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Theorem rprmval 31072
 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 19462 . 2 RPrime = (𝑟 ∈ V ↦ (Base‘𝑟) / 𝑏{𝑝 ∈ (𝑏 ∖ ((Unit‘𝑟) ∪ {(0g𝑟)})) ∣ ∀𝑥𝑏𝑦𝑏 [(∥r𝑟) / 𝑑](𝑝𝑑(𝑥(.r𝑟)𝑦) → (𝑝𝑑𝑥𝑝𝑑𝑦))})
2 fvexd 6664 . . 3 (𝑟 = 𝑅 → (Base‘𝑟) ∈ V)
3 simpr 488 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → 𝑏 = (Base‘𝑟))
4 fveq2 6649 . . . . . . . 8 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
54adantr 484 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (Base‘𝑟) = (Base‘𝑅))
63, 5eqtrd 2836 . . . . . 6 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → 𝑏 = (Base‘𝑅))
7 rprmval.b . . . . . 6 𝐵 = (Base‘𝑅)
86, 7eqtr4di 2854 . . . . 5 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → 𝑏 = 𝐵)
9 fveq2 6649 . . . . . . . 8 (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
10 rprmval.u . . . . . . . 8 𝑈 = (Unit‘𝑅)
119, 10eqtr4di 2854 . . . . . . 7 (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
12 fveq2 6649 . . . . . . . . 9 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
13 rprmval.1 . . . . . . . . 9 0 = (0g𝑅)
1412, 13eqtr4di 2854 . . . . . . . 8 (𝑟 = 𝑅 → (0g𝑟) = 0 )
1514sneqd 4540 . . . . . . 7 (𝑟 = 𝑅 → {(0g𝑟)} = { 0 })
1611, 15uneq12d 4094 . . . . . 6 (𝑟 = 𝑅 → ((Unit‘𝑟) ∪ {(0g𝑟)}) = (𝑈 ∪ { 0 }))
1716adantr 484 . . . . 5 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → ((Unit‘𝑟) ∪ {(0g𝑟)}) = (𝑈 ∪ { 0 }))
188, 17difeq12d 4054 . . . 4 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (𝑏 ∖ ((Unit‘𝑟) ∪ {(0g𝑟)})) = (𝐵 ∖ (𝑈 ∪ { 0 })))
19 fvexd 6664 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (∥r𝑟) ∈ V)
20 eqidd 2802 . . . . . . . . 9 (((𝑟 = 𝑅𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r𝑟)) → 𝑝 = 𝑝)
21 simpr 488 . . . . . . . . . . 11 (((𝑟 = 𝑅𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r𝑟)) → 𝑑 = (∥r𝑟))
22 fveq2 6649 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (∥r𝑟) = (∥r𝑅))
2322ad2antrr 725 . . . . . . . . . . 11 (((𝑟 = 𝑅𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r𝑟)) → (∥r𝑟) = (∥r𝑅))
2421, 23eqtrd 2836 . . . . . . . . . 10 (((𝑟 = 𝑅𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r𝑟)) → 𝑑 = (∥r𝑅))
25 rprmval.d . . . . . . . . . 10 = (∥r𝑅)
2624, 25eqtr4di 2854 . . . . . . . . 9 (((𝑟 = 𝑅𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r𝑟)) → 𝑑 = )
27 fveq2 6649 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
28 rprmval.m . . . . . . . . . . . 12 · = (.r𝑅)
2927, 28eqtr4di 2854 . . . . . . . . . . 11 (𝑟 = 𝑅 → (.r𝑟) = · )
3029ad2antrr 725 . . . . . . . . . 10 (((𝑟 = 𝑅𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r𝑟)) → (.r𝑟) = · )
3130oveqd 7156 . . . . . . . . 9 (((𝑟 = 𝑅𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r𝑟)) → (𝑥(.r𝑟)𝑦) = (𝑥 · 𝑦))
3220, 26, 31breq123d 5047 . . . . . . . 8 (((𝑟 = 𝑅𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r𝑟)) → (𝑝𝑑(𝑥(.r𝑟)𝑦) ↔ 𝑝 (𝑥 · 𝑦)))
3326breqd 5044 . . . . . . . . 9 (((𝑟 = 𝑅𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r𝑟)) → (𝑝𝑑𝑥𝑝 𝑥))
3426breqd 5044 . . . . . . . . 9 (((𝑟 = 𝑅𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r𝑟)) → (𝑝𝑑𝑦𝑝 𝑦))
3533, 34orbi12d 916 . . . . . . . 8 (((𝑟 = 𝑅𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r𝑟)) → ((𝑝𝑑𝑥𝑝𝑑𝑦) ↔ (𝑝 𝑥𝑝 𝑦)))
3632, 35imbi12d 348 . . . . . . 7 (((𝑟 = 𝑅𝑏 = (Base‘𝑟)) ∧ 𝑑 = (∥r𝑟)) → ((𝑝𝑑(𝑥(.r𝑟)𝑦) → (𝑝𝑑𝑥𝑝𝑑𝑦)) ↔ (𝑝 (𝑥 · 𝑦) → (𝑝 𝑥𝑝 𝑦))))
3719, 36sbcied 3765 . . . . . 6 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → ([(∥r𝑟) / 𝑑](𝑝𝑑(𝑥(.r𝑟)𝑦) → (𝑝𝑑𝑥𝑝𝑑𝑦)) ↔ (𝑝 (𝑥 · 𝑦) → (𝑝 𝑥𝑝 𝑦))))
388, 37raleqbidv 3357 . . . . 5 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (∀𝑦𝑏 [(∥r𝑟) / 𝑑](𝑝𝑑(𝑥(.r𝑟)𝑦) → (𝑝𝑑𝑥𝑝𝑑𝑦)) ↔ ∀𝑦𝐵 (𝑝 (𝑥 · 𝑦) → (𝑝 𝑥𝑝 𝑦))))
398, 38raleqbidv 3357 . . . 4 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (∀𝑥𝑏𝑦𝑏 [(∥r𝑟) / 𝑑](𝑝𝑑(𝑥(.r𝑟)𝑦) → (𝑝𝑑𝑥𝑝𝑑𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑝 (𝑥 · 𝑦) → (𝑝 𝑥𝑝 𝑦))))
4018, 39rabeqbidv 3436 . . 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 3867 . 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 3462 . 2 (𝑅𝑉𝑅 ∈ V)
437fvexi 6663 . . . . 5 𝐵 ∈ V
4443difexi 5199 . . . 4 (𝐵 ∖ (𝑈 ∪ { 0 })) ∈ V
4544rabex 5202 . . 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 6772 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 399   ∨ wo 844   = wceq 1538   ∈ wcel 2112  ∀wral 3109  {crab 3113  Vcvv 3444  [wsbc 3723  ⦋csb 3831   ∖ cdif 3881   ∪ cun 3882  {csn 4528   class class class wbr 5033  ‘cfv 6328  (class class class)co 7139  Basecbs 16478  .rcmulr 16561  0gc0g 16708  ∥rcdsr 19387  Unitcui 19388  RPrimecrpm 19461 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-rprm 19462 This theorem is referenced by:  isrprm  31073
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