Theorem rprmnz 33613

Description: A ring prime is nonzero. (Contributed by Thierry Arnoux, 18-May-2025.)

Hypotheses

Ref	Expression
rprmnz.p	⊢ 𝑃 = (RPrime‘𝑅)
rprmnz.0	⊢ 0 = (0g‘𝑅)
rprmnz.r	⊢ (𝜑 → 𝑅 ∈ 𝑉)
rprmnz.q	⊢ (𝜑 → 𝑄 ∈ 𝑃)

Assertion

Ref	Expression
rprmnz	⊢ (𝜑 → 𝑄 ≠ 0 )

Proof of Theorem rprmnz

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

Step	Hyp	Ref	Expression
1		eqidd 2738	. . 3 ⊢ (𝜑 → ((Unit‘𝑅) ∪ { 0 }) = ((Unit‘𝑅) ∪ { 0 }))
2		rprmnz.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝑉)
3		rprmnz.q	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ 𝑃)
4		rprmnz.p	. . . . . 6 ⊢ 𝑃 = (RPrime‘𝑅)
5	3, 4	eleqtrdi 2847	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ (RPrime‘𝑅))
6		eqid 2737	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
7		eqid 2737	. . . . . . 7 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
8		rprmnz.0	. . . . . . 7 ⊢ 0 = (0g‘𝑅)
9		eqid 2737	. . . . . . 7 ⊢ (∥r‘𝑅) = (∥r‘𝑅)
10		eqid 2737	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
11	6, 7, 8, 9, 10	isrprm 33610	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (𝑄 ∈ (RPrime‘𝑅) ↔ (𝑄 ∈ ((Base‘𝑅) ∖ ((Unit‘𝑅) ∪ { 0 })) ∧ ∀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	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑄 ∈ (RPrime‘𝑅)) → 𝑄 ∈ ((Base‘𝑅) ∖ ((Unit‘𝑅) ∪ { 0 })))
13	2, 5, 12	syl2anc 585	. . . 4 ⊢ (𝜑 → 𝑄 ∈ ((Base‘𝑅) ∖ ((Unit‘𝑅) ∪ { 0 })))
14	13	eldifbd 3916	. . 3 ⊢ (𝜑 → ¬ 𝑄 ∈ ((Unit‘𝑅) ∪ { 0 }))
15		nelun 32600	. . . 4 ⊢ (((Unit‘𝑅) ∪ { 0 }) = ((Unit‘𝑅) ∪ { 0 }) → (¬ 𝑄 ∈ ((Unit‘𝑅) ∪ { 0 }) ↔ (¬ 𝑄 ∈ (Unit‘𝑅) ∧ ¬ 𝑄 ∈ { 0 })))
16	15	simplbda 499	. . 3 ⊢ ((((Unit‘𝑅) ∪ { 0 }) = ((Unit‘𝑅) ∪ { 0 }) ∧ ¬ 𝑄 ∈ ((Unit‘𝑅) ∪ { 0 })) → ¬ 𝑄 ∈ { 0 })
17	1, 14, 16	syl2anc 585	. 2 ⊢ (𝜑 → ¬ 𝑄 ∈ { 0 })
18		elsng 4596	. . . 4 ⊢ (𝑄 ∈ 𝑃 → (𝑄 ∈ { 0 } ↔ 𝑄 = 0 ))
19	3, 18	syl 17	. . 3 ⊢ (𝜑 → (𝑄 ∈ { 0 } ↔ 𝑄 = 0 ))
20	19	necon3bbid 2970	. 2 ⊢ (𝜑 → (¬ 𝑄 ∈ { 0 } ↔ 𝑄 ≠ 0 ))
21	17, 20	mpbid 232	1 ⊢ (𝜑 → 𝑄 ≠ 0 )

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052   ∖ cdif 3900   ∪ cun 3901  {csn 4582   class class class wbr 5100  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  ._rcmulr 17190  0_gc0g 17371  ∥_rcdsr 20302  Unitcui 20303  RPrimecrpm 20380

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-rprm 20381

This theorem is referenced by:  rprmasso  33618  rprmasso2  33619  rprmirred  33624  1arithidomlem1  33628  1arithufdlem3  33639  dfufd2lem  33642