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Theorem oppr1 15729
Description: Multiplicative identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
Hypotheses
Ref Expression
opprbas.1  |-  O  =  (oppr
`  R )
oppr1.2  |-  .1.  =  ( 1r `  R )
Assertion
Ref Expression
oppr1  |-  .1.  =  ( 1r `  O )

Proof of Theorem oppr1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2435 . . . . . . . . 9  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2435 . . . . . . . . 9  |-  ( .r
`  R )  =  ( .r `  R
)
3 opprbas.1 . . . . . . . . 9  |-  O  =  (oppr
`  R )
4 eqid 2435 . . . . . . . . 9  |-  ( .r
`  O )  =  ( .r `  O
)
51, 2, 3, 4opprmul 15721 . . . . . . . 8  |-  ( x ( .r `  O
) y )  =  ( y ( .r
`  R ) x )
65eqeq1i 2442 . . . . . . 7  |-  ( ( x ( .r `  O ) y )  =  y  <->  ( y
( .r `  R
) x )  =  y )
71, 2, 3, 4opprmul 15721 . . . . . . . 8  |-  ( y ( .r `  O
) x )  =  ( x ( .r
`  R ) y )
87eqeq1i 2442 . . . . . . 7  |-  ( ( y ( .r `  O ) x )  =  y  <->  ( x
( .r `  R
) y )  =  y )
96, 8anbi12ci 680 . . . . . 6  |-  ( ( ( x ( .r
`  O ) y )  =  y  /\  ( y ( .r
`  O ) x )  =  y )  <-> 
( ( x ( .r `  R ) y )  =  y  /\  ( y ( .r `  R ) x )  =  y ) )
109ralbii 2721 . . . . 5  |-  ( A. y  e.  ( Base `  R ) ( ( x ( .r `  O ) y )  =  y  /\  (
y ( .r `  O ) x )  =  y )  <->  A. y  e.  ( Base `  R
) ( ( x ( .r `  R
) y )  =  y  /\  ( y ( .r `  R
) x )  =  y ) )
1110anbi2i 676 . . . 4  |-  ( ( x  e.  ( Base `  R )  /\  A. y  e.  ( Base `  R ) ( ( x ( .r `  O ) y )  =  y  /\  (
y ( .r `  O ) x )  =  y ) )  <-> 
( x  e.  (
Base `  R )  /\  A. y  e.  (
Base `  R )
( ( x ( .r `  R ) y )  =  y  /\  ( y ( .r `  R ) x )  =  y ) ) )
1211iotabii 5432 . . 3  |-  ( iota
x ( x  e.  ( Base `  R
)  /\  A. y  e.  ( Base `  R
) ( ( x ( .r `  O
) y )  =  y  /\  ( y ( .r `  O
) x )  =  y ) ) )  =  ( iota x
( x  e.  (
Base `  R )  /\  A. y  e.  (
Base `  R )
( ( x ( .r `  R ) y )  =  y  /\  ( y ( .r `  R ) x )  =  y ) ) )
13 eqid 2435 . . . . 5  |-  (mulGrp `  O )  =  (mulGrp `  O )
143, 1opprbas 15724 . . . . 5  |-  ( Base `  R )  =  (
Base `  O )
1513, 14mgpbas 15644 . . . 4  |-  ( Base `  R )  =  (
Base `  (mulGrp `  O
) )
1613, 4mgpplusg 15642 . . . 4  |-  ( .r
`  O )  =  ( +g  `  (mulGrp `  O ) )
17 eqid 2435 . . . 4  |-  ( 0g
`  (mulGrp `  O )
)  =  ( 0g
`  (mulGrp `  O )
)
1815, 16, 17grpidval 14697 . . 3  |-  ( 0g
`  (mulGrp `  O )
)  =  ( iota
x ( x  e.  ( Base `  R
)  /\  A. y  e.  ( Base `  R
) ( ( x ( .r `  O
) y )  =  y  /\  ( y ( .r `  O
) x )  =  y ) ) )
19 eqid 2435 . . . . 5  |-  (mulGrp `  R )  =  (mulGrp `  R )
2019, 1mgpbas 15644 . . . 4  |-  ( Base `  R )  =  (
Base `  (mulGrp `  R
) )
2119, 2mgpplusg 15642 . . . 4  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
22 eqid 2435 . . . 4  |-  ( 0g
`  (mulGrp `  R )
)  =  ( 0g
`  (mulGrp `  R )
)
2320, 21, 22grpidval 14697 . . 3  |-  ( 0g
`  (mulGrp `  R )
)  =  ( iota
x ( x  e.  ( Base `  R
)  /\  A. y  e.  ( Base `  R
) ( ( x ( .r `  R
) y )  =  y  /\  ( y ( .r `  R
) x )  =  y ) ) )
2412, 18, 233eqtr4i 2465 . 2  |-  ( 0g
`  (mulGrp `  O )
)  =  ( 0g
`  (mulGrp `  R )
)
25 eqid 2435 . . 3  |-  ( 1r
`  O )  =  ( 1r `  O
)
2613, 25rngidval 15656 . 2  |-  ( 1r
`  O )  =  ( 0g `  (mulGrp `  O ) )
27 oppr1.2 . . 3  |-  .1.  =  ( 1r `  R )
2819, 27rngidval 15656 . 2  |-  .1.  =  ( 0g `  (mulGrp `  R ) )
2924, 26, 283eqtr4ri 2466 1  |-  .1.  =  ( 1r `  O )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   iotacio 5408   ` cfv 5446  (class class class)co 6073   Basecbs 13459   .rcmulr 13520   0gc0g 13713  mulGrpcmgp 15638   1rcur 15652  opprcoppr 15717
This theorem is referenced by:  opprunit  15756  isdrngrd  15851  opprsubrg  15879  srng1  15937  issrngd  15939  fidomndrng  16357  rhmopp  24247  ldual1  29847  lduallmodlem  29851  ldualvsub  29854  lcd1  32308  lcdvsub  32316
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-nn 9991  df-2 10048  df-3 10049  df-ndx 13462  df-slot 13463  df-base 13464  df-sets 13465  df-plusg 13532  df-mulr 13533  df-0g 13717  df-mgp 15639  df-ur 15655  df-oppr 15718
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