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Theorem vsfval 25204
Description: Value of the function for the vector subtraction operation on a normed complex vector space. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 27-Dec-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
vsfval.2  |-  G  =  ( +v `  U
)
vsfval.3  |-  M  =  ( -v `  U
)
Assertion
Ref Expression
vsfval  |-  M  =  (  /g  `  G
)

Proof of Theorem vsfval
Dummy variables  g  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-vs 25168 . . . . 5  |-  -v  =  (  /g  o.  +v )
21fveq1i 5865 . . . 4  |-  ( -v
`  U )  =  ( (  /g  o.  +v ) `  U )
3 fo1st 6801 . . . . . . . 8  |-  1st : _V -onto-> _V
4 fof 5793 . . . . . . . 8  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
53, 4ax-mp 5 . . . . . . 7  |-  1st : _V
--> _V
6 fco 5739 . . . . . . 7  |-  ( ( 1st : _V --> _V  /\  1st : _V --> _V )  ->  ( 1st  o.  1st ) : _V --> _V )
75, 5, 6mp2an 672 . . . . . 6  |-  ( 1st 
o.  1st ) : _V --> _V
8 df-va 25164 . . . . . . 7  |-  +v  =  ( 1st  o.  1st )
98feq1i 5721 . . . . . 6  |-  ( +v : _V --> _V  <->  ( 1st  o. 
1st ) : _V --> _V )
107, 9mpbir 209 . . . . 5  |-  +v : _V
--> _V
11 fvco3 5942 . . . . 5  |-  ( ( +v : _V --> _V  /\  U  e.  _V )  ->  ( (  /g  o.  +v ) `  U )  =  (  /g  `  ( +v `  U ) ) )
1210, 11mpan 670 . . . 4  |-  ( U  e.  _V  ->  (
(  /g  o.  +v ) `  U )  =  (  /g  `  ( +v `  U ) ) )
132, 12syl5eq 2520 . . 3  |-  ( U  e.  _V  ->  ( -v `  U )  =  (  /g  `  ( +v `  U ) ) )
14 0ngrp 24889 . . . . . 6  |-  -.  (/)  e.  GrpOp
15 vex 3116 . . . . . . . . . 10  |-  g  e. 
_V
1615rnex 6715 . . . . . . . . 9  |-  ran  g  e.  _V
1716, 16mpt2ex 6857 . . . . . . . 8  |-  ( x  e.  ran  g ,  y  e.  ran  g  |->  ( x g ( ( inv `  g
) `  y )
) )  e.  _V
18 df-gdiv 24872 . . . . . . . 8  |-  /g  =  ( g  e.  GrpOp  |->  ( x  e.  ran  g ,  y  e.  ran  g  |->  ( x g ( ( inv `  g ) `  y
) ) ) )
1917, 18dmmpti 5708 . . . . . . 7  |-  dom  /g  =  GrpOp
2019eleq2i 2545 . . . . . 6  |-  ( (/)  e.  dom  /g  <->  (/)  e.  GrpOp )
2114, 20mtbir 299 . . . . 5  |-  -.  (/)  e.  dom  /g
22 ndmfv 5888 . . . . 5  |-  ( -.  (/)  e.  dom  /g  ->  (  /g  `  (/) )  =  (/) )
2321, 22mp1i 12 . . . 4  |-  ( -.  U  e.  _V  ->  (  /g  `  (/) )  =  (/) )
24 fvprc 5858 . . . . 5  |-  ( -.  U  e.  _V  ->  ( +v `  U )  =  (/) )
2524fveq2d 5868 . . . 4  |-  ( -.  U  e.  _V  ->  (  /g  `  ( +v
`  U ) )  =  (  /g  `  (/) ) )
26 fvprc 5858 . . . 4  |-  ( -.  U  e.  _V  ->  ( -v `  U )  =  (/) )
2723, 25, 263eqtr4rd 2519 . . 3  |-  ( -.  U  e.  _V  ->  ( -v `  U )  =  (  /g  `  ( +v `  U ) ) )
2813, 27pm2.61i 164 . 2  |-  ( -v
`  U )  =  (  /g  `  ( +v `  U ) )
29 vsfval.3 . 2  |-  M  =  ( -v `  U
)
30 vsfval.2 . . 3  |-  G  =  ( +v `  U
)
3130fveq2i 5867 . 2  |-  (  /g  `  G )  =  (  /g  `  ( +v
`  U ) )
3228, 29, 313eqtr4i 2506 1  |-  M  =  (  /g  `  G
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1379    e. wcel 1767   _Vcvv 3113   (/)c0 3785   dom cdm 4999   ran crn 5000    o. ccom 5003   -->wf 5582   -onto->wfo 5584   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   1stc1st 6779   GrpOpcgr 24864   invcgn 24866    /g cgs 24867   +vcpv 25154   -vcnsb 25158
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-grpo 24869  df-gdiv 24872  df-va 25164  df-vs 25168
This theorem is referenced by:  nvm  25212  nvmfval  25215  nvnnncan1  25219  nvnnncan2  25220  nvaddsubass  25229  nvaddsub  25230  nvmtri2  25251
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