Theorem rprmnz 33716

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 2763	. . 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 2872	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ (RPrime‘𝑅))
6		eqid 2762	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
7		eqid 2762	. . . . . . 7 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
8		rprmnz.0	. . . . . . 7 ⊢ 0 = (0g‘𝑅)
9		eqid 2762	. . . . . . 7 ⊢ (∥r‘𝑅) = (∥r‘𝑅)
10		eqid 2762	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
11	6, 7, 8, 9, 10	isrprm 33713	. . . . . 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 502	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑄 ∈ (RPrime‘𝑅)) → 𝑄 ∈ ((Base‘𝑅) ∖ ((Unit‘𝑅) ∪ { 0 })))
13	2, 5, 12	syl2anc 593	. . . 4 ⊢ (𝜑 → 𝑄 ∈ ((Base‘𝑅) ∖ ((Unit‘𝑅) ∪ { 0 })))
14	13	eldifbd 3917	. . 3 ⊢ (𝜑 → ¬ 𝑄 ∈ ((Unit‘𝑅) ∪ { 0 }))
15		nelun 32712	. . . 4 ⊢ (((Unit‘𝑅) ∪ { 0 }) = ((Unit‘𝑅) ∪ { 0 }) → (¬ 𝑄 ∈ ((Unit‘𝑅) ∪ { 0 }) ↔ (¬ 𝑄 ∈ (Unit‘𝑅) ∧ ¬ 𝑄 ∈ { 0 })))
16	15	simplbda 503	. . 3 ⊢ ((((Unit‘𝑅) ∪ { 0 }) = ((Unit‘𝑅) ∪ { 0 }) ∧ ¬ 𝑄 ∈ ((Unit‘𝑅) ∪ { 0 })) → ¬ 𝑄 ∈ { 0 })
17	1, 14, 16	syl2anc 593	. 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 2994	. 2 ⊢ (𝜑 → (¬ 𝑄 ∈ { 0 } ↔ 𝑄 ≠ 0 ))
21	17, 20	mpbid 234	1 ⊢ (𝜑 → 𝑄 ≠ 0 )

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∨ wo 858   = wceq 1560   ∈ wcel 2142   ≠ wne 2957  ∀wral 3076   ∖ cdif 3901   ∪ cun 3902  {csn 4582   class class class wbr 5100  ‘cfv 6521  (class class class)co 7396  Basecbs 17245  ._rcmulr 17287  0_gc0g 17468  ∥_rcdsr 20403  Unitcui 20404  RPrimecrpm 20481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-iota 6477  df-fun 6523  df-fv 6529  df-ov 7399  df-rprm 20482

This theorem is referenced by:  rprmasso  33721  rprmasso2  33722  rprmirred  33727  1arithidomlem1  33731  1arithufdlem3  33742  dfufd2lem  33745