Theorem rprmdvds 33609

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 7370	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 · 𝑦) = (𝑋 · 𝑦))
3	2	breq2d 5091	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑄 ∥ (𝑥 · 𝑦) ↔ 𝑄 ∥ (𝑋 · 𝑦)))
4		breq2 5083	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑄 ∥ 𝑥 ↔ 𝑄 ∥ 𝑋))
5	4	orbi1d 922	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑄 ∥ 𝑥 ∨ 𝑄 ∥ 𝑦) ↔ (𝑄 ∥ 𝑋 ∨ 𝑄 ∥ 𝑦)))
6	3, 5	imbi12d 345	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑄 ∥ (𝑥 · 𝑦) → (𝑄 ∥ 𝑥 ∨ 𝑄 ∥ 𝑦)) ↔ (𝑄 ∥ (𝑋 · 𝑦) → (𝑄 ∥ 𝑋 ∨ 𝑄 ∥ 𝑦))))
7		oveq2 7371	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌))
8	7	breq2d 5091	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑄 ∥ (𝑋 · 𝑦) ↔ 𝑄 ∥ (𝑋 · 𝑌)))
9		breq2 5083	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑄 ∥ 𝑦 ↔ 𝑄 ∥ 𝑌))
10	9	orbi2d 921	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑄 ∥ 𝑋 ∨ 𝑄 ∥ 𝑦) ↔ (𝑄 ∥ 𝑋 ∨ 𝑄 ∥ 𝑌)))
11	8, 10	imbi12d 345	. . 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 2850	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (RPrime‘𝑅))
16		rprmdvds.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
17		eqid 2740	. . . . . 6 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
18		eqid 2740	. . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅)
19		rprmdvds.d	. . . . . 6 ⊢ ∥ = (∥r‘𝑅)
20		rprmdvds.t	. . . . . 6 ⊢ · = (.r‘𝑅)
21	16, 17, 18, 19, 20	isrprm 33607	. . . . 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 500	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑄 ∈ (RPrime‘𝑅)) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑄 ∥ (𝑥 · 𝑦) → (𝑄 ∥ 𝑥 ∨ 𝑄 ∥ 𝑦)))
23	12, 15, 22	syl2anc 590	. . 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 3582	. 2 ⊢ (𝜑 → (𝑄 ∥ (𝑋 · 𝑌) → (𝑄 ∥ 𝑋 ∨ 𝑄 ∥ 𝑌)))
27	1, 26	mpd 15	1 ⊢ (𝜑 → (𝑄 ∥ 𝑋 ∨ 𝑄 ∥ 𝑌))

Syntax hints:   → wi 4   ∨ wo 853   = wceq 1547   ∈ wcel 2119  ∀wral 3054   ∖ cdif 3887   ∪ cun 3888  {csn 4562   class class class wbr 5079  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  ._rcmulr 17219  0_gc0g 17400  ∥_rcdsr 20332  Unitcui 20333  RPrimecrpm 20410

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 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

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 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7366  df-rprm 20411

This theorem is referenced by:  rsprprmprmidl  33612  rprmasso2  33616  rprmirred  33621  rprmdvdspow  33623  rprmdvdsprod  33624  1arithidom  33627  1arithufdlem3  33636