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Theorem isnirred 20234
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 2826 . . . . . 6 (𝑋 ∈ 𝑁 ↔ 𝑋 ∈ (𝐡 βˆ– π‘ˆ))
3 eldif 3959 . . . . . 6 (𝑋 ∈ (𝐡 βˆ– π‘ˆ) ↔ (𝑋 ∈ 𝐡 ∧ Β¬ 𝑋 ∈ π‘ˆ))
42, 3bitri 275 . . . . 5 (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ 𝐡 ∧ Β¬ 𝑋 ∈ π‘ˆ))
54baibr 538 . . . 4 (𝑋 ∈ 𝐡 β†’ (Β¬ 𝑋 ∈ π‘ˆ ↔ 𝑋 ∈ 𝑁))
6 df-ne 2942 . . . . . . . . 9 ((π‘₯ Β· 𝑦) β‰  𝑋 ↔ Β¬ (π‘₯ Β· 𝑦) = 𝑋)
76ralbii 3094 . . . . . . . 8 (βˆ€π‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) β‰  𝑋 ↔ βˆ€π‘¦ ∈ 𝑁 Β¬ (π‘₯ Β· 𝑦) = 𝑋)
8 ralnex 3073 . . . . . . . 8 (βˆ€π‘¦ ∈ 𝑁 Β¬ (π‘₯ Β· 𝑦) = 𝑋 ↔ Β¬ βˆƒπ‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) = 𝑋)
97, 8bitri 275 . . . . . . 7 (βˆ€π‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) β‰  𝑋 ↔ Β¬ βˆƒπ‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) = 𝑋)
109ralbii 3094 . . . . . 6 (βˆ€π‘₯ ∈ 𝑁 βˆ€π‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) β‰  𝑋 ↔ βˆ€π‘₯ ∈ 𝑁 Β¬ βˆƒπ‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) = 𝑋)
11 ralnex 3073 . . . . . 6 (βˆ€π‘₯ ∈ 𝑁 Β¬ βˆƒπ‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) = 𝑋 ↔ Β¬ βˆƒπ‘₯ ∈ 𝑁 βˆƒπ‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) = 𝑋)
1210, 11bitr2i 276 . . . . 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 632 . . 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 983 . . 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 20233 . . 3 (𝑋 ∈ 𝐼 ↔ (𝑋 ∈ 𝑁 ∧ βˆ€π‘₯ ∈ 𝑁 βˆ€π‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) β‰  𝑋))
2114, 15, 203bitr4g 314 . 2 (𝑋 ∈ 𝐡 β†’ (Β¬ (𝑋 ∈ π‘ˆ ∨ βˆƒπ‘₯ ∈ 𝑁 βˆƒπ‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) = 𝑋) ↔ 𝑋 ∈ 𝐼))
2221con1bid 356 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 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071   βˆ– cdif 3946  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:  irredn0  20237  irredrmul  20241
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