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Theorem invrfval 15504
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 5563 . . . . . . 7  |-  ( r  =  R  ->  (mulGrp `  r )  =  (mulGrp `  R ) )
3 fveq2 5563 . . . . . . . 8  |-  ( r  =  R  ->  (Unit `  r )  =  (Unit `  R ) )
4 invrfval.u . . . . . . . 8  |-  U  =  (Unit `  R )
53, 4syl6eqr 2366 . . . . . . 7  |-  ( r  =  R  ->  (Unit `  r )  =  U )
62, 5oveq12d 5918 . . . . . 6  |-  ( r  =  R  ->  (
(mulGrp `  r )s  (Unit `  r ) )  =  ( (mulGrp `  R
)s 
U ) )
7 invrfval.g . . . . . 6  |-  G  =  ( (mulGrp `  R
)s 
U )
86, 7syl6eqr 2366 . . . . 5  |-  ( r  =  R  ->  (
(mulGrp `  r )s  (Unit `  r ) )  =  G )
98fveq2d 5567 . . . 4  |-  ( r  =  R  ->  ( inv g `  ( (mulGrp `  r )s  (Unit `  r )
) )  =  ( inv g `  G
) )
10 df-invr 15503 . . . 4  |-  invr  =  ( r  e.  _V  |->  ( inv g `  (
(mulGrp `  r )s  (Unit `  r ) ) ) )
11 fvex 5577 . . . 4  |-  ( inv g `  G )  e.  _V
129, 10, 11fvmpt 5640 . . 3  |-  ( R  e.  _V  ->  ( invr `  R )  =  ( inv g `  G ) )
13 fvprc 5557 . . . . 5  |-  ( -.  R  e.  _V  ->  (
invr `  R )  =  (/) )
14 base0 13232 . . . . . . 7  |-  (/)  =  (
Base `  (/) )
15 eqid 2316 . . . . . . 7  |-  ( inv g `  (/) )  =  ( inv g `  (/) )
1614, 15grpinvfn 14571 . . . . . 6  |-  ( inv g `  (/) )  Fn  (/)
17 fn0 5400 . . . . . 6  |-  ( ( inv g `  (/) )  Fn  (/) 
<->  ( inv g `  (/) )  =  (/) )
1816, 17mpbi 199 . . . . 5  |-  ( inv g `  (/) )  =  (/)
1913, 18syl6eqr 2366 . . . 4  |-  ( -.  R  e.  _V  ->  (
invr `  R )  =  ( inv g `  (/) ) )
20 fvprc 5557 . . . . . . . 8  |-  ( -.  R  e.  _V  ->  (mulGrp `  R )  =  (/) )
2120oveq1d 5915 . . . . . . 7  |-  ( -.  R  e.  _V  ->  ( (mulGrp `  R )s  U
)  =  ( (/)s  U ) )
227, 21syl5eq 2360 . . . . . 6  |-  ( -.  R  e.  _V  ->  G  =  ( (/)s  U ) )
23 ress0 13249 . . . . . 6  |-  ( (/)s  U )  =  (/)
2422, 23syl6eq 2364 . . . . 5  |-  ( -.  R  e.  _V  ->  G  =  (/) )
2524fveq2d 5567 . . . 4  |-  ( -.  R  e.  _V  ->  ( inv g `  G
)  =  ( inv g `  (/) ) )
2619, 25eqtr4d 2351 . . 3  |-  ( -.  R  e.  _V  ->  (
invr `  R )  =  ( inv g `  G ) )
2712, 26pm2.61i 156 . 2  |-  ( invr `  R )  =  ( inv g `  G
)
281, 27eqtri 2336 1  |-  I  =  ( inv g `  G )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1633    e. wcel 1701   _Vcvv 2822   (/)c0 3489    Fn wfn 5287   ` cfv 5292  (class class class)co 5900   ↾s cress 13196   inv gcminusg 14412  mulGrpcmgp 15374  Unitcui 15470   invrcinvr 15502
This theorem is referenced by:  unitinvcl  15505  unitinvinv  15506  unitlinv  15508  unitrinv  15509  invrpropd  15529  subrgugrp  15613  cnmsubglem  16490  invrcn2  17914  nrginvrcn  18254  nrgtdrg  18255  sum2dchr  20566  rdivmuldivd  23517  rnginvval  23518  dvrcan5  23519  cntzsdrg  26658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-riota 6346  df-slot 13199  df-base 13200  df-ress 13202  df-minusg 14539  df-invr 15503
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