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Theorem opprirred 15809
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 2870 . . . . 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 15733 . . . . . . 7  |-  ( y ( .r `  S
) z )  =  ( z ( .r
`  R ) y )
76neeq1i 2613 . . . . . 6  |-  ( ( y ( .r `  S ) z )  =/=  x  <->  ( z
( .r `  R
) y )  =/=  x )
872ralbii 2733 . . . . 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 245 . . . 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 677 . . 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 15806 . . 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 15736 . . . 4  |-  ( Base `  R )  =  (
Base `  S )
1611, 4opprunit 15768 . . . 4  |-  (Unit `  R )  =  (Unit `  S )
17 eqid 2438 . . . 4  |-  (Irred `  S )  =  (Irred `  S )
1815, 16, 17, 13, 5isirred 15806 . . 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 270 . 2  |-  ( x  e.  I  <->  x  e.  (Irred `  S ) )
2019eqriv 2435 1  |-  I  =  (Irred `  S )
Colors of variables: wff set class
Syntax hints:    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707    \ cdif 3319   ` cfv 5456  (class class class)co 6083   Basecbs 13471   .rcmulr 13532  opprcoppr 15729  Unitcui 15746  Irredcir 15747
This theorem is referenced by:  irredlmul  15815
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-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-tpos 6481  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-3 10061  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-plusg 13544  df-mulr 13545  df-0g 13729  df-mgp 15651  df-ur 15667  df-oppr 15730  df-dvdsr 15748  df-unit 15749  df-irred 15750
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