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Theorem rprmnunit 33549
Description: A ring prime is not a unit. (Contributed by Thierry Arnoux, 18-May-2025.)
Hypotheses
Ref Expression
rprmdvds.2 𝑃 = (RPrime‘𝑅)
rprmdvds.3 𝑈 = (Unit‘𝑅)
rprmdvds.5 (𝜑𝑅𝑉)
rprmdvds.6 (𝜑𝑄𝑃)
Assertion
Ref Expression
rprmnunit (𝜑 → ¬ 𝑄𝑈)

Proof of Theorem rprmnunit
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2738 . 2 (𝜑 → (𝑈 ∪ {(0g𝑅)}) = (𝑈 ∪ {(0g𝑅)}))
2 rprmdvds.5 . . . 4 (𝜑𝑅𝑉)
3 rprmdvds.6 . . . . 5 (𝜑𝑄𝑃)
4 rprmdvds.2 . . . . 5 𝑃 = (RPrime‘𝑅)
53, 4eleqtrdi 2851 . . . 4 (𝜑𝑄 ∈ (RPrime‘𝑅))
6 eqid 2737 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
7 rprmdvds.3 . . . . . 6 𝑈 = (Unit‘𝑅)
8 eqid 2737 . . . . . 6 (0g𝑅) = (0g𝑅)
9 eqid 2737 . . . . . 6 (∥r𝑅) = (∥r𝑅)
10 eqid 2737 . . . . . 6 (.r𝑅) = (.r𝑅)
116, 7, 8, 9, 10isrprm 33545 . . . . 5 (𝑅𝑉 → (𝑄 ∈ (RPrime‘𝑅) ↔ (𝑄 ∈ ((Base‘𝑅) ∖ (𝑈 ∪ {(0g𝑅)})) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝑄(∥r𝑅)(𝑥(.r𝑅)𝑦) → (𝑄(∥r𝑅)𝑥𝑄(∥r𝑅)𝑦)))))
1211simprbda 498 . . . 4 ((𝑅𝑉𝑄 ∈ (RPrime‘𝑅)) → 𝑄 ∈ ((Base‘𝑅) ∖ (𝑈 ∪ {(0g𝑅)})))
132, 5, 12syl2anc 584 . . 3 (𝜑𝑄 ∈ ((Base‘𝑅) ∖ (𝑈 ∪ {(0g𝑅)})))
1413eldifbd 3964 . 2 (𝜑 → ¬ 𝑄 ∈ (𝑈 ∪ {(0g𝑅)}))
15 nelun 32532 . . 3 ((𝑈 ∪ {(0g𝑅)}) = (𝑈 ∪ {(0g𝑅)}) → (¬ 𝑄 ∈ (𝑈 ∪ {(0g𝑅)}) ↔ (¬ 𝑄𝑈 ∧ ¬ 𝑄 ∈ {(0g𝑅)})))
1615simprbda 498 . 2 (((𝑈 ∪ {(0g𝑅)}) = (𝑈 ∪ {(0g𝑅)}) ∧ ¬ 𝑄 ∈ (𝑈 ∪ {(0g𝑅)})) → ¬ 𝑄𝑈)
171, 14, 16syl2anc 584 1 (𝜑 → ¬ 𝑄𝑈)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 848   = wceq 1540  wcel 2108  wral 3061  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:  rsprprmprmidl  33550  rprmndvdsr1  33552  rprmirred  33559  1arithidom  33565
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