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Theorem irredmul 20243
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 = (Irredβ€˜π‘…)
irredmul.b 𝐡 = (Baseβ€˜π‘…)
irredmul.u π‘ˆ = (Unitβ€˜π‘…)
irredmul.t Β· = (.rβ€˜π‘…)
Assertion
Ref Expression
irredmul ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ (𝑋 Β· π‘Œ) ∈ 𝐼) β†’ (𝑋 ∈ π‘ˆ ∨ π‘Œ ∈ π‘ˆ))
Proof of Theorem irredmul
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 irredmul.b . . . . 5 𝐡 = (Baseβ€˜π‘…)
2 irredmul.u . . . . 5 π‘ˆ = (Unitβ€˜π‘…)
3 irredn0.i . . . . 5 𝐼 = (Irredβ€˜π‘…)
4 irredmul.t . . . . 5 Β· = (.rβ€˜π‘…)
51, 2, 3, 4isirred2 20235 . . . 4 ((𝑋 Β· π‘Œ) ∈ 𝐼 ↔ ((𝑋 Β· π‘Œ) ∈ 𝐡 ∧ Β¬ (𝑋 Β· π‘Œ) ∈ π‘ˆ ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = (𝑋 Β· π‘Œ) β†’ (π‘₯ ∈ π‘ˆ ∨ 𝑦 ∈ π‘ˆ))))
65simp3bi 1148 . . 3 ((𝑋 Β· π‘Œ) ∈ 𝐼 β†’ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = (𝑋 Β· π‘Œ) β†’ (π‘₯ ∈ π‘ˆ ∨ 𝑦 ∈ π‘ˆ)))
7 eqid 2733 . . . 4 (𝑋 Β· π‘Œ) = (𝑋 Β· π‘Œ)
8 oveq1 7416 . . . . . . 7 (π‘₯ = 𝑋 β†’ (π‘₯ Β· 𝑦) = (𝑋 Β· 𝑦))
98eqeq1d 2735 . . . . . 6 (π‘₯ = 𝑋 β†’ ((π‘₯ Β· 𝑦) = (𝑋 Β· π‘Œ) ↔ (𝑋 Β· 𝑦) = (𝑋 Β· π‘Œ)))
10 eleq1 2822 . . . . . . 7 (π‘₯ = 𝑋 β†’ (π‘₯ ∈ π‘ˆ ↔ 𝑋 ∈ π‘ˆ))
1110orbi1d 916 . . . . . 6 (π‘₯ = 𝑋 β†’ ((π‘₯ ∈ π‘ˆ ∨ 𝑦 ∈ π‘ˆ) ↔ (𝑋 ∈ π‘ˆ ∨ 𝑦 ∈ π‘ˆ)))
129, 11imbi12d 345 . . . . 5 (π‘₯ = 𝑋 β†’ (((π‘₯ Β· 𝑦) = (𝑋 Β· π‘Œ) β†’ (π‘₯ ∈ π‘ˆ ∨ 𝑦 ∈ π‘ˆ)) ↔ ((𝑋 Β· 𝑦) = (𝑋 Β· π‘Œ) β†’ (𝑋 ∈ π‘ˆ ∨ 𝑦 ∈ π‘ˆ))))
13 oveq2 7417 . . . . . . 7 (𝑦 = π‘Œ β†’ (𝑋 Β· 𝑦) = (𝑋 Β· π‘Œ))
1413eqeq1d 2735 . . . . . 6 (𝑦 = π‘Œ β†’ ((𝑋 Β· 𝑦) = (𝑋 Β· π‘Œ) ↔ (𝑋 Β· π‘Œ) = (𝑋 Β· π‘Œ)))
15 eleq1 2822 . . . . . . 7 (𝑦 = π‘Œ β†’ (𝑦 ∈ π‘ˆ ↔ π‘Œ ∈ π‘ˆ))
1615orbi2d 915 . . . . . 6 (𝑦 = π‘Œ β†’ ((𝑋 ∈ π‘ˆ ∨ 𝑦 ∈ π‘ˆ) ↔ (𝑋 ∈ π‘ˆ ∨ π‘Œ ∈ π‘ˆ)))
1714, 16imbi12d 345 . . . . 5 (𝑦 = π‘Œ β†’ (((𝑋 Β· 𝑦) = (𝑋 Β· π‘Œ) β†’ (𝑋 ∈ π‘ˆ ∨ 𝑦 ∈ π‘ˆ)) ↔ ((𝑋 Β· π‘Œ) = (𝑋 Β· π‘Œ) β†’ (𝑋 ∈ π‘ˆ ∨ π‘Œ ∈ π‘ˆ))))
1812, 17rspc2v 3623 . . . 4 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = (𝑋 Β· π‘Œ) β†’ (π‘₯ ∈ π‘ˆ ∨ 𝑦 ∈ π‘ˆ)) β†’ ((𝑋 Β· π‘Œ) = (𝑋 Β· π‘Œ) β†’ (𝑋 ∈ π‘ˆ ∨ π‘Œ ∈ π‘ˆ))))
197, 18mpii 46 . . 3 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = (𝑋 Β· π‘Œ) β†’ (π‘₯ ∈ π‘ˆ ∨ 𝑦 ∈ π‘ˆ)) β†’ (𝑋 ∈ π‘ˆ ∨ π‘Œ ∈ π‘ˆ)))
206, 19syl5 34 . 2 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 Β· π‘Œ) ∈ 𝐼 β†’ (𝑋 ∈ π‘ˆ ∨ π‘Œ ∈ π‘ˆ)))
21203impia 1118 1 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ (𝑋 Β· π‘Œ) ∈ 𝐼) β†’ (𝑋 ∈ π‘ˆ ∨ π‘Œ ∈ π‘ˆ))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  .rcmulr 17198  Unitcui 20169  Irredcir 20170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-irred 20173
This theorem is referenced by:  prmirredlem  21042  mxidlirred  32588
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