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Theorem irredcl 15772
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 2412 . . 3  |-  (Unit `  R )  =  (Unit `  R )
3 irredn0.i . . 3  |-  I  =  (Irred `  R )
4 eqid 2412 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
51, 2, 3, 4isirred2 15769 . 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 972 1  |-  ( X  e.  I  ->  X  e.  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    = wceq 1649    e. wcel 1721   A.wral 2674   ` cfv 5421  (class class class)co 6048   Basecbs 13432   .rcmulr 13493  Unitcui 15707  Irredcir 15708
This theorem is referenced by:  irredrmul  15775  irredneg  15778  prmirred  16738
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5385  df-fun 5423  df-fv 5429  df-ov 6051  df-irred 15711
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