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Theorem irredcl 17134
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 2467 . . 3  |-  (Unit `  R )  =  (Unit `  R )
3 irredn0.i . . 3  |-  I  =  (Irred `  R )
4 eqid 2467 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
51, 2, 3, 4isirred2 17131 . 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 1011 1  |-  ( X  e.  I  ->  X  e.  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    = wceq 1379    e. wcel 1767   A.wral 2814   ` cfv 5586  (class class class)co 6282   Basecbs 14483   .rcmulr 14549  Unitcui 17069  Irredcir 17070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-ov 6285  df-irred 17073
This theorem is referenced by:  irredrmul  17137  irredneg  17140  prmirred  18289  prmirredOLD  18292
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