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Theorem irredn0 15810
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 2438 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
2 irredn0.z . . . . . . . . . 10  |-  .0.  =  ( 0g `  R )
31, 2rng0cl 15687 . . . . . . . . 9  |-  ( R  e.  Ring  ->  .0.  e.  ( Base `  R )
)
43anim1i 553 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  -.  .0.  e.  (Unit `  R
) )  ->  (  .0.  e.  ( Base `  R
)  /\  -.  .0.  e.  (Unit `  R )
) )
5 eldif 3332 . . . . . . . 8  |-  (  .0. 
e.  ( ( Base `  R )  \  (Unit `  R ) )  <->  (  .0.  e.  ( Base `  R
)  /\  -.  .0.  e.  (Unit `  R )
) )
64, 5sylibr 205 . . . . . . 7  |-  ( ( R  e.  Ring  /\  -.  .0.  e.  (Unit `  R
) )  ->  .0.  e.  ( ( Base `  R
)  \  (Unit `  R
) ) )
7 eqid 2438 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( .r `  R
)
81, 7, 2rnglz 15702 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  .0.  e.  ( Base `  R
) )  ->  (  .0.  ( .r `  R
)  .0.  )  =  .0.  )
93, 8mpdan 651 . . . . . . . 8  |-  ( R  e.  Ring  ->  (  .0.  ( .r `  R
)  .0.  )  =  .0.  )
109adantr 453 . . . . . . 7  |-  ( ( R  e.  Ring  /\  -.  .0.  e.  (Unit `  R
) )  ->  (  .0.  ( .r `  R
)  .0.  )  =  .0.  )
11 oveq1 6090 . . . . . . . . 9  |-  ( x  =  .0.  ->  (
x ( .r `  R ) y )  =  (  .0.  ( .r `  R ) y ) )
1211eqeq1d 2446 . . . . . . . 8  |-  ( x  =  .0.  ->  (
( x ( .r
`  R ) y )  =  .0.  <->  (  .0.  ( .r `  R ) y )  =  .0.  ) )
13 oveq2 6091 . . . . . . . . 9  |-  ( y  =  .0.  ->  (  .0.  ( .r `  R
) y )  =  (  .0.  ( .r
`  R )  .0.  ) )
1413eqeq1d 2446 . . . . . . . 8  |-  ( y  =  .0.  ->  (
(  .0.  ( .r
`  R ) y )  =  .0.  <->  (  .0.  ( .r `  R )  .0.  )  =  .0.  ) )
1512, 14rspc2ev 3062 . . . . . . 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 1185 . . . . . 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 425 . . . . 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 369 . . . 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 2438 . . . . . 6  |-  (Unit `  R )  =  (Unit `  R )
20 irredn0.i . . . . . 6  |-  I  =  (Irred `  R )
21 eqid 2438 . . . . . 6  |-  ( (
Base `  R )  \  (Unit `  R )
)  =  ( (
Base `  R )  \  (Unit `  R )
)
221, 19, 20, 21, 7isnirred 15807 . . . . 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 225 . . 3  |-  ( R  e.  Ring  ->  -.  .0.  e.  I )
2524adantr 453 . 2  |-  ( ( R  e.  Ring  /\  X  e.  I )  ->  -.  .0.  e.  I )
26 simpr 449 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  I )  ->  X  e.  I )
27 eleq1 2498 . . . 4  |-  ( X  =  .0.  ->  ( X  e.  I  <->  .0.  e.  I ) )
2826, 27syl5ibcom 213 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  I )  ->  ( X  =  .0.  ->  .0. 
e.  I ) )
2928necon3bd 2640 . 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708    \ cdif 3319   ` cfv 5456  (class class class)co 6083   Basecbs 13471   .rcmulr 13532   0gc0g 13725   Ringcrg 15662  Unitcui 15746  Irredcir 15747
This theorem is referenced by:  prmirred  16777
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-plusg 13544  df-0g 13729  df-mnd 14692  df-grp 14814  df-minusg 14815  df-mgp 15651  df-rng 15665  df-irred 15750
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