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Theorem invrfval 15778
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 5728 . . . . . . 7  |-  ( r  =  R  ->  (mulGrp `  r )  =  (mulGrp `  R ) )
3 fveq2 5728 . . . . . . . 8  |-  ( r  =  R  ->  (Unit `  r )  =  (Unit `  R ) )
4 invrfval.u . . . . . . . 8  |-  U  =  (Unit `  R )
53, 4syl6eqr 2486 . . . . . . 7  |-  ( r  =  R  ->  (Unit `  r )  =  U )
62, 5oveq12d 6099 . . . . . 6  |-  ( r  =  R  ->  (
(mulGrp `  r )s  (Unit `  r ) )  =  ( (mulGrp `  R
)s 
U ) )
7 invrfval.g . . . . . 6  |-  G  =  ( (mulGrp `  R
)s 
U )
86, 7syl6eqr 2486 . . . . 5  |-  ( r  =  R  ->  (
(mulGrp `  r )s  (Unit `  r ) )  =  G )
98fveq2d 5732 . . . 4  |-  ( r  =  R  ->  ( inv g `  ( (mulGrp `  r )s  (Unit `  r )
) )  =  ( inv g `  G
) )
10 df-invr 15777 . . . 4  |-  invr  =  ( r  e.  _V  |->  ( inv g `  (
(mulGrp `  r )s  (Unit `  r ) ) ) )
11 fvex 5742 . . . 4  |-  ( inv g `  G )  e.  _V
129, 10, 11fvmpt 5806 . . 3  |-  ( R  e.  _V  ->  ( invr `  R )  =  ( inv g `  G ) )
13 fvprc 5722 . . . . 5  |-  ( -.  R  e.  _V  ->  (
invr `  R )  =  (/) )
14 base0 13506 . . . . . . 7  |-  (/)  =  (
Base `  (/) )
15 eqid 2436 . . . . . . 7  |-  ( inv g `  (/) )  =  ( inv g `  (/) )
1614, 15grpinvfn 14845 . . . . . 6  |-  ( inv g `  (/) )  Fn  (/)
17 fn0 5564 . . . . . 6  |-  ( ( inv g `  (/) )  Fn  (/) 
<->  ( inv g `  (/) )  =  (/) )
1816, 17mpbi 200 . . . . 5  |-  ( inv g `  (/) )  =  (/)
1913, 18syl6eqr 2486 . . . 4  |-  ( -.  R  e.  _V  ->  (
invr `  R )  =  ( inv g `  (/) ) )
20 fvprc 5722 . . . . . . . 8  |-  ( -.  R  e.  _V  ->  (mulGrp `  R )  =  (/) )
2120oveq1d 6096 . . . . . . 7  |-  ( -.  R  e.  _V  ->  ( (mulGrp `  R )s  U
)  =  ( (/)s  U ) )
227, 21syl5eq 2480 . . . . . 6  |-  ( -.  R  e.  _V  ->  G  =  ( (/)s  U ) )
23 ress0 13523 . . . . . 6  |-  ( (/)s  U )  =  (/)
2422, 23syl6eq 2484 . . . . 5  |-  ( -.  R  e.  _V  ->  G  =  (/) )
2524fveq2d 5732 . . . 4  |-  ( -.  R  e.  _V  ->  ( inv g `  G
)  =  ( inv g `  (/) ) )
2619, 25eqtr4d 2471 . . 3  |-  ( -.  R  e.  _V  ->  (
invr `  R )  =  ( inv g `  G ) )
2712, 26pm2.61i 158 . 2  |-  ( invr `  R )  =  ( inv g `  G
)
281, 27eqtri 2456 1  |-  I  =  ( inv g `  G )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1652    e. wcel 1725   _Vcvv 2956   (/)c0 3628    Fn wfn 5449   ` cfv 5454  (class class class)co 6081   ↾s cress 13470   inv gcminusg 14686  mulGrpcmgp 15648  Unitcui 15744   invrcinvr 15776
This theorem is referenced by:  unitinvcl  15779  unitinvinv  15780  unitlinv  15782  unitrinv  15783  invrpropd  15803  subrgugrp  15887  cnmsubglem  16761  invrcn2  18209  nrginvrcn  18727  nrgtdrg  18728  sum2dchr  21058  rdivmuldivd  24227  rnginvval  24228  dvrcan5  24229  cntzsdrg  27487
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-slot 13473  df-base 13474  df-ress 13476  df-minusg 14813  df-invr 15777
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