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Theorem opprneg 17097
Description: The negative function in an opposite ring. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
opprbas.1  |-  O  =  (oppr
`  R )
opprneg.2  |-  N  =  ( invg `  R )
Assertion
Ref Expression
opprneg  |-  N  =  ( invg `  O )

Proof of Theorem opprneg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2467 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
3 eqid 2467 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
4 opprneg.2 . . 3  |-  N  =  ( invg `  R )
51, 2, 3, 4grpinvfval 15902 . 2  |-  N  =  ( x  e.  (
Base `  R )  |->  ( iota_ y  e.  (
Base `  R )
( y ( +g  `  R ) x )  =  ( 0g `  R ) ) )
6 opprbas.1 . . . 4  |-  O  =  (oppr
`  R )
76, 1opprbas 17091 . . 3  |-  ( Base `  R )  =  (
Base `  O )
86, 2oppradd 17092 . . 3  |-  ( +g  `  R )  =  ( +g  `  O )
96, 3oppr0 17095 . . 3  |-  ( 0g
`  R )  =  ( 0g `  O
)
10 eqid 2467 . . 3  |-  ( invg `  O )  =  ( invg `  O )
117, 8, 9, 10grpinvfval 15902 . 2  |-  ( invg `  O )  =  ( x  e.  ( Base `  R
)  |->  ( iota_ y  e.  ( Base `  R
) ( y ( +g  `  R ) x )  =  ( 0g `  R ) ) )
125, 11eqtr4i 2499 1  |-  N  =  ( invg `  O )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    |-> cmpt 4505   ` cfv 5588   iota_crio 6245  (class class class)co 6285   Basecbs 14493   +g cplusg 14558   0gc0g 14698   invgcminusg 15731  opprcoppr 17084
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 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-tpos 6956  df-recs 7043  df-rdg 7077  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-3 10596  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-plusg 14571  df-mulr 14572  df-0g 14700  df-minusg 15872  df-oppr 17085
This theorem is referenced by:  unitnegcl  17143  ldualneg  34163  ldualvsub  34169  lcdneg  36624  lcdvsub  36631
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