Theorem rprmdvds 33840

Description: If a ring prime 𝑄 divides a product 𝑋 · 𝑌, then it divides either 𝑋 or 𝑌. (Contributed by Thierry Arnoux, 18-May-2025.)

Hypotheses

Ref	Expression
rprmdvds.b	⊢ 𝐵 = (Base‘𝑅)
rprmdvds.p	⊢ 𝑃 = (RPrime‘𝑅)
rprmdvds.d	⊢ ∥ = (∥r‘𝑅)
rprmdvds.t	⊢ · = (.r‘𝑅)
rprmdvds.r	⊢ (𝜑 → 𝑅 ∈ 𝑉)
rprmdvds.q	⊢ (𝜑 → 𝑄 ∈ 𝑃)
rprmdvds.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
rprmdvds.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
rprmdvds.1	⊢ (𝜑 → 𝑄 ∥ (𝑋 · 𝑌))

Assertion

Ref	Expression
rprmdvds	⊢ (𝜑 → (𝑄 ∥ 𝑋 ∨ 𝑄 ∥ 𝑌))

Proof of Theorem rprmdvds

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

Step	Hyp	Ref	Expression
1		rprmdvds.1	. 2 ⊢ (𝜑 → 𝑄 ∥ (𝑋 · 𝑌))
2		oveq1 7430	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 · 𝑦) = (𝑋 · 𝑦))
3	2	breq2d 5126	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑄 ∥ (𝑥 · 𝑦) ↔ 𝑄 ∥ (𝑋 · 𝑦)))
4		breq2 5118	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑄 ∥ 𝑥 ↔ 𝑄 ∥ 𝑋))
5	4	orbi1d 930	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑄 ∥ 𝑥 ∨ 𝑄 ∥ 𝑦) ↔ (𝑄 ∥ 𝑋 ∨ 𝑄 ∥ 𝑦)))
6	3, 5	imbi12d 347	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑄 ∥ (𝑥 · 𝑦) → (𝑄 ∥ 𝑥 ∨ 𝑄 ∥ 𝑦)) ↔ (𝑄 ∥ (𝑋 · 𝑦) → (𝑄 ∥ 𝑋 ∨ 𝑄 ∥ 𝑦))))
7		oveq2 7431	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌))
8	7	breq2d 5126	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑄 ∥ (𝑋 · 𝑦) ↔ 𝑄 ∥ (𝑋 · 𝑌)))
9		breq2 5118	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑄 ∥ 𝑦 ↔ 𝑄 ∥ 𝑌))
10	9	orbi2d 929	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑄 ∥ 𝑋 ∨ 𝑄 ∥ 𝑦) ↔ (𝑄 ∥ 𝑋 ∨ 𝑄 ∥ 𝑌)))
11	8, 10	imbi12d 347	. . 3 ⊢ (𝑦 = 𝑌 → ((𝑄 ∥ (𝑋 · 𝑦) → (𝑄 ∥ 𝑋 ∨ 𝑄 ∥ 𝑦)) ↔ (𝑄 ∥ (𝑋 · 𝑌) → (𝑄 ∥ 𝑋 ∨ 𝑄 ∥ 𝑌))))
12		rprmdvds.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑉)
13		rprmdvds.q	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝑃)
14		rprmdvds.p	. . . . 5 ⊢ 𝑃 = (RPrime‘𝑅)
15	13, 14	eleqtrdi 2876	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (RPrime‘𝑅))
16		rprmdvds.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
17		eqid 2766	. . . . . 6 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
18		eqid 2766	. . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅)
19		rprmdvds.d	. . . . . 6 ⊢ ∥ = (∥r‘𝑅)
20		rprmdvds.t	. . . . . 6 ⊢ · = (.r‘𝑅)
21	16, 17, 18, 19, 20	isrprm 33838	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (𝑄 ∈ (RPrime‘𝑅) ↔ (𝑄 ∈ (𝐵 ∖ ((Unit‘𝑅) ∪ {(0g‘𝑅)})) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑄 ∥ (𝑥 · 𝑦) → (𝑄 ∥ 𝑥 ∨ 𝑄 ∥ 𝑦)))))
22	21	simplbda 505	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑄 ∈ (RPrime‘𝑅)) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑄 ∥ (𝑥 · 𝑦) → (𝑄 ∥ 𝑥 ∨ 𝑄 ∥ 𝑦)))
23	12, 15, 22	syl2anc 596	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑄 ∥ (𝑥 · 𝑦) → (𝑄 ∥ 𝑥 ∨ 𝑄 ∥ 𝑦)))
24		rprmdvds.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
25		rprmdvds.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
26	6, 11, 23, 24, 25	rspc2dv 3599	. 2 ⊢ (𝜑 → (𝑄 ∥ (𝑋 · 𝑌) → (𝑄 ∥ 𝑋 ∨ 𝑄 ∥ 𝑌)))
27	1, 26	mpd 16	1 ⊢ (𝜑 → (𝑄 ∥ 𝑋 ∨ 𝑄 ∥ 𝑌))

This proof depends on syntax axioms:   → wi 4   ∨ wo 861   = wceq 1570   ∈ wcel 2146  ∀wral 3082   ∖ cdif 3905   ∪ cun 3906  {csn 4594   class class class wbr 5114  ‘cfv 6543  (class class class)co 7423  Basecbs 17294  ._rcmulr 17336  0_gc0g 17517  ∥_rcdsr 20469  Unitcui 20470  RPrimecrpm 20547

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6499  df-fun 6545  df-fv 6551  df-ov 7426  df-rprm 20548

This theorem is used by:  rsprprmprmidl  33843  rprmasso2  33847  rprmirred  33852  rprmdvdspow  33854  rprmdvdsprod  33855  1arithidom  33858  1arithufdlem3  33867