Theorem rprmnunit 33811

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 2764	. 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 2873	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (RPrime‘𝑅))
6		eqid 2763	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
7		rprmdvds.3	. . . . . 6 ⊢ 𝑈 = (Unit‘𝑅)
8		eqid 2763	. . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅)
9		eqid 2763	. . . . . 6 ⊢ (∥r‘𝑅) = (∥r‘𝑅)
10		eqid 2763	. . . . . 6 ⊢ (.r‘𝑅) = (.r‘𝑅)
11	6, 7, 8, 9, 10	isrprm 33807	. . . . 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 503	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑄 ∈ (RPrime‘𝑅)) → 𝑄 ∈ ((Base‘𝑅) ∖ (𝑈 ∪ {(0g‘𝑅)})))
13	2, 5, 12	syl2anc 595	. . 3 ⊢ (𝜑 → 𝑄 ∈ ((Base‘𝑅) ∖ (𝑈 ∪ {(0g‘𝑅)})))
14	13	eldifbd 3918	. 2 ⊢ (𝜑 → ¬ 𝑄 ∈ (𝑈 ∪ {(0g‘𝑅)}))
15		nelun 32859	. . 3 ⊢ ((𝑈 ∪ {(0g‘𝑅)}) = (𝑈 ∪ {(0g‘𝑅)}) → (¬ 𝑄 ∈ (𝑈 ∪ {(0g‘𝑅)}) ↔ (¬ 𝑄 ∈ 𝑈 ∧ ¬ 𝑄 ∈ {(0g‘𝑅)})))
16	15	simprbda 503	. 2 ⊢ (((𝑈 ∪ {(0g‘𝑅)}) = (𝑈 ∪ {(0g‘𝑅)}) ∧ ¬ 𝑄 ∈ (𝑈 ∪ {(0g‘𝑅)})) → ¬ 𝑄 ∈ 𝑈)
17	1, 14, 16	syl2anc 595	1 ⊢ (𝜑 → ¬ 𝑄 ∈ 𝑈)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 860   = wceq 1570   ∈ wcel 2143  ∀wral 3079   ∖ cdif 3902   ∪ cun 3903  {csn 4589   class class class wbr 5109  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  ._rcmulr 17306  0_gc0g 17487  ∥_rcdsr 20432  Unitcui 20433  RPrimecrpm 20510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-rprm 20511

This theorem is referenced by:  rsprprmprmidl  33812  rprmndvdsr1  33814  rprmirred  33821  1arithidom  33827