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Theorem irredcl 16899
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 2451 . . 3  |-  (Unit `  R )  =  (Unit `  R )
3 irredn0.i . . 3  |-  I  =  (Irred `  R )
4 eqid 2451 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
51, 2, 3, 4isirred2 16896 . 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 1003 1  |-  ( X  e.  I  ->  X  e.  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    = wceq 1370    e. wcel 1758   A.wral 2793   ` cfv 5513  (class class class)co 6187   Basecbs 14273   .rcmulr 14338  Unitcui 16834  Irredcir 16835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-iota 5476  df-fun 5515  df-fv 5521  df-ov 6190  df-irred 16838
This theorem is referenced by:  irredrmul  16902  irredneg  16905  prmirred  18025  prmirredOLD  18028
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