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Theorem irredmul 19070
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 19062 . . . 4 ((𝑋 · 𝑌) ∈ 𝐼 ↔ ((𝑋 · 𝑌) ∈ 𝐵 ∧ ¬ (𝑋 · 𝑌) ∈ 𝑈 ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = (𝑋 · 𝑌) → (𝑥𝑈𝑦𝑈))))
65simp3bi 1181 . . 3 ((𝑋 · 𝑌) ∈ 𝐼 → ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = (𝑋 · 𝑌) → (𝑥𝑈𝑦𝑈)))
7 eqid 2825 . . . 4 (𝑋 · 𝑌) = (𝑋 · 𝑌)
8 oveq1 6917 . . . . . . 7 (𝑥 = 𝑋 → (𝑥 · 𝑦) = (𝑋 · 𝑦))
98eqeq1d 2827 . . . . . 6 (𝑥 = 𝑋 → ((𝑥 · 𝑦) = (𝑋 · 𝑌) ↔ (𝑋 · 𝑦) = (𝑋 · 𝑌)))
10 eleq1 2894 . . . . . . 7 (𝑥 = 𝑋 → (𝑥𝑈𝑋𝑈))
1110orbi1d 945 . . . . . 6 (𝑥 = 𝑋 → ((𝑥𝑈𝑦𝑈) ↔ (𝑋𝑈𝑦𝑈)))
129, 11imbi12d 336 . . . . 5 (𝑥 = 𝑋 → (((𝑥 · 𝑦) = (𝑋 · 𝑌) → (𝑥𝑈𝑦𝑈)) ↔ ((𝑋 · 𝑦) = (𝑋 · 𝑌) → (𝑋𝑈𝑦𝑈))))
13 oveq2 6918 . . . . . . 7 (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌))
1413eqeq1d 2827 . . . . . 6 (𝑦 = 𝑌 → ((𝑋 · 𝑦) = (𝑋 · 𝑌) ↔ (𝑋 · 𝑌) = (𝑋 · 𝑌)))
15 eleq1 2894 . . . . . . 7 (𝑦 = 𝑌 → (𝑦𝑈𝑌𝑈))
1615orbi2d 944 . . . . . 6 (𝑦 = 𝑌 → ((𝑋𝑈𝑦𝑈) ↔ (𝑋𝑈𝑌𝑈)))
1714, 16imbi12d 336 . . . . 5 (𝑦 = 𝑌 → (((𝑋 · 𝑦) = (𝑋 · 𝑌) → (𝑋𝑈𝑦𝑈)) ↔ ((𝑋 · 𝑌) = (𝑋 · 𝑌) → (𝑋𝑈𝑌𝑈))))
1812, 17rspc2v 3539 . . . 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 1149 1 ((𝑋𝐵𝑌𝐵 ∧ (𝑋 · 𝑌) ∈ 𝐼) → (𝑋𝑈𝑌𝑈))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386  wo 878  w3a 1111   = wceq 1656  wcel 2164  wral 3117  cfv 6127  (class class class)co 6910  Basecbs 16229  .rcmulr 16313  Unitcui 19000  Irredcir 19001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-iota 6090  df-fun 6129  df-fv 6135  df-ov 6913  df-irred 19004
This theorem is referenced by:  prmirredlem  20208
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