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

Theorem isnirred 17544
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  |-  B  =  ( Base `  R
)
irred.2  |-  U  =  (Unit `  R )
irred.3  |-  I  =  (Irred `  R )
irred.4  |-  N  =  ( B  \  U
)
irred.5  |-  .x.  =  ( .r `  R )
Assertion
Ref Expression
isnirred  |-  ( X  e.  B  ->  ( -.  X  e.  I  <->  ( X  e.  U  \/  E. x  e.  N  E. y  e.  N  (
x  .x.  y )  =  X ) ) )
Distinct variable groups:    x, y, N    x, R, y    x, X, y
Allowed substitution hints:    B( x, y)    .x. ( x, y)    U( x, y)    I( x, y)

Proof of Theorem isnirred
StepHypRef Expression
1 irred.4 . . . . . . 7  |-  N  =  ( B  \  U
)
21eleq2i 2532 . . . . . 6  |-  ( X  e.  N  <->  X  e.  ( B  \  U ) )
3 eldif 3471 . . . . . 6  |-  ( X  e.  ( B  \  U )  <->  ( X  e.  B  /\  -.  X  e.  U ) )
42, 3bitri 249 . . . . 5  |-  ( X  e.  N  <->  ( X  e.  B  /\  -.  X  e.  U ) )
54baibr 902 . . . 4  |-  ( X  e.  B  ->  ( -.  X  e.  U  <->  X  e.  N ) )
6 df-ne 2651 . . . . . . . . 9  |-  ( ( x  .x.  y )  =/=  X  <->  -.  (
x  .x.  y )  =  X )
76ralbii 2885 . . . . . . . 8  |-  ( A. y  e.  N  (
x  .x.  y )  =/=  X  <->  A. y  e.  N  -.  ( x  .x.  y
)  =  X )
8 ralnex 2900 . . . . . . . 8  |-  ( A. y  e.  N  -.  ( x  .x.  y )  =  X  <->  -.  E. y  e.  N  ( x  .x.  y )  =  X )
97, 8bitri 249 . . . . . . 7  |-  ( A. y  e.  N  (
x  .x.  y )  =/=  X  <->  -.  E. y  e.  N  ( x  .x.  y )  =  X )
109ralbii 2885 . . . . . 6  |-  ( A. x  e.  N  A. y  e.  N  (
x  .x.  y )  =/=  X  <->  A. x  e.  N  -.  E. y  e.  N  ( x  .x.  y )  =  X )
11 ralnex 2900 . . . . . 6  |-  ( A. x  e.  N  -.  E. y  e.  N  ( x  .x.  y )  =  X  <->  -.  E. x  e.  N  E. y  e.  N  ( x  .x.  y )  =  X )
1210, 11bitr2i 250 . . . . 5  |-  ( -. 
E. x  e.  N  E. y  e.  N  ( x  .x.  y )  =  X  <->  A. x  e.  N  A. y  e.  N  ( x  .x.  y )  =/=  X
)
1312a1i 11 . . . 4  |-  ( X  e.  B  ->  ( -.  E. x  e.  N  E. y  e.  N  ( x  .x.  y )  =  X  <->  A. x  e.  N  A. y  e.  N  ( x  .x.  y )  =/=  X
) )
145, 13anbi12d 708 . . 3  |-  ( X  e.  B  ->  (
( -.  X  e.  U  /\  -.  E. x  e.  N  E. y  e.  N  (
x  .x.  y )  =  X )  <->  ( X  e.  N  /\  A. x  e.  N  A. y  e.  N  ( x  .x.  y )  =/=  X
) ) )
15 ioran 488 . . 3  |-  ( -.  ( X  e.  U  \/  E. x  e.  N  E. y  e.  N  ( x  .x.  y )  =  X )  <->  ( -.  X  e.  U  /\  -.  E. x  e.  N  E. y  e.  N  ( x  .x.  y )  =  X ) )
16 irred.1 . . . 4  |-  B  =  ( Base `  R
)
17 irred.2 . . . 4  |-  U  =  (Unit `  R )
18 irred.3 . . . 4  |-  I  =  (Irred `  R )
19 irred.5 . . . 4  |-  .x.  =  ( .r `  R )
2016, 17, 18, 1, 19isirred 17543 . . 3  |-  ( X  e.  I  <->  ( X  e.  N  /\  A. x  e.  N  A. y  e.  N  ( x  .x.  y )  =/=  X
) )
2114, 15, 203bitr4g 288 . 2  |-  ( X  e.  B  ->  ( -.  ( X  e.  U  \/  E. x  e.  N  E. y  e.  N  ( x  .x.  y )  =  X )  <->  X  e.  I ) )
2221con1bid 328 1  |-  ( X  e.  B  ->  ( -.  X  e.  I  <->  ( X  e.  U  \/  E. x  e.  N  E. y  e.  N  (
x  .x.  y )  =  X ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805    \ cdif 3458   ` cfv 5570  (class class class)co 6270   Basecbs 14716   .rcmulr 14785  Unitcui 17483  Irredcir 17484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-irred 17487
This theorem is referenced by:  irredn0  17547  irredrmul  17551
  Copyright terms: Public domain W3C validator