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Theorem isnirred 20238
Description: The property of being a non-irreducible (reducible) element in a ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypotheses
Ref Expression
irred.1 𝐡 = (Baseβ€˜π‘…)
irred.2 π‘ˆ = (Unitβ€˜π‘…)
irred.3 𝐼 = (Irredβ€˜π‘…)
irred.4 𝑁 = (𝐡 βˆ– π‘ˆ)
irred.5 Β· = (.rβ€˜π‘…)
Assertion
Ref Expression
isnirred (𝑋 ∈ 𝐡 β†’ (Β¬ 𝑋 ∈ 𝐼 ↔ (𝑋 ∈ π‘ˆ ∨ βˆƒπ‘₯ ∈ 𝑁 βˆƒπ‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) = 𝑋)))
Distinct variable groups:   π‘₯,𝑦,𝑁   π‘₯,𝑅,𝑦   π‘₯,𝑋,𝑦
Allowed substitution hints:   𝐡(π‘₯,𝑦)   Β· (π‘₯,𝑦)   π‘ˆ(π‘₯,𝑦)   𝐼(π‘₯,𝑦)

Proof of Theorem isnirred
StepHypRef Expression
1 irred.4 . . . . . . 7 𝑁 = (𝐡 βˆ– π‘ˆ)
21eleq2i 2825 . . . . . 6 (𝑋 ∈ 𝑁 ↔ 𝑋 ∈ (𝐡 βˆ– π‘ˆ))
3 eldif 3958 . . . . . 6 (𝑋 ∈ (𝐡 βˆ– π‘ˆ) ↔ (𝑋 ∈ 𝐡 ∧ Β¬ 𝑋 ∈ π‘ˆ))
42, 3bitri 274 . . . . 5 (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ 𝐡 ∧ Β¬ 𝑋 ∈ π‘ˆ))
54baibr 537 . . . 4 (𝑋 ∈ 𝐡 β†’ (Β¬ 𝑋 ∈ π‘ˆ ↔ 𝑋 ∈ 𝑁))
6 df-ne 2941 . . . . . . . . 9 ((π‘₯ Β· 𝑦) β‰  𝑋 ↔ Β¬ (π‘₯ Β· 𝑦) = 𝑋)
76ralbii 3093 . . . . . . . 8 (βˆ€π‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) β‰  𝑋 ↔ βˆ€π‘¦ ∈ 𝑁 Β¬ (π‘₯ Β· 𝑦) = 𝑋)
8 ralnex 3072 . . . . . . . 8 (βˆ€π‘¦ ∈ 𝑁 Β¬ (π‘₯ Β· 𝑦) = 𝑋 ↔ Β¬ βˆƒπ‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) = 𝑋)
97, 8bitri 274 . . . . . . 7 (βˆ€π‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) β‰  𝑋 ↔ Β¬ βˆƒπ‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) = 𝑋)
109ralbii 3093 . . . . . 6 (βˆ€π‘₯ ∈ 𝑁 βˆ€π‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) β‰  𝑋 ↔ βˆ€π‘₯ ∈ 𝑁 Β¬ βˆƒπ‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) = 𝑋)
11 ralnex 3072 . . . . . 6 (βˆ€π‘₯ ∈ 𝑁 Β¬ βˆƒπ‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) = 𝑋 ↔ Β¬ βˆƒπ‘₯ ∈ 𝑁 βˆƒπ‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) = 𝑋)
1210, 11bitr2i 275 . . . . 5 (Β¬ βˆƒπ‘₯ ∈ 𝑁 βˆƒπ‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) = 𝑋 ↔ βˆ€π‘₯ ∈ 𝑁 βˆ€π‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) β‰  𝑋)
1312a1i 11 . . . 4 (𝑋 ∈ 𝐡 β†’ (Β¬ βˆƒπ‘₯ ∈ 𝑁 βˆƒπ‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) = 𝑋 ↔ βˆ€π‘₯ ∈ 𝑁 βˆ€π‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) β‰  𝑋))
145, 13anbi12d 631 . . 3 (𝑋 ∈ 𝐡 β†’ ((Β¬ 𝑋 ∈ π‘ˆ ∧ Β¬ βˆƒπ‘₯ ∈ 𝑁 βˆƒπ‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) = 𝑋) ↔ (𝑋 ∈ 𝑁 ∧ βˆ€π‘₯ ∈ 𝑁 βˆ€π‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) β‰  𝑋)))
15 ioran 982 . . 3 (Β¬ (𝑋 ∈ π‘ˆ ∨ βˆƒπ‘₯ ∈ 𝑁 βˆƒπ‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) = 𝑋) ↔ (Β¬ 𝑋 ∈ π‘ˆ ∧ Β¬ βˆƒπ‘₯ ∈ 𝑁 βˆƒπ‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) = 𝑋))
16 irred.1 . . . 4 𝐡 = (Baseβ€˜π‘…)
17 irred.2 . . . 4 π‘ˆ = (Unitβ€˜π‘…)
18 irred.3 . . . 4 𝐼 = (Irredβ€˜π‘…)
19 irred.5 . . . 4 Β· = (.rβ€˜π‘…)
2016, 17, 18, 1, 19isirred 20237 . . 3 (𝑋 ∈ 𝐼 ↔ (𝑋 ∈ 𝑁 ∧ βˆ€π‘₯ ∈ 𝑁 βˆ€π‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) β‰  𝑋))
2114, 15, 203bitr4g 313 . 2 (𝑋 ∈ 𝐡 β†’ (Β¬ (𝑋 ∈ π‘ˆ ∨ βˆƒπ‘₯ ∈ 𝑁 βˆƒπ‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) = 𝑋) ↔ 𝑋 ∈ 𝐼))
2221con1bid 355 1 (𝑋 ∈ 𝐡 β†’ (Β¬ 𝑋 ∈ 𝐼 ↔ (𝑋 ∈ π‘ˆ ∨ βˆƒπ‘₯ ∈ 𝑁 βˆƒπ‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) = 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070   βˆ– cdif 3945  β€˜cfv 6543  (class class class)co 7411  Basecbs 17146  .rcmulr 17200  Unitcui 20173  Irredcir 20174
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7414  df-irred 20177
This theorem is referenced by:  irredn0  20241  irredrmul  20245
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