Theorem rprmnunit 33551

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	⊢ (𝜑 → 𝑅 ∈ 𝑉)
rprmdvds.6	⊢ (𝜑 → 𝑄 ∈ 𝑃)

Assertion

Ref	Expression
rprmnunit	⊢ (𝜑 → ¬ 𝑄 ∈ 𝑈)

Proof of Theorem rprmnunit

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2735	. 2 ⊢ (𝜑 → (𝑈 ∪ {(0g‘𝑅)}) = (𝑈 ∪ {(0g‘𝑅)}))
2		rprmdvds.5	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑉)
3		rprmdvds.6	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝑃)
4		rprmdvds.2	. . . . 5 ⊢ 𝑃 = (RPrime‘𝑅)
5	3, 4	eleqtrdi 2844	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (RPrime‘𝑅))
6		eqid 2734	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
7		rprmdvds.3	. . . . . 6 ⊢ 𝑈 = (Unit‘𝑅)
8		eqid 2734	. . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅)
9		eqid 2734	. . . . . 6 ⊢ (∥r‘𝑅) = (∥r‘𝑅)
10		eqid 2734	. . . . . 6 ⊢ (.r‘𝑅) = (.r‘𝑅)
11	6, 7, 8, 9, 10	isrprm 33547	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (𝑄 ∈ (RPrime‘𝑅) ↔ (𝑄 ∈ ((Base‘𝑅) ∖ (𝑈 ∪ {(0g‘𝑅)})) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝑄(∥r‘𝑅)(𝑥(.r‘𝑅)𝑦) → (𝑄(∥r‘𝑅)𝑥 ∨ 𝑄(∥r‘𝑅)𝑦)))))
12	11	simprbda 498	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑄 ∈ (RPrime‘𝑅)) → 𝑄 ∈ ((Base‘𝑅) ∖ (𝑈 ∪ {(0g‘𝑅)})))
13	2, 5, 12	syl2anc 584	. . 3 ⊢ (𝜑 → 𝑄 ∈ ((Base‘𝑅) ∖ (𝑈 ∪ {(0g‘𝑅)})))
14	13	eldifbd 3912	. 2 ⊢ (𝜑 → ¬ 𝑄 ∈ (𝑈 ∪ {(0g‘𝑅)}))
15		nelun 32537	. . 3 ⊢ ((𝑈 ∪ {(0g‘𝑅)}) = (𝑈 ∪ {(0g‘𝑅)}) → (¬ 𝑄 ∈ (𝑈 ∪ {(0g‘𝑅)}) ↔ (¬ 𝑄 ∈ 𝑈 ∧ ¬ 𝑄 ∈ {(0g‘𝑅)})))
16	15	simprbda 498	. 2 ⊢ (((𝑈 ∪ {(0g‘𝑅)}) = (𝑈 ∪ {(0g‘𝑅)}) ∧ ¬ 𝑄 ∈ (𝑈 ∪ {(0g‘𝑅)})) → ¬ 𝑄 ∈ 𝑈)
17	1, 14, 16	syl2anc 584	1 ⊢ (𝜑 → ¬ 𝑄 ∈ 𝑈)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 847   = wceq 1541   ∈ wcel 2113  ∀wral 3049   ∖ cdif 3896   ∪ cun 3897  {csn 4578   class class class wbr 5096  ‘cfv 6490  (class class class)co 7356  Basecbs 17134  ._rcmulr 17176  0_gc0g 17357  ∥_rcdsr 20288  Unitcui 20289  RPrimecrpm 20366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-iota 6446  df-fun 6492  df-fv 6498  df-ov 7359  df-rprm 20367

This theorem is referenced by:  rsprprmprmidl  33552  rprmndvdsr1  33554  rprmirred  33561  1arithidom  33567