MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isnirred Structured version   Visualization version   GIF version

Theorem isnirred 20348
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 2820 . . . . . 6 (𝑋 ∈ 𝑁 ↔ 𝑋 ∈ (𝐡 βˆ– π‘ˆ))
3 eldif 3954 . . . . . 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 536 . . . 4 (𝑋 ∈ 𝐡 β†’ (Β¬ 𝑋 ∈ π‘ˆ ↔ 𝑋 ∈ 𝑁))
6 df-ne 2936 . . . . . . . . 9 ((π‘₯ Β· 𝑦) β‰  𝑋 ↔ Β¬ (π‘₯ Β· 𝑦) = 𝑋)
76ralbii 3088 . . . . . . . 8 (βˆ€π‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) β‰  𝑋 ↔ βˆ€π‘¦ ∈ 𝑁 Β¬ (π‘₯ Β· 𝑦) = 𝑋)
8 ralnex 3067 . . . . . . . 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 3088 . . . . . 6 (βˆ€π‘₯ ∈ 𝑁 βˆ€π‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) β‰  𝑋 ↔ βˆ€π‘₯ ∈ 𝑁 Β¬ βˆƒπ‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) = 𝑋)
11 ralnex 3067 . . . . . 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 630 . . 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 20347 . . 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 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 395   ∨ wo 846   = wceq 1534   ∈ wcel 2099   β‰  wne 2935  βˆ€wral 3056  βˆƒwrex 3065   βˆ– cdif 3941  β€˜cfv 6542  (class class class)co 7414  Basecbs 17171  .rcmulr 17225  Unitcui 20283  Irredcir 20284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7417  df-irred 20287
This theorem is referenced by:  irredn0  20351  irredrmul  20355
  Copyright terms: Public domain W3C validator