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Theorem vsfval 24158
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 24122 . . . . 5  |-  -v  =  (  /g  o.  +v )
21fveq1i 5793 . . . 4  |-  ( -v
`  U )  =  ( (  /g  o.  +v ) `  U )
3 fo1st 6699 . . . . . . . 8  |-  1st : _V -onto-> _V
4 fof 5721 . . . . . . . 8  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
53, 4ax-mp 5 . . . . . . 7  |-  1st : _V
--> _V
6 fco 5669 . . . . . . 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 24118 . . . . . . 7  |-  +v  =  ( 1st  o.  1st )
98feq1i 5652 . . . . . 6  |-  ( +v : _V --> _V  <->  ( 1st  o. 
1st ) : _V --> _V )
107, 9mpbir 209 . . . . 5  |-  +v : _V
--> _V
11 fvco3 5870 . . . . 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 2504 . . 3  |-  ( U  e.  _V  ->  ( -v `  U )  =  (  /g  `  ( +v `  U ) ) )
14 0ngrp 23843 . . . . . 6  |-  -.  (/)  e.  GrpOp
15 vex 3074 . . . . . . . . . 10  |-  g  e. 
_V
1615rnex 6615 . . . . . . . . 9  |-  ran  g  e.  _V
1716, 16mpt2ex 6753 . . . . . . . 8  |-  ( x  e.  ran  g ,  y  e.  ran  g  |->  ( x g ( ( inv `  g
) `  y )
) )  e.  _V
18 df-gdiv 23826 . . . . . . . 8  |-  /g  =  ( g  e.  GrpOp  |->  ( x  e.  ran  g ,  y  e.  ran  g  |->  ( x g ( ( inv `  g ) `  y
) ) ) )
1917, 18dmmpti 5641 . . . . . . 7  |-  dom  /g  =  GrpOp
2019eleq2i 2529 . . . . . 6  |-  ( (/)  e.  dom  /g  <->  (/)  e.  GrpOp )
2114, 20mtbir 299 . . . . 5  |-  -.  (/)  e.  dom  /g
22 ndmfv 5816 . . . . 5  |-  ( -.  (/)  e.  dom  /g  ->  (  /g  `  (/) )  =  (/) )
2321, 22mp1i 12 . . . 4  |-  ( -.  U  e.  _V  ->  (  /g  `  (/) )  =  (/) )
24 fvprc 5786 . . . . 5  |-  ( -.  U  e.  _V  ->  ( +v `  U )  =  (/) )
2524fveq2d 5796 . . . 4  |-  ( -.  U  e.  _V  ->  (  /g  `  ( +v
`  U ) )  =  (  /g  `  (/) ) )
26 fvprc 5786 . . . 4  |-  ( -.  U  e.  _V  ->  ( -v `  U )  =  (/) )
2723, 25, 263eqtr4rd 2503 . . 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 5795 . 2  |-  (  /g  `  G )  =  (  /g  `  ( +v
`  U ) )
3228, 29, 313eqtr4i 2490 1  |-  M  =  (  /g  `  G
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1370    e. wcel 1758   _Vcvv 3071   (/)c0 3738   dom cdm 4941   ran crn 4942    o. ccom 4945   -->wf 5515   -onto->wfo 5517   ` cfv 5519  (class class class)co 6193    |-> cmpt2 6195   1stc1st 6678   GrpOpcgr 23818   invcgn 23820    /g cgs 23821   +vcpv 24108   -vcnsb 24112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-1st 6680  df-2nd 6681  df-grpo 23823  df-gdiv 23826  df-va 24118  df-vs 24122
This theorem is referenced by:  nvm  24166  nvmfval  24169  nvnnncan1  24173  nvnnncan2  24174  nvaddsubass  24183  nvaddsub  24184  nvmtri2  24205
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