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Theorem opprirred 17132
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 3022 . . . . 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 2467 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
3 eqid 2467 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
4 opprirred.1 . . . . . . . 8  |-  S  =  (oppr
`  R )
5 eqid 2467 . . . . . . . 8  |-  ( .r
`  S )  =  ( .r `  S
)
62, 3, 4, 5opprmul 17056 . . . . . . 7  |-  ( y ( .r `  S
) z )  =  ( z ( .r
`  R ) y )
76neeq1i 2752 . . . . . 6  |-  ( ( y ( .r `  S ) z )  =/=  x  <->  ( z
( .r `  R
) y )  =/=  x )
872ralbii 2896 . . . . 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 2467 . . . 4  |-  (Unit `  R )  =  (Unit `  R )
12 opprirred.2 . . . 4  |-  I  =  (Irred `  R )
13 eqid 2467 . . . 4  |-  ( (
Base `  R )  \  (Unit `  R )
)  =  ( (
Base `  R )  \  (Unit `  R )
)
142, 11, 12, 13, 3isirred 17129 . . 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 17059 . . . 4  |-  ( Base `  R )  =  (
Base `  S )
1611, 4opprunit 17091 . . . 4  |-  (Unit `  R )  =  (Unit `  S )
17 eqid 2467 . . . 4  |-  (Irred `  S )  =  (Irred `  S )
1815, 16, 17, 13, 5isirred 17129 . . 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 2463 1  |-  I  =  (Irred `  S )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814    \ cdif 3473   ` cfv 5586  (class class class)co 6282   Basecbs 14483   .rcmulr 14549  opprcoppr 17052  Unitcui 17069  Irredcir 17070
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-tpos 6952  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-plusg 14561  df-mulr 14562  df-0g 14690  df-mgp 16929  df-ur 16941  df-oppr 17053  df-dvdsr 17071  df-unit 17072  df-irred 17073
This theorem is referenced by:  irredlmul  17138
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