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Theorem irredn0 17201
Description: The additive identity is not irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypotheses
Ref Expression
irredn0.i  |-  I  =  (Irred `  R )
irredn0.z  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
irredn0  |-  ( ( R  e.  Ring  /\  X  e.  I )  ->  X  =/=  .0.  )

Proof of Theorem irredn0
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
2 irredn0.z . . . . . . . . . 10  |-  .0.  =  ( 0g `  R )
31, 2ring0cl 17069 . . . . . . . . 9  |-  ( R  e.  Ring  ->  .0.  e.  ( Base `  R )
)
43anim1i 568 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  -.  .0.  e.  (Unit `  R
) )  ->  (  .0.  e.  ( Base `  R
)  /\  -.  .0.  e.  (Unit `  R )
) )
5 eldif 3491 . . . . . . . 8  |-  (  .0. 
e.  ( ( Base `  R )  \  (Unit `  R ) )  <->  (  .0.  e.  ( Base `  R
)  /\  -.  .0.  e.  (Unit `  R )
) )
64, 5sylibr 212 . . . . . . 7  |-  ( ( R  e.  Ring  /\  -.  .0.  e.  (Unit `  R
) )  ->  .0.  e.  ( ( Base `  R
)  \  (Unit `  R
) ) )
7 eqid 2467 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( .r `  R
)
81, 7, 2ringlz 17084 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  .0.  e.  ( Base `  R
) )  ->  (  .0.  ( .r `  R
)  .0.  )  =  .0.  )
93, 8mpdan 668 . . . . . . . 8  |-  ( R  e.  Ring  ->  (  .0.  ( .r `  R
)  .0.  )  =  .0.  )
109adantr 465 . . . . . . 7  |-  ( ( R  e.  Ring  /\  -.  .0.  e.  (Unit `  R
) )  ->  (  .0.  ( .r `  R
)  .0.  )  =  .0.  )
11 oveq1 6301 . . . . . . . . 9  |-  ( x  =  .0.  ->  (
x ( .r `  R ) y )  =  (  .0.  ( .r `  R ) y ) )
1211eqeq1d 2469 . . . . . . . 8  |-  ( x  =  .0.  ->  (
( x ( .r
`  R ) y )  =  .0.  <->  (  .0.  ( .r `  R ) y )  =  .0.  ) )
13 oveq2 6302 . . . . . . . . 9  |-  ( y  =  .0.  ->  (  .0.  ( .r `  R
) y )  =  (  .0.  ( .r
`  R )  .0.  ) )
1413eqeq1d 2469 . . . . . . . 8  |-  ( y  =  .0.  ->  (
(  .0.  ( .r
`  R ) y )  =  .0.  <->  (  .0.  ( .r `  R )  .0.  )  =  .0.  ) )
1512, 14rspc2ev 3230 . . . . . . 7  |-  ( (  .0.  e.  ( (
Base `  R )  \  (Unit `  R )
)  /\  .0.  e.  ( ( Base `  R
)  \  (Unit `  R
) )  /\  (  .0.  ( .r `  R
)  .0.  )  =  .0.  )  ->  E. x  e.  ( ( Base `  R
)  \  (Unit `  R
) ) E. y  e.  ( ( Base `  R
)  \  (Unit `  R
) ) ( x ( .r `  R
) y )  =  .0.  )
166, 6, 10, 15syl3anc 1228 . . . . . 6  |-  ( ( R  e.  Ring  /\  -.  .0.  e.  (Unit `  R
) )  ->  E. x  e.  ( ( Base `  R
)  \  (Unit `  R
) ) E. y  e.  ( ( Base `  R
)  \  (Unit `  R
) ) ( x ( .r `  R
) y )  =  .0.  )
1716ex 434 . . . . 5  |-  ( R  e.  Ring  ->  ( -.  .0.  e.  (Unit `  R )  ->  E. x  e.  ( ( Base `  R
)  \  (Unit `  R
) ) E. y  e.  ( ( Base `  R
)  \  (Unit `  R
) ) ( x ( .r `  R
) y )  =  .0.  ) )
1817orrd 378 . . . 4  |-  ( R  e.  Ring  ->  (  .0. 
e.  (Unit `  R
)  \/  E. x  e.  ( ( Base `  R
)  \  (Unit `  R
) ) E. y  e.  ( ( Base `  R
)  \  (Unit `  R
) ) ( x ( .r `  R
) y )  =  .0.  ) )
19 eqid 2467 . . . . . 6  |-  (Unit `  R )  =  (Unit `  R )
20 irredn0.i . . . . . 6  |-  I  =  (Irred `  R )
21 eqid 2467 . . . . . 6  |-  ( (
Base `  R )  \  (Unit `  R )
)  =  ( (
Base `  R )  \  (Unit `  R )
)
221, 19, 20, 21, 7isnirred 17198 . . . . 5  |-  (  .0. 
e.  ( Base `  R
)  ->  ( -.  .0.  e.  I  <->  (  .0.  e.  (Unit `  R )  \/  E. x  e.  ( ( Base `  R
)  \  (Unit `  R
) ) E. y  e.  ( ( Base `  R
)  \  (Unit `  R
) ) ( x ( .r `  R
) y )  =  .0.  ) ) )
233, 22syl 16 . . . 4  |-  ( R  e.  Ring  ->  ( -.  .0.  e.  I  <->  (  .0.  e.  (Unit `  R )  \/  E. x  e.  ( ( Base `  R
)  \  (Unit `  R
) ) E. y  e.  ( ( Base `  R
)  \  (Unit `  R
) ) ( x ( .r `  R
) y )  =  .0.  ) ) )
2418, 23mpbird 232 . . 3  |-  ( R  e.  Ring  ->  -.  .0.  e.  I )
2524adantr 465 . 2  |-  ( ( R  e.  Ring  /\  X  e.  I )  ->  -.  .0.  e.  I )
26 simpr 461 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  I )  ->  X  e.  I )
27 eleq1 2539 . . . 4  |-  ( X  =  .0.  ->  ( X  e.  I  <->  .0.  e.  I ) )
2826, 27syl5ibcom 220 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  I )  ->  ( X  =  .0.  ->  .0. 
e.  I ) )
2928necon3bd 2679 . 2  |-  ( ( R  e.  Ring  /\  X  e.  I )  ->  ( -.  .0.  e.  I  ->  X  =/=  .0.  ) )
3025, 29mpd 15 1  |-  ( ( R  e.  Ring  /\  X  e.  I )  ->  X  =/=  .0.  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2818    \ cdif 3478   ` cfv 5593  (class class class)co 6294   Basecbs 14502   .rcmulr 14568   0gc0g 14707   Ringcrg 17047  Unitcui 17137  Irredcir 17138
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-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  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-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-recs 7052  df-rdg 7086  df-er 7321  df-en 7527  df-dom 7528  df-sdom 7529  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-nn 10547  df-2 10604  df-ndx 14505  df-slot 14506  df-base 14507  df-sets 14508  df-plusg 14580  df-0g 14709  df-mgm 15741  df-sgrp 15764  df-mnd 15774  df-grp 15906  df-minusg 15907  df-mgp 16991  df-ring 17049  df-irred 17141
This theorem is referenced by:  prmirred  18371  prmirredOLD  18374
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