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Theorem irredmul 20380
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 20372 . . . 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 2725 . . . 4 (𝑋 · 𝑌) = (𝑋 · 𝑌)
8 oveq1 7426 . . . . . . 7 (𝑥 = 𝑋 → (𝑥 · 𝑦) = (𝑋 · 𝑦))
98eqeq1d 2727 . . . . . 6 (𝑥 = 𝑋 → ((𝑥 · 𝑦) = (𝑋 · 𝑌) ↔ (𝑋 · 𝑦) = (𝑋 · 𝑌)))
10 eleq1 2813 . . . . . . 7 (𝑥 = 𝑋 → (𝑥𝑈𝑋𝑈))
1110orbi1d 914 . . . . . 6 (𝑥 = 𝑋 → ((𝑥𝑈𝑦𝑈) ↔ (𝑋𝑈𝑦𝑈)))
129, 11imbi12d 343 . . . . 5 (𝑥 = 𝑋 → (((𝑥 · 𝑦) = (𝑋 · 𝑌) → (𝑥𝑈𝑦𝑈)) ↔ ((𝑋 · 𝑦) = (𝑋 · 𝑌) → (𝑋𝑈𝑦𝑈))))
13 oveq2 7427 . . . . . . 7 (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌))
1413eqeq1d 2727 . . . . . 6 (𝑦 = 𝑌 → ((𝑋 · 𝑦) = (𝑋 · 𝑌) ↔ (𝑋 · 𝑌) = (𝑋 · 𝑌)))
15 eleq1 2813 . . . . . . 7 (𝑦 = 𝑌 → (𝑦𝑈𝑌𝑈))
1615orbi2d 913 . . . . . 6 (𝑦 = 𝑌 → ((𝑋𝑈𝑦𝑈) ↔ (𝑋𝑈𝑌𝑈)))
1714, 16imbi12d 343 . . . . 5 (𝑦 = 𝑌 → (((𝑋 · 𝑦) = (𝑋 · 𝑌) → (𝑋𝑈𝑦𝑈)) ↔ ((𝑋 · 𝑌) = (𝑋 · 𝑌) → (𝑋𝑈𝑌𝑈))))
1812, 17rspc2v 3617 . . . 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 394  wo 845  w3a 1084   = wceq 1533  wcel 2098  wral 3050  cfv 6549  (class class class)co 7419  Basecbs 17183  .rcmulr 17237  Unitcui 20306  Irredcir 20307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fv 6557  df-ov 7422  df-irred 20310
This theorem is referenced by:  prmirredlem  21415  mxidlirred  33284  irredminply  33515
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