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Theorem invrfval 16777
Description: Multiplicative inverse function for a division ring. (Contributed by NM, 21-Sep-2011.) (Revised by Mario Carneiro, 25-Dec-2014.)
Hypotheses
Ref Expression
invrfval.u  |-  U  =  (Unit `  R )
invrfval.g  |-  G  =  ( (mulGrp `  R
)s 
U )
invrfval.i  |-  I  =  ( invr `  R
)
Assertion
Ref Expression
invrfval  |-  I  =  ( invg `  G )

Proof of Theorem invrfval
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 invrfval.i . 2  |-  I  =  ( invr `  R
)
2 fveq2 5703 . . . . . . 7  |-  ( r  =  R  ->  (mulGrp `  r )  =  (mulGrp `  R ) )
3 fveq2 5703 . . . . . . . 8  |-  ( r  =  R  ->  (Unit `  r )  =  (Unit `  R ) )
4 invrfval.u . . . . . . . 8  |-  U  =  (Unit `  R )
53, 4syl6eqr 2493 . . . . . . 7  |-  ( r  =  R  ->  (Unit `  r )  =  U )
62, 5oveq12d 6121 . . . . . 6  |-  ( r  =  R  ->  (
(mulGrp `  r )s  (Unit `  r ) )  =  ( (mulGrp `  R
)s 
U ) )
7 invrfval.g . . . . . 6  |-  G  =  ( (mulGrp `  R
)s 
U )
86, 7syl6eqr 2493 . . . . 5  |-  ( r  =  R  ->  (
(mulGrp `  r )s  (Unit `  r ) )  =  G )
98fveq2d 5707 . . . 4  |-  ( r  =  R  ->  ( invg `  ( (mulGrp `  r )s  (Unit `  r )
) )  =  ( invg `  G
) )
10 df-invr 16776 . . . 4  |-  invr  =  ( r  e.  _V  |->  ( invg `  (
(mulGrp `  r )s  (Unit `  r ) ) ) )
11 fvex 5713 . . . 4  |-  ( invg `  G )  e.  _V
129, 10, 11fvmpt 5786 . . 3  |-  ( R  e.  _V  ->  ( invr `  R )  =  ( invg `  G ) )
13 fvprc 5697 . . . . 5  |-  ( -.  R  e.  _V  ->  (
invr `  R )  =  (/) )
14 base0 14225 . . . . . . 7  |-  (/)  =  (
Base `  (/) )
15 eqid 2443 . . . . . . 7  |-  ( invg `  (/) )  =  ( invg `  (/) )
1614, 15grpinvfn 15590 . . . . . 6  |-  ( invg `  (/) )  Fn  (/)
17 fn0 5542 . . . . . 6  |-  ( ( invg `  (/) )  Fn  (/) 
<->  ( invg `  (/) )  =  (/) )
1816, 17mpbi 208 . . . . 5  |-  ( invg `  (/) )  =  (/)
1913, 18syl6eqr 2493 . . . 4  |-  ( -.  R  e.  _V  ->  (
invr `  R )  =  ( invg `  (/) ) )
20 fvprc 5697 . . . . . . . 8  |-  ( -.  R  e.  _V  ->  (mulGrp `  R )  =  (/) )
2120oveq1d 6118 . . . . . . 7  |-  ( -.  R  e.  _V  ->  ( (mulGrp `  R )s  U
)  =  ( (/)s  U ) )
227, 21syl5eq 2487 . . . . . 6  |-  ( -.  R  e.  _V  ->  G  =  ( (/)s  U ) )
23 ress0 14244 . . . . . 6  |-  ( (/)s  U )  =  (/)
2422, 23syl6eq 2491 . . . . 5  |-  ( -.  R  e.  _V  ->  G  =  (/) )
2524fveq2d 5707 . . . 4  |-  ( -.  R  e.  _V  ->  ( invg `  G
)  =  ( invg `  (/) ) )
2619, 25eqtr4d 2478 . . 3  |-  ( -.  R  e.  _V  ->  (
invr `  R )  =  ( invg `  G ) )
2712, 26pm2.61i 164 . 2  |-  ( invr `  R )  =  ( invg `  G
)
281, 27eqtri 2463 1  |-  I  =  ( invg `  G )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1369    e. wcel 1756   _Vcvv 2984   (/)c0 3649    Fn wfn 5425   ` cfv 5430  (class class class)co 6103   ↾s cress 14187   invgcminusg 15423  mulGrpcmgp 16603  Unitcui 16743   invrcinvr 16775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-slot 14190  df-base 14191  df-ress 14193  df-minusg 15558  df-invr 16776
This theorem is referenced by:  unitinvcl  16778  unitinvinv  16779  unitlinv  16781  unitrinv  16782  invrpropd  16802  subrgugrp  16896  cnmsubglem  17887  psgninv  18024  invrvald  18494  invrcn2  19766  nrginvrcn  20284  nrgtdrg  20285  sum2dchr  22625  rdivmuldivd  26271  rnginvval  26272  dvrcan5  26273  cntzsdrg  29571
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