝑝𝑑𝑦)) ↔ ∀𝑦𝐵 (𝑝 (𝑥 · 𝑦) → (𝑝 𝑥𝑝 𝑦))))
398, 38raleqbidv 3346 . . . 4 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (∀𝑥𝑏𝑦𝑏 [(∥r𝑟) / 𝑑](𝑝𝑑(𝑥(.r𝑟)𝑦) → (𝑝𝑑𝑥𝑝𝑑𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑝 (𝑥 · 𝑦) → (𝑝 𝑥𝑝 𝑦))))
4018, 39rabeqbidv 3455 . . 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 3935 . 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 3501 . 2 (𝑅𝑉𝑅 ∈ V)
437fvexi 6920 . . . . 5 𝐵 ∈ V
4443difexi 5330 . . . 4 (𝐵 ∖ (𝑈 ∪ { 0 })) ∈ V
4544rabex 5339 . . 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 848   = wceq 1540  wcel 2108  wral 3061  {crab 3436  Vcvv 3480  [wsbc 3788  csb 3899  cdif 3948  cun 3949  {csn 4626   class class class wbr 5143  cfv 6561  (class class class)co 7431  Basecbs 17247  .rcmulr 17298  0gc0g 17484  rcdsr 20354  Unitcui 20355  RPrimecrpm 20432
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-rprm 20433
This theorem is referenced by:  isrprm  33545
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