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Theorem irredn0 16800
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 2443 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
2 irredn0.z . . . . . . . . . 10  |-  .0.  =  ( 0g `  R )
31, 2rng0cl 16671 . . . . . . . . 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 3343 . . . . . . . 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 2443 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( .r `  R
)
81, 7, 2rnglz 16686 . . . . . . . . 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 6103 . . . . . . . . 9  |-  ( x  =  .0.  ->  (
x ( .r `  R ) y )  =  (  .0.  ( .r `  R ) y ) )
1211eqeq1d 2451 . . . . . . . 8  |-  ( x  =  .0.  ->  (
( x ( .r
`  R ) y )  =  .0.  <->  (  .0.  ( .r `  R ) y )  =  .0.  ) )
13 oveq2 6104 . . . . . . . . 9  |-  ( y  =  .0.  ->  (  .0.  ( .r `  R
) y )  =  (  .0.  ( .r
`  R )  .0.  ) )
1413eqeq1d 2451 . . . . . . . 8  |-  ( y  =  .0.  ->  (
(  .0.  ( .r
`  R ) y )  =  .0.  <->  (  .0.  ( .r `  R )  .0.  )  =  .0.  ) )
1512, 14rspc2ev 3086 . . . . . . 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 1218 . . . . . 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 2443 . . . . . 6  |-  (Unit `  R )  =  (Unit `  R )
20 irredn0.i . . . . . 6  |-  I  =  (Irred `  R )
21 eqid 2443 . . . . . 6  |-  ( (
Base `  R )  \  (Unit `  R )
)  =  ( (
Base `  R )  \  (Unit `  R )
)
221, 19, 20, 21, 7isnirred 16797 . . . . 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 2503 . . . 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 2650 . 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 1369    e. wcel 1756    =/= wne 2611   E.wrex 2721    \ cdif 3330   ` cfv 5423  (class class class)co 6096   Basecbs 14179   .rcmulr 14244   0gc0g 14383   Ringcrg 16650  Unitcui 16736  Irredcir 16737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-plusg 14256  df-0g 14385  df-mnd 15420  df-grp 15550  df-minusg 15551  df-mgp 16597  df-rng 16652  df-irred 16740
This theorem is referenced by:  prmirred  17924  prmirredOLD  17927
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