Theorem rprmnunit 33465

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 2730	. 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 2838	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (RPrime‘𝑅))
6		eqid 2729	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
7		rprmdvds.3	. . . . . 6 ⊢ 𝑈 = (Unit‘𝑅)
8		eqid 2729	. . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅)
9		eqid 2729	. . . . . 6 ⊢ (∥r‘𝑅) = (∥r‘𝑅)
10		eqid 2729	. . . . . 6 ⊢ (.r‘𝑅) = (.r‘𝑅)
11	6, 7, 8, 9, 10	isrprm 33461	. . . . 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 3924	. 2 ⊢ (𝜑 → ¬ 𝑄 ∈ (𝑈 ∪ {(0g‘𝑅)}))
15		nelun 32415	. . 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 3044   ∖ cdif 3908   ∪ cun 3909  {csn 4585   class class class wbr 5102  ‘cfv 6499  (class class class)co 7369  Basecbs 17155  ._rcmulr 17197  0_gc0g 17378  ∥_rcdsr 20239  Unitcui 20240  RPrimecrpm 20317

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 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6452  df-fun 6501  df-fv 6507  df-ov 7372  df-rprm 20318

This theorem is referenced by:  rsprprmprmidl  33466  rprmndvdsr1  33468  rprmirred  33475  1arithidom  33481