MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  invrfval Structured version   Unicode version

Theorem invrfval 17192
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 5872 . . . . . . 7  |-  ( r  =  R  ->  (mulGrp `  r )  =  (mulGrp `  R ) )
3 fveq2 5872 . . . . . . . 8  |-  ( r  =  R  ->  (Unit `  r )  =  (Unit `  R ) )
4 invrfval.u . . . . . . . 8  |-  U  =  (Unit `  R )
53, 4syl6eqr 2526 . . . . . . 7  |-  ( r  =  R  ->  (Unit `  r )  =  U )
62, 5oveq12d 6313 . . . . . 6  |-  ( r  =  R  ->  (
(mulGrp `  r )s  (Unit `  r ) )  =  ( (mulGrp `  R
)s 
U ) )
7 invrfval.g . . . . . 6  |-  G  =  ( (mulGrp `  R
)s 
U )
86, 7syl6eqr 2526 . . . . 5  |-  ( r  =  R  ->  (
(mulGrp `  r )s  (Unit `  r ) )  =  G )
98fveq2d 5876 . . . 4  |-  ( r  =  R  ->  ( invg `  ( (mulGrp `  r )s  (Unit `  r )
) )  =  ( invg `  G
) )
10 df-invr 17191 . . . 4  |-  invr  =  ( r  e.  _V  |->  ( invg `  (
(mulGrp `  r )s  (Unit `  r ) ) ) )
11 fvex 5882 . . . 4  |-  ( invg `  G )  e.  _V
129, 10, 11fvmpt 5957 . . 3  |-  ( R  e.  _V  ->  ( invr `  R )  =  ( invg `  G ) )
13 fvprc 5866 . . . . 5  |-  ( -.  R  e.  _V  ->  (
invr `  R )  =  (/) )
14 base0 14545 . . . . . . 7  |-  (/)  =  (
Base `  (/) )
15 eqid 2467 . . . . . . 7  |-  ( invg `  (/) )  =  ( invg `  (/) )
1614, 15grpinvfn 15961 . . . . . 6  |-  ( invg `  (/) )  Fn  (/)
17 fn0 5706 . . . . . 6  |-  ( ( invg `  (/) )  Fn  (/) 
<->  ( invg `  (/) )  =  (/) )
1816, 17mpbi 208 . . . . 5  |-  ( invg `  (/) )  =  (/)
1913, 18syl6eqr 2526 . . . 4  |-  ( -.  R  e.  _V  ->  (
invr `  R )  =  ( invg `  (/) ) )
20 fvprc 5866 . . . . . . . 8  |-  ( -.  R  e.  _V  ->  (mulGrp `  R )  =  (/) )
2120oveq1d 6310 . . . . . . 7  |-  ( -.  R  e.  _V  ->  ( (mulGrp `  R )s  U
)  =  ( (/)s  U ) )
227, 21syl5eq 2520 . . . . . 6  |-  ( -.  R  e.  _V  ->  G  =  ( (/)s  U ) )
23 ress0 14565 . . . . . 6  |-  ( (/)s  U )  =  (/)
2422, 23syl6eq 2524 . . . . 5  |-  ( -.  R  e.  _V  ->  G  =  (/) )
2524fveq2d 5876 . . . 4  |-  ( -.  R  e.  _V  ->  ( invg `  G
)  =  ( invg `  (/) ) )
2619, 25eqtr4d 2511 . . 3  |-  ( -.  R  e.  _V  ->  (
invr `  R )  =  ( invg `  G ) )
2712, 26pm2.61i 164 . 2  |-  ( invr `  R )  =  ( invg `  G
)
281, 27eqtri 2496 1  |-  I  =  ( invg `  G )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1379    e. wcel 1767   _Vcvv 3118   (/)c0 3790    Fn wfn 5589   ` cfv 5594  (class class class)co 6295   ↾s cress 14507   invgcminusg 15925  mulGrpcmgp 17011  Unitcui 17158   invrcinvr 17190
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-slot 14510  df-base 14511  df-ress 14513  df-minusg 15929  df-invr 17191
This theorem is referenced by:  unitinvcl  17193  unitinvinv  17194  unitlinv  17196  unitrinv  17197  invrpropd  17217  subrgugrp  17317  cnmsubglem  18348  psgninv  18485  invrvald  19045  invrcn2  20548  nrginvrcn  21066  nrgtdrg  21067  sum2dchr  23413  rdivmuldivd  27613  ringinvval  27614  dvrcan5  27615  cntzsdrg  31086
  Copyright terms: Public domain W3C validator