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Theorem irredmul 19458
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 19450 . . . 4 ((𝑋 · 𝑌) ∈ 𝐼 ↔ ((𝑋 · 𝑌) ∈ 𝐵 ∧ ¬ (𝑋 · 𝑌) ∈ 𝑈 ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = (𝑋 · 𝑌) → (𝑥𝑈𝑦𝑈))))
65simp3bi 1144 . . 3 ((𝑋 · 𝑌) ∈ 𝐼 → ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = (𝑋 · 𝑌) → (𝑥𝑈𝑦𝑈)))
7 eqid 2801 . . . 4 (𝑋 · 𝑌) = (𝑋 · 𝑌)
8 oveq1 7146 . . . . . . 7 (𝑥 = 𝑋 → (𝑥 · 𝑦) = (𝑋 · 𝑦))
98eqeq1d 2803 . . . . . 6 (𝑥 = 𝑋 → ((𝑥 · 𝑦) = (𝑋 · 𝑌) ↔ (𝑋 · 𝑦) = (𝑋 · 𝑌)))
10 eleq1 2880 . . . . . . 7 (𝑥 = 𝑋 → (𝑥𝑈𝑋𝑈))
1110orbi1d 914 . . . . . 6 (𝑥 = 𝑋 → ((𝑥𝑈𝑦𝑈) ↔ (𝑋𝑈𝑦𝑈)))
129, 11imbi12d 348 . . . . 5 (𝑥 = 𝑋 → (((𝑥 · 𝑦) = (𝑋 · 𝑌) → (𝑥𝑈𝑦𝑈)) ↔ ((𝑋 · 𝑦) = (𝑋 · 𝑌) → (𝑋𝑈𝑦𝑈))))
13 oveq2 7147 . . . . . . 7 (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌))
1413eqeq1d 2803 . . . . . 6 (𝑦 = 𝑌 → ((𝑋 · 𝑦) = (𝑋 · 𝑌) ↔ (𝑋 · 𝑌) = (𝑋 · 𝑌)))
15 eleq1 2880 . . . . . . 7 (𝑦 = 𝑌 → (𝑦𝑈𝑌𝑈))
1615orbi2d 913 . . . . . 6 (𝑦 = 𝑌 → ((𝑋𝑈𝑦𝑈) ↔ (𝑋𝑈𝑌𝑈)))
1714, 16imbi12d 348 . . . . 5 (𝑦 = 𝑌 → (((𝑋 · 𝑦) = (𝑋 · 𝑌) → (𝑋𝑈𝑦𝑈)) ↔ ((𝑋 · 𝑌) = (𝑋 · 𝑌) → (𝑋𝑈𝑌𝑈))))
1812, 17rspc2v 3584 . . . 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 1114 1 ((𝑋𝐵𝑌𝐵 ∧ (𝑋 · 𝑌) ∈ 𝐼) → (𝑋𝑈𝑌𝑈))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2112  wral 3109  cfv 6328  (class class class)co 7139  Basecbs 16478  .rcmulr 16561  Unitcui 19388  Irredcir 19389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-irred 19392
This theorem is referenced by:  prmirredlem  20189
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