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Theorem irredmul 20331
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 20323 . . . 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 1144 . . 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 2726 . . . 4 (𝑋 Β· π‘Œ) = (𝑋 Β· π‘Œ)
8 oveq1 7412 . . . . . . 7 (π‘₯ = 𝑋 β†’ (π‘₯ Β· 𝑦) = (𝑋 Β· 𝑦))
98eqeq1d 2728 . . . . . 6 (π‘₯ = 𝑋 β†’ ((π‘₯ Β· 𝑦) = (𝑋 Β· π‘Œ) ↔ (𝑋 Β· 𝑦) = (𝑋 Β· π‘Œ)))
10 eleq1 2815 . . . . . . 7 (π‘₯ = 𝑋 β†’ (π‘₯ ∈ π‘ˆ ↔ 𝑋 ∈ π‘ˆ))
1110orbi1d 913 . . . . . 6 (π‘₯ = 𝑋 β†’ ((π‘₯ ∈ π‘ˆ ∨ 𝑦 ∈ π‘ˆ) ↔ (𝑋 ∈ π‘ˆ ∨ 𝑦 ∈ π‘ˆ)))
129, 11imbi12d 344 . . . . 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 7413 . . . . . . 7 (𝑦 = π‘Œ β†’ (𝑋 Β· 𝑦) = (𝑋 Β· π‘Œ))
1413eqeq1d 2728 . . . . . 6 (𝑦 = π‘Œ β†’ ((𝑋 Β· 𝑦) = (𝑋 Β· π‘Œ) ↔ (𝑋 Β· π‘Œ) = (𝑋 Β· π‘Œ)))
15 eleq1 2815 . . . . . . 7 (𝑦 = π‘Œ β†’ (𝑦 ∈ π‘ˆ ↔ π‘Œ ∈ π‘ˆ))
1615orbi2d 912 . . . . . 6 (𝑦 = π‘Œ β†’ ((𝑋 ∈ π‘ˆ ∨ 𝑦 ∈ π‘ˆ) ↔ (𝑋 ∈ π‘ˆ ∨ π‘Œ ∈ π‘ˆ)))
1714, 16imbi12d 344 . . . . 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 3617 . . . 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 1114 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 395   ∨ wo 844   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153  .rcmulr 17207  Unitcui 20257  Irredcir 20258
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fv 6545  df-ov 7408  df-irred 20261
This theorem is referenced by:  prmirredlem  21359  mxidlirred  33094  irredminply  33293
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