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Theorem invrfval 15455
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  =  ( inv g `  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 5525 . . . . . . 7  |-  ( r  =  R  ->  (mulGrp `  r )  =  (mulGrp `  R ) )
3 fveq2 5525 . . . . . . . 8  |-  ( r  =  R  ->  (Unit `  r )  =  (Unit `  R ) )
4 invrfval.u . . . . . . . 8  |-  U  =  (Unit `  R )
53, 4syl6eqr 2333 . . . . . . 7  |-  ( r  =  R  ->  (Unit `  r )  =  U )
62, 5oveq12d 5876 . . . . . 6  |-  ( r  =  R  ->  (
(mulGrp `  r )s  (Unit `  r ) )  =  ( (mulGrp `  R
)s 
U ) )
7 invrfval.g . . . . . 6  |-  G  =  ( (mulGrp `  R
)s 
U )
86, 7syl6eqr 2333 . . . . 5  |-  ( r  =  R  ->  (
(mulGrp `  r )s  (Unit `  r ) )  =  G )
98fveq2d 5529 . . . 4  |-  ( r  =  R  ->  ( inv g `  ( (mulGrp `  r )s  (Unit `  r )
) )  =  ( inv g `  G
) )
10 df-invr 15454 . . . 4  |-  invr  =  ( r  e.  _V  |->  ( inv g `  (
(mulGrp `  r )s  (Unit `  r ) ) ) )
11 fvex 5539 . . . 4  |-  ( inv g `  G )  e.  _V
129, 10, 11fvmpt 5602 . . 3  |-  ( R  e.  _V  ->  ( invr `  R )  =  ( inv g `  G ) )
13 fvprc 5519 . . . . 5  |-  ( -.  R  e.  _V  ->  (
invr `  R )  =  (/) )
14 base0 13185 . . . . . . 7  |-  (/)  =  (
Base `  (/) )
15 eqid 2283 . . . . . . 7  |-  ( inv g `  (/) )  =  ( inv g `  (/) )
1614, 15grpinvfn 14522 . . . . . 6  |-  ( inv g `  (/) )  Fn  (/)
17 fn0 5363 . . . . . 6  |-  ( ( inv g `  (/) )  Fn  (/) 
<->  ( inv g `  (/) )  =  (/) )
1816, 17mpbi 199 . . . . 5  |-  ( inv g `  (/) )  =  (/)
1913, 18syl6eqr 2333 . . . 4  |-  ( -.  R  e.  _V  ->  (
invr `  R )  =  ( inv g `  (/) ) )
20 fvprc 5519 . . . . . . . 8  |-  ( -.  R  e.  _V  ->  (mulGrp `  R )  =  (/) )
2120oveq1d 5873 . . . . . . 7  |-  ( -.  R  e.  _V  ->  ( (mulGrp `  R )s  U
)  =  ( (/)s  U ) )
227, 21syl5eq 2327 . . . . . 6  |-  ( -.  R  e.  _V  ->  G  =  ( (/)s  U ) )
23 ress0 13202 . . . . . 6  |-  ( (/)s  U )  =  (/)
2422, 23syl6eq 2331 . . . . 5  |-  ( -.  R  e.  _V  ->  G  =  (/) )
2524fveq2d 5529 . . . 4  |-  ( -.  R  e.  _V  ->  ( inv g `  G
)  =  ( inv g `  (/) ) )
2619, 25eqtr4d 2318 . . 3  |-  ( -.  R  e.  _V  ->  (
invr `  R )  =  ( inv g `  G ) )
2712, 26pm2.61i 156 . 2  |-  ( invr `  R )  =  ( inv g `  G
)
281, 27eqtri 2303 1  |-  I  =  ( inv g `  G )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455    Fn wfn 5250   ` cfv 5255  (class class class)co 5858   ↾s cress 13149   inv gcminusg 14363  mulGrpcmgp 15325  Unitcui 15421   invrcinvr 15453
This theorem is referenced by:  unitinvcl  15456  unitinvinv  15457  unitlinv  15459  unitrinv  15460  invrpropd  15480  subrgugrp  15564  cnmsubglem  16434  invrcn2  17862  nrginvrcn  18202  nrgtdrg  18203  sum2dchr  20513  cntzsdrg  27510
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-slot 13152  df-base 13153  df-ress 13155  df-minusg 14490  df-invr 15454
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