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Theorem opprirred 16784
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 2876 . . . . 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 2438 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
3 eqid 2438 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
4 opprirred.1 . . . . . . . 8  |-  S  =  (oppr
`  R )
5 eqid 2438 . . . . . . . 8  |-  ( .r
`  S )  =  ( .r `  S
)
62, 3, 4, 5opprmul 16708 . . . . . . 7  |-  ( y ( .r `  S
) z )  =  ( z ( .r
`  R ) y )
76neeq1i 2613 . . . . . 6  |-  ( ( y ( .r `  S ) z )  =/=  x  <->  ( z
( .r `  R
) y )  =/=  x )
872ralbii 2736 . . . . 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 2438 . . . 4  |-  (Unit `  R )  =  (Unit `  R )
12 opprirred.2 . . . 4  |-  I  =  (Irred `  R )
13 eqid 2438 . . . 4  |-  ( (
Base `  R )  \  (Unit `  R )
)  =  ( (
Base `  R )  \  (Unit `  R )
)
142, 11, 12, 13, 3isirred 16781 . . 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 16711 . . . 4  |-  ( Base `  R )  =  (
Base `  S )
1611, 4opprunit 16743 . . . 4  |-  (Unit `  R )  =  (Unit `  S )
17 eqid 2438 . . . 4  |-  (Irred `  S )  =  (Irred `  S )
1815, 16, 17, 13, 5isirred 16781 . . 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 2435 1  |-  I  =  (Irred `  S )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2601   A.wral 2710    \ cdif 3320   ` cfv 5413  (class class class)co 6086   Basecbs 14166   .rcmulr 14231  opprcoppr 16704  Unitcui 16721  Irredcir 16722
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-tpos 6740  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-plusg 14243  df-mulr 14244  df-0g 14372  df-mgp 16582  df-ur 16594  df-oppr 16705  df-dvdsr 16723  df-unit 16724  df-irred 16725
This theorem is referenced by:  irredlmul  16790
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