Theorem rprmnunit 33499

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 2731	. 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 2839	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (RPrime‘𝑅))
6		eqid 2730	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
7		rprmdvds.3	. . . . . 6 ⊢ 𝑈 = (Unit‘𝑅)
8		eqid 2730	. . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅)
9		eqid 2730	. . . . . 6 ⊢ (∥r‘𝑅) = (∥r‘𝑅)
10		eqid 2730	. . . . . 6 ⊢ (.r‘𝑅) = (.r‘𝑅)
11	6, 7, 8, 9, 10	isrprm 33495	. . . . 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 3930	. 2 ⊢ (𝜑 → ¬ 𝑄 ∈ (𝑈 ∪ {(0g‘𝑅)}))
15		nelun 32449	. . 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 1540   ∈ wcel 2109  ∀wral 3045   ∖ cdif 3914   ∪ cun 3915  {csn 4592   class class class wbr 5110  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  ._rcmulr 17228  0_gc0g 17409  ∥_rcdsr 20270  Unitcui 20271  RPrimecrpm 20348

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522  df-ov 7393  df-rprm 20349

This theorem is referenced by:  rsprprmprmidl  33500  rprmndvdsr1  33502  rprmirred  33509  1arithidom  33515