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Theorem irredcl 15486
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 2283 . . 3  |-  (Unit `  R )  =  (Unit `  R )
3 irredn0.i . . 3  |-  I  =  (Irred `  R )
4 eqid 2283 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
51, 2, 3, 4isirred2 15483 . 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 970 1  |-  ( X  e.  I  ->  X  e.  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    = wceq 1623    e. wcel 1684   A.wral 2543   ` cfv 5255  (class class class)co 5858   Basecbs 13148   .rcmulr 13209  Unitcui 15421  Irredcir 15422
This theorem is referenced by:  irredrmul  15489  irredneg  15492  prmirred  16448
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-irred 15425
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