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

Theorem irredcl 17673
Description: An irreducible element is in the ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypotheses
Ref Expression
irredn0.i  |-  I  =  (Irred `  R )
irredcl.b  |-  B  =  ( Base `  R
)
Assertion
Ref Expression
irredcl  |-  ( X  e.  I  ->  X  e.  B )

Proof of Theorem irredcl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 irredcl.b . . 3  |-  B  =  ( Base `  R
)
2 eqid 2402 . . 3  |-  (Unit `  R )  =  (Unit `  R )
3 irredn0.i . . 3  |-  I  =  (Irred `  R )
4 eqid 2402 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
51, 2, 3, 4isirred2 17670 . 2  |-  ( X  e.  I  <->  ( X  e.  B  /\  -.  X  e.  (Unit `  R )  /\  A. x  e.  B  A. y  e.  B  ( ( x ( .r `  R ) y )  =  X  ->  ( x  e.  (Unit `  R )  \/  y  e.  (Unit `  R ) ) ) ) )
65simp1bi 1012 1  |-  ( X  e.  I  ->  X  e.  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 366    = wceq 1405    e. wcel 1842   A.wral 2754   ` cfv 5569  (class class class)co 6278   Basecbs 14841   .rcmulr 14910  Unitcui 17608  Irredcir 17609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-iota 5533  df-fun 5571  df-fv 5577  df-ov 6281  df-irred 17612
This theorem is referenced by:  irredrmul  17676  irredneg  17679  prmirred  18832
  Copyright terms: Public domain W3C validator