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Theorem opprirred 16916
Description: Irreducibility is symmetric, so the irreducible elements of the opposite ring are the same as the original ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypotheses
Ref Expression
opprirred.1  |-  S  =  (oppr
`  R )
opprirred.2  |-  I  =  (Irred `  R )
Assertion
Ref Expression
opprirred  |-  I  =  (Irred `  S )

Proof of Theorem opprirred
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ralcom 2985 . . . . 5  |-  ( A. z  e.  ( ( Base `  R )  \ 
(Unit `  R )
) A. y  e.  ( ( Base `  R
)  \  (Unit `  R
) ) ( z ( .r `  R
) y )  =/=  x  <->  A. y  e.  ( ( Base `  R
)  \  (Unit `  R
) ) A. z  e.  ( ( Base `  R
)  \  (Unit `  R
) ) ( z ( .r `  R
) y )  =/=  x )
2 eqid 2454 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
3 eqid 2454 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
4 opprirred.1 . . . . . . . 8  |-  S  =  (oppr
`  R )
5 eqid 2454 . . . . . . . 8  |-  ( .r
`  S )  =  ( .r `  S
)
62, 3, 4, 5opprmul 16840 . . . . . . 7  |-  ( y ( .r `  S
) z )  =  ( z ( .r
`  R ) y )
76neeq1i 2736 . . . . . 6  |-  ( ( y ( .r `  S ) z )  =/=  x  <->  ( z
( .r `  R
) y )  =/=  x )
872ralbii 2839 . . . . 5  |-  ( A. y  e.  ( ( Base `  R )  \ 
(Unit `  R )
) A. z  e.  ( ( Base `  R
)  \  (Unit `  R
) ) ( y ( .r `  S
) z )  =/=  x  <->  A. y  e.  ( ( Base `  R
)  \  (Unit `  R
) ) A. z  e.  ( ( Base `  R
)  \  (Unit `  R
) ) ( z ( .r `  R
) y )  =/=  x )
91, 8bitr4i 252 . . . 4  |-  ( A. z  e.  ( ( Base `  R )  \ 
(Unit `  R )
) A. y  e.  ( ( Base `  R
)  \  (Unit `  R
) ) ( z ( .r `  R
) y )  =/=  x  <->  A. y  e.  ( ( Base `  R
)  \  (Unit `  R
) ) A. z  e.  ( ( Base `  R
)  \  (Unit `  R
) ) ( y ( .r `  S
) z )  =/=  x )
109anbi2i 694 . . 3  |-  ( ( x  e.  ( (
Base `  R )  \  (Unit `  R )
)  /\  A. z  e.  ( ( Base `  R
)  \  (Unit `  R
) ) A. y  e.  ( ( Base `  R
)  \  (Unit `  R
) ) ( z ( .r `  R
) y )  =/=  x )  <->  ( x  e.  ( ( Base `  R
)  \  (Unit `  R
) )  /\  A. y  e.  ( ( Base `  R )  \ 
(Unit `  R )
) A. z  e.  ( ( Base `  R
)  \  (Unit `  R
) ) ( y ( .r `  S
) z )  =/=  x ) )
11 eqid 2454 . . . 4  |-  (Unit `  R )  =  (Unit `  R )
12 opprirred.2 . . . 4  |-  I  =  (Irred `  R )
13 eqid 2454 . . . 4  |-  ( (
Base `  R )  \  (Unit `  R )
)  =  ( (
Base `  R )  \  (Unit `  R )
)
142, 11, 12, 13, 3isirred 16913 . . 3  |-  ( x  e.  I  <->  ( x  e.  ( ( Base `  R
)  \  (Unit `  R
) )  /\  A. z  e.  ( ( Base `  R )  \ 
(Unit `  R )
) A. y  e.  ( ( Base `  R
)  \  (Unit `  R
) ) ( z ( .r `  R
) y )  =/=  x ) )
154, 2opprbas 16843 . . . 4  |-  ( Base `  R )  =  (
Base `  S )
1611, 4opprunit 16875 . . . 4  |-  (Unit `  R )  =  (Unit `  S )
17 eqid 2454 . . . 4  |-  (Irred `  S )  =  (Irred `  S )
1815, 16, 17, 13, 5isirred 16913 . . 3  |-  ( x  e.  (Irred `  S
)  <->  ( x  e.  ( ( Base `  R
)  \  (Unit `  R
) )  /\  A. y  e.  ( ( Base `  R )  \ 
(Unit `  R )
) A. z  e.  ( ( Base `  R
)  \  (Unit `  R
) ) ( y ( .r `  S
) z )  =/=  x ) )
1910, 14, 183bitr4i 277 . 2  |-  ( x  e.  I  <->  x  e.  (Irred `  S ) )
2019eqriv 2450 1  |-  I  =  (Irred `  S )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2647   A.wral 2798    \ cdif 3432   ` cfv 5525  (class class class)co 6199   Basecbs 14291   .rcmulr 14357  opprcoppr 16836  Unitcui 16853  Irredcir 16854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-tpos 6854  df-recs 6941  df-rdg 6975  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-nn 10433  df-2 10490  df-3 10491  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-plusg 14369  df-mulr 14370  df-0g 14498  df-mgp 16713  df-ur 16725  df-oppr 16837  df-dvdsr 16855  df-unit 16856  df-irred 16857
This theorem is referenced by:  irredlmul  16922
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