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Theorem rprmdvds 33547
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 = (∥r𝑅)
rprmdvds.t · = (.r𝑅)
rprmdvds.r (𝜑𝑅𝑉)
rprmdvds.q (𝜑𝑄𝑃)
rprmdvds.x (𝜑𝑋𝐵)
rprmdvds.y (𝜑𝑌𝐵)
rprmdvds.1 (𝜑𝑄 (𝑋 · 𝑌))
Assertion
Ref Expression
rprmdvds (𝜑 → (𝑄 𝑋𝑄 𝑌))

Proof of Theorem rprmdvds
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rprmdvds.1 . 2 (𝜑𝑄 (𝑋 · 𝑌))
2 oveq1 7438 . . . . 5 (𝑥 = 𝑋 → (𝑥 · 𝑦) = (𝑋 · 𝑦))
32breq2d 5155 . . . 4 (𝑥 = 𝑋 → (𝑄 (𝑥 · 𝑦) ↔ 𝑄 (𝑋 · 𝑦)))
4 breq2 5147 . . . . 5 (𝑥 = 𝑋 → (𝑄 𝑥𝑄 𝑋))
54orbi1d 917 . . . 4 (𝑥 = 𝑋 → ((𝑄 𝑥𝑄 𝑦) ↔ (𝑄 𝑋𝑄 𝑦)))
63, 5imbi12d 344 . . 3 (𝑥 = 𝑋 → ((𝑄 (𝑥 · 𝑦) → (𝑄 𝑥𝑄 𝑦)) ↔ (𝑄 (𝑋 · 𝑦) → (𝑄 𝑋𝑄 𝑦))))
7 oveq2 7439 . . . . 5 (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌))
87breq2d 5155 . . . 4 (𝑦 = 𝑌 → (𝑄 (𝑋 · 𝑦) ↔ 𝑄 (𝑋 · 𝑌)))
9 breq2 5147 . . . . 5 (𝑦 = 𝑌 → (𝑄 𝑦𝑄 𝑌))
109orbi2d 916 . . . 4 (𝑦 = 𝑌 → ((𝑄 𝑋𝑄 𝑦) ↔ (𝑄 𝑋𝑄 𝑌)))
118, 10imbi12d 344 . . 3 (𝑦 = 𝑌 → ((𝑄 (𝑋 · 𝑦) → (𝑄 𝑋𝑄 𝑦)) ↔ (𝑄 (𝑋 · 𝑌) → (𝑄 𝑋𝑄 𝑌))))
12 rprmdvds.r . . . 4 (𝜑𝑅𝑉)
13 rprmdvds.q . . . . 5 (𝜑𝑄𝑃)
14 rprmdvds.p . . . . 5 𝑃 = (RPrime‘𝑅)
1513, 14eleqtrdi 2851 . . . 4 (𝜑𝑄 ∈ (RPrime‘𝑅))
16 rprmdvds.b . . . . . 6 𝐵 = (Base‘𝑅)
17 eqid 2737 . . . . . 6 (Unit‘𝑅) = (Unit‘𝑅)
18 eqid 2737 . . . . . 6 (0g𝑅) = (0g𝑅)
19 rprmdvds.d . . . . . 6 = (∥r𝑅)
20 rprmdvds.t . . . . . 6 · = (.r𝑅)
2116, 17, 18, 19, 20isrprm 33545 . . . . 5 (𝑅𝑉 → (𝑄 ∈ (RPrime‘𝑅) ↔ (𝑄 ∈ (𝐵 ∖ ((Unit‘𝑅) ∪ {(0g𝑅)})) ∧ ∀𝑥𝐵𝑦𝐵 (𝑄 (𝑥 · 𝑦) → (𝑄 𝑥𝑄 𝑦)))))
2221simplbda 499 . . . 4 ((𝑅𝑉𝑄 ∈ (RPrime‘𝑅)) → ∀𝑥𝐵𝑦𝐵 (𝑄 (𝑥 · 𝑦) → (𝑄 𝑥𝑄 𝑦)))
2312, 15, 22syl2anc 584 . . 3 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑄 (𝑥 · 𝑦) → (𝑄 𝑥𝑄 𝑦)))
24 rprmdvds.x . . 3 (𝜑𝑋𝐵)
25 rprmdvds.y . . 3 (𝜑𝑌𝐵)
266, 11, 23, 24, 25rspc2dv 3637 . 2 (𝜑 → (𝑄 (𝑋 · 𝑌) → (𝑄 𝑋𝑄 𝑌)))
271, 26mpd 15 1 (𝜑 → (𝑄 𝑋𝑄 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 848   = wceq 1540  wcel 2108  wral 3061  cdif 3948  cun 3949  {csn 4626   class class class wbr 5143  cfv 6561  (class class class)co 7431  Basecbs 17247  .rcmulr 17298  0gc0g 17484  rcdsr 20354  Unitcui 20355  RPrimecrpm 20432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-rprm 20433
This theorem is referenced by:  rsprprmprmidl  33550  rprmasso2  33554  rprmirred  33559  rprmdvdspow  33561  rprmdvdsprod  33562  1arithidom  33565  1arithufdlem3  33574
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