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Theorem vsfval 25729
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 25693 . . . . 5  |-  -v  =  (  /g  o.  +v )
21fveq1i 5849 . . . 4  |-  ( -v
`  U )  =  ( (  /g  o.  +v ) `  U )
3 fo1st 6793 . . . . . . . 8  |-  1st : _V -onto-> _V
4 fof 5777 . . . . . . . 8  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
53, 4ax-mp 5 . . . . . . 7  |-  1st : _V
--> _V
6 fco 5723 . . . . . . 7  |-  ( ( 1st : _V --> _V  /\  1st : _V --> _V )  ->  ( 1st  o.  1st ) : _V --> _V )
75, 5, 6mp2an 670 . . . . . 6  |-  ( 1st 
o.  1st ) : _V --> _V
8 df-va 25689 . . . . . . 7  |-  +v  =  ( 1st  o.  1st )
98feq1i 5705 . . . . . 6  |-  ( +v : _V --> _V  <->  ( 1st  o. 
1st ) : _V --> _V )
107, 9mpbir 209 . . . . 5  |-  +v : _V
--> _V
11 fvco3 5925 . . . . 5  |-  ( ( +v : _V --> _V  /\  U  e.  _V )  ->  ( (  /g  o.  +v ) `  U )  =  (  /g  `  ( +v `  U ) ) )
1210, 11mpan 668 . . . 4  |-  ( U  e.  _V  ->  (
(  /g  o.  +v ) `  U )  =  (  /g  `  ( +v `  U ) ) )
132, 12syl5eq 2507 . . 3  |-  ( U  e.  _V  ->  ( -v `  U )  =  (  /g  `  ( +v `  U ) ) )
14 0ngrp 25414 . . . . . 6  |-  -.  (/)  e.  GrpOp
15 vex 3109 . . . . . . . . . 10  |-  g  e. 
_V
1615rnex 6707 . . . . . . . . 9  |-  ran  g  e.  _V
1716, 16mpt2ex 6850 . . . . . . . 8  |-  ( x  e.  ran  g ,  y  e.  ran  g  |->  ( x g ( ( inv `  g
) `  y )
) )  e.  _V
18 df-gdiv 25397 . . . . . . . 8  |-  /g  =  ( g  e.  GrpOp  |->  ( x  e.  ran  g ,  y  e.  ran  g  |->  ( x g ( ( inv `  g ) `  y
) ) ) )
1917, 18dmmpti 5692 . . . . . . 7  |-  dom  /g  =  GrpOp
2019eleq2i 2532 . . . . . 6  |-  ( (/)  e.  dom  /g  <->  (/)  e.  GrpOp )
2114, 20mtbir 297 . . . . 5  |-  -.  (/)  e.  dom  /g
22 ndmfv 5872 . . . . 5  |-  ( -.  (/)  e.  dom  /g  ->  (  /g  `  (/) )  =  (/) )
2321, 22mp1i 12 . . . 4  |-  ( -.  U  e.  _V  ->  (  /g  `  (/) )  =  (/) )
24 fvprc 5842 . . . . 5  |-  ( -.  U  e.  _V  ->  ( +v `  U )  =  (/) )
2524fveq2d 5852 . . . 4  |-  ( -.  U  e.  _V  ->  (  /g  `  ( +v
`  U ) )  =  (  /g  `  (/) ) )
26 fvprc 5842 . . . 4  |-  ( -.  U  e.  _V  ->  ( -v `  U )  =  (/) )
2723, 25, 263eqtr4rd 2506 . . 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 5851 . 2  |-  (  /g  `  G )  =  (  /g  `  ( +v
`  U ) )
3228, 29, 313eqtr4i 2493 1  |-  M  =  (  /g  `  G
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1398    e. wcel 1823   _Vcvv 3106   (/)c0 3783   dom cdm 4988   ran crn 4989    o. ccom 4992   -->wf 5566   -onto->wfo 5568   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   1stc1st 6771   GrpOpcgr 25389   invcgn 25391    /g cgs 25392   +vcpv 25679   -vcnsb 25683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-grpo 25394  df-gdiv 25397  df-va 25689  df-vs 25693
This theorem is referenced by:  nvm  25737  nvmfval  25740  nvnnncan1  25744  nvnnncan2  25745  nvaddsubass  25754  nvaddsub  25755  nvmtri2  25776
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