Theorem rprmdvds 34033

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 7419	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 · 𝑦) = (𝑋 · 𝑦))
3	2	breq2d 5115	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑄 ∥ (𝑥 · 𝑦) ↔ 𝑄 ∥ (𝑋 · 𝑦)))
4		breq2 5107	. . . . 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 7420	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌))
8	7	breq2d 5115	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑄 ∥ (𝑋 · 𝑦) ↔ 𝑄 ∥ (𝑋 · 𝑌)))
9		breq2 5107	. . . . 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 2871	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (RPrime‘𝑅))
16		rprmdvds.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
17		eqid 2761	. . . . . 6 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
18		eqid 2761	. . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅)
19		rprmdvds.d	. . . . . 6 ⊢ ∥ = (∥r‘𝑅)
20		rprmdvds.t	. . . . . 6 ⊢ · = (.r‘𝑅)
21	16, 17, 18, 19, 20	isrprm 34031	. . . . 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 3591	. 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 2145  ∀wral 3077   ∖ cdif 3896   ∪ cun 3897  {csn 4584   class class class wbr 5103  ‘cfv 6531  (class class class)co 7412  Basecbs 17367  ._rcmulr 17409  0_gc0g 17590  ∥_rcdsr 20564  Unitcui 20565  RPrimecrpm 20642

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6487  df-fun 6533  df-fv 6539  df-ov 7415  df-rprm 20643

This theorem is used by:  rsprprmprmidl  34036  rprmasso2  34040  rprmirred  34045  rprmdvdspow  34047  rprmdvdsprod  34048  1arithidom  34051  1arithufdlem3  34060