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Theorem opprirred 17464
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 2943 . . . . 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 2382 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
3 eqid 2382 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
4 opprirred.1 . . . . . . . 8  |-  S  =  (oppr
`  R )
5 eqid 2382 . . . . . . . 8  |-  ( .r
`  S )  =  ( .r `  S
)
62, 3, 4, 5opprmul 17388 . . . . . . 7  |-  ( y ( .r `  S
) z )  =  ( z ( .r
`  R ) y )
76neeq1i 2667 . . . . . 6  |-  ( ( y ( .r `  S ) z )  =/=  x  <->  ( z
( .r `  R
) y )  =/=  x )
872ralbii 2814 . . . . 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 692 . . 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 2382 . . . 4  |-  (Unit `  R )  =  (Unit `  R )
12 opprirred.2 . . . 4  |-  I  =  (Irred `  R )
13 eqid 2382 . . . 4  |-  ( (
Base `  R )  \  (Unit `  R )
)  =  ( (
Base `  R )  \  (Unit `  R )
)
142, 11, 12, 13, 3isirred 17461 . . 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 17391 . . . 4  |-  ( Base `  R )  =  (
Base `  S )
1611, 4opprunit 17423 . . . 4  |-  (Unit `  R )  =  (Unit `  S )
17 eqid 2382 . . . 4  |-  (Irred `  S )  =  (Irred `  S )
1815, 16, 17, 13, 5isirred 17461 . . 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 2378 1  |-  I  =  (Irred `  S )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1399    e. wcel 1826    =/= wne 2577   A.wral 2732    \ cdif 3386   ` cfv 5496  (class class class)co 6196   Basecbs 14634   .rcmulr 14703  opprcoppr 17384  Unitcui 17401  Irredcir 17402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-tpos 6873  df-recs 6960  df-rdg 6994  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-3 10512  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-plusg 14715  df-mulr 14716  df-0g 14849  df-mgp 17255  df-ur 17267  df-oppr 17385  df-dvdsr 17403  df-unit 17404  df-irred 17405
This theorem is referenced by:  irredlmul  17470
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