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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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