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Theorem irredmul 20446
Description: If product of two elements is irreducible, then one of the elements must be a unit. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypotheses
Ref Expression
irredn0.i 𝐼 = (Irred‘𝑅)
irredmul.b 𝐵 = (Base‘𝑅)
irredmul.u 𝑈 = (Unit‘𝑅)
irredmul.t · = (.r𝑅)
Assertion
Ref Expression
irredmul ((𝑋𝐵𝑌𝐵 ∧ (𝑋 · 𝑌) ∈ 𝐼) → (𝑋𝑈𝑌𝑈))

Proof of Theorem irredmul
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 irredmul.b . . . . 5 𝐵 = (Base‘𝑅)
2 irredmul.u . . . . 5 𝑈 = (Unit‘𝑅)
3 irredn0.i . . . . 5 𝐼 = (Irred‘𝑅)
4 irredmul.t . . . . 5 · = (.r𝑅)
51, 2, 3, 4isirred2 20438 . . . 4 ((𝑋 · 𝑌) ∈ 𝐼 ↔ ((𝑋 · 𝑌) ∈ 𝐵 ∧ ¬ (𝑋 · 𝑌) ∈ 𝑈 ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = (𝑋 · 𝑌) → (𝑥𝑈𝑦𝑈))))
65simp3bi 1146 . . 3 ((𝑋 · 𝑌) ∈ 𝐼 → ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = (𝑋 · 𝑌) → (𝑥𝑈𝑦𝑈)))
7 eqid 2735 . . . 4 (𝑋 · 𝑌) = (𝑋 · 𝑌)
8 oveq1 7438 . . . . . . 7 (𝑥 = 𝑋 → (𝑥 · 𝑦) = (𝑋 · 𝑦))
98eqeq1d 2737 . . . . . 6 (𝑥 = 𝑋 → ((𝑥 · 𝑦) = (𝑋 · 𝑌) ↔ (𝑋 · 𝑦) = (𝑋 · 𝑌)))
10 eleq1 2827 . . . . . . 7 (𝑥 = 𝑋 → (𝑥𝑈𝑋𝑈))
1110orbi1d 916 . . . . . 6 (𝑥 = 𝑋 → ((𝑥𝑈𝑦𝑈) ↔ (𝑋𝑈𝑦𝑈)))
129, 11imbi12d 344 . . . . 5 (𝑥 = 𝑋 → (((𝑥 · 𝑦) = (𝑋 · 𝑌) → (𝑥𝑈𝑦𝑈)) ↔ ((𝑋 · 𝑦) = (𝑋 · 𝑌) → (𝑋𝑈𝑦𝑈))))
13 oveq2 7439 . . . . . . 7 (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌))
1413eqeq1d 2737 . . . . . 6 (𝑦 = 𝑌 → ((𝑋 · 𝑦) = (𝑋 · 𝑌) ↔ (𝑋 · 𝑌) = (𝑋 · 𝑌)))
15 eleq1 2827 . . . . . . 7 (𝑦 = 𝑌 → (𝑦𝑈𝑌𝑈))
1615orbi2d 915 . . . . . 6 (𝑦 = 𝑌 → ((𝑋𝑈𝑦𝑈) ↔ (𝑋𝑈𝑌𝑈)))
1714, 16imbi12d 344 . . . . 5 (𝑦 = 𝑌 → (((𝑋 · 𝑦) = (𝑋 · 𝑌) → (𝑋𝑈𝑦𝑈)) ↔ ((𝑋 · 𝑌) = (𝑋 · 𝑌) → (𝑋𝑈𝑌𝑈))))
1812, 17rspc2v 3633 . . . 4 ((𝑋𝐵𝑌𝐵) → (∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = (𝑋 · 𝑌) → (𝑥𝑈𝑦𝑈)) → ((𝑋 · 𝑌) = (𝑋 · 𝑌) → (𝑋𝑈𝑌𝑈))))
197, 18mpii 46 . . 3 ((𝑋𝐵𝑌𝐵) → (∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = (𝑋 · 𝑌) → (𝑥𝑈𝑦𝑈)) → (𝑋𝑈𝑌𝑈)))
206, 19syl5 34 . 2 ((𝑋𝐵𝑌𝐵) → ((𝑋 · 𝑌) ∈ 𝐼 → (𝑋𝑈𝑌𝑈)))
21203impia 1116 1 ((𝑋𝐵𝑌𝐵 ∧ (𝑋 · 𝑌) ∈ 𝐼) → (𝑋𝑈𝑌𝑈))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  wral 3059  cfv 6563  (class class class)co 7431  Basecbs 17245  .rcmulr 17299  Unitcui 20372  Irredcir 20373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-irred 20376
This theorem is referenced by:  prmirredlem  21501  mxidlirred  33480  irredminply  33722
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