Theorem rprmnz 33603

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 2740	. . 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 2849	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ (RPrime‘𝑅))
6		eqid 2739	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
7		eqid 2739	. . . . . . 7 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
8		rprmnz.0	. . . . . . 7 ⊢ 0 = (0g‘𝑅)
9		eqid 2739	. . . . . . 7 ⊢ (∥r‘𝑅) = (∥r‘𝑅)
10		eqid 2739	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
11	6, 7, 8, 9, 10	isrprm 33600	. . . . . 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 499	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑄 ∈ (RPrime‘𝑅)) → 𝑄 ∈ ((Base‘𝑅) ∖ ((Unit‘𝑅) ∪ { 0 })))
13	2, 5, 12	syl2anc 590	. . . 4 ⊢ (𝜑 → 𝑄 ∈ ((Base‘𝑅) ∖ ((Unit‘𝑅) ∪ { 0 })))
14	13	eldifbd 3896	. . 3 ⊢ (𝜑 → ¬ 𝑄 ∈ ((Unit‘𝑅) ∪ { 0 }))
15		nelun 32601	. . . 4 ⊢ (((Unit‘𝑅) ∪ { 0 }) = ((Unit‘𝑅) ∪ { 0 }) → (¬ 𝑄 ∈ ((Unit‘𝑅) ∪ { 0 }) ↔ (¬ 𝑄 ∈ (Unit‘𝑅) ∧ ¬ 𝑄 ∈ { 0 })))
16	15	simplbda 500	. . 3 ⊢ ((((Unit‘𝑅) ∪ { 0 }) = ((Unit‘𝑅) ∪ { 0 }) ∧ ¬ 𝑄 ∈ ((Unit‘𝑅) ∪ { 0 })) → ¬ 𝑄 ∈ { 0 })
17	1, 14, 16	syl2anc 590	. 2 ⊢ (𝜑 → ¬ 𝑄 ∈ { 0 })
18		elsng 4569	. . . 4 ⊢ (𝑄 ∈ 𝑃 → (𝑄 ∈ { 0 } ↔ 𝑄 = 0 ))
19	3, 18	syl 17	. . 3 ⊢ (𝜑 → (𝑄 ∈ { 0 } ↔ 𝑄 = 0 ))
20	19	necon3bbid 2971	. 2 ⊢ (𝜑 → (¬ 𝑄 ∈ { 0 } ↔ 𝑄 ≠ 0 ))
21	17, 20	mpbid 233	1 ⊢ (𝜑 → 𝑄 ≠ 0 )

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∨ wo 853   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  ∀wral 3053   ∖ cdif 3880   ∪ cun 3881  {csn 4555   class class class wbr 5072  ‘cfv 6485  (class class class)co 7356  Basecbs 17170  ._rcmulr 17212  0_gc0g 17393  ∥_rcdsr 20325  Unitcui 20326  RPrimecrpm 20403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-iota 6441  df-fun 6487  df-fv 6493  df-ov 7359  df-rprm 20404

This theorem is referenced by:  rprmasso  33608  rprmasso2  33609  rprmirred  33614  1arithidomlem1  33618  1arithufdlem3  33629  dfufd2lem  33632