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Theorem nvinvfval 25199
Description: Function for the negative of a vector on a normed complex vector space, in terms of the underlying addition group inverse. (We currently do not have a separate notation for the negative of a vector.) (Contributed by NM, 27-Mar-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvinvfval.2  |-  G  =  ( +v `  U
)
nvinvfval.4  |-  S  =  ( .sOLD `  U )
nvinvfval.3  |-  N  =  ( S  o.  `' ( 2nd  |`  ( { -u 1 }  X.  _V ) ) )
Assertion
Ref Expression
nvinvfval  |-  ( U  e.  NrmCVec  ->  N  =  ( inv `  G ) )

Proof of Theorem nvinvfval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2462 . . . . 5  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
2 nvinvfval.4 . . . . 5  |-  S  =  ( .sOLD `  U )
31, 2nvsf 25176 . . . 4  |-  ( U  e.  NrmCVec  ->  S : ( CC  X.  ( BaseSet `  U ) ) --> (
BaseSet `  U ) )
4 neg1cn 10630 . . . 4  |-  -u 1  e.  CC
5 nvinvfval.3 . . . . 5  |-  N  =  ( S  o.  `' ( 2nd  |`  ( { -u 1 }  X.  _V ) ) )
65curry1f 6869 . . . 4  |-  ( ( S : ( CC 
X.  ( BaseSet `  U
) ) --> ( BaseSet `  U )  /\  -u 1  e.  CC )  ->  N : ( BaseSet `  U
) --> ( BaseSet `  U
) )
73, 4, 6sylancl 662 . . 3  |-  ( U  e.  NrmCVec  ->  N : (
BaseSet `  U ) --> (
BaseSet `  U ) )
8 ffn 5724 . . 3  |-  ( N : ( BaseSet `  U
) --> ( BaseSet `  U
)  ->  N  Fn  ( BaseSet `  U )
)
97, 8syl 16 . 2  |-  ( U  e.  NrmCVec  ->  N  Fn  ( BaseSet
`  U ) )
10 nvinvfval.2 . . . 4  |-  G  =  ( +v `  U
)
1110nvgrp 25174 . . 3  |-  ( U  e.  NrmCVec  ->  G  e.  GrpOp )
121, 10bafval 25161 . . . 4  |-  ( BaseSet `  U )  =  ran  G
13 eqid 2462 . . . 4  |-  ( inv `  G )  =  ( inv `  G )
1412, 13grpoinvf 24906 . . 3  |-  ( G  e.  GrpOp  ->  ( inv `  G ) : (
BaseSet `  U ) -1-1-onto-> ( BaseSet `  U ) )
15 f1ofn 5810 . . 3  |-  ( ( inv `  G ) : ( BaseSet `  U
)
-1-1-onto-> ( BaseSet `  U )  ->  ( inv `  G
)  Fn  ( BaseSet `  U ) )
1611, 14, 153syl 20 . 2  |-  ( U  e.  NrmCVec  ->  ( inv `  G
)  Fn  ( BaseSet `  U ) )
17 ffn 5724 . . . . . 6  |-  ( S : ( CC  X.  ( BaseSet `  U )
) --> ( BaseSet `  U
)  ->  S  Fn  ( CC  X.  ( BaseSet
`  U ) ) )
183, 17syl 16 . . . . 5  |-  ( U  e.  NrmCVec  ->  S  Fn  ( CC  X.  ( BaseSet `  U
) ) )
1918adantr 465 . . . 4  |-  ( ( U  e.  NrmCVec  /\  x  e.  ( BaseSet `  U )
)  ->  S  Fn  ( CC  X.  ( BaseSet
`  U ) ) )
205curry1val 6868 . . . 4  |-  ( ( S  Fn  ( CC 
X.  ( BaseSet `  U
) )  /\  -u 1  e.  CC )  ->  ( N `  x )  =  ( -u 1 S x ) )
2119, 4, 20sylancl 662 . . 3  |-  ( ( U  e.  NrmCVec  /\  x  e.  ( BaseSet `  U )
)  ->  ( N `  x )  =  (
-u 1 S x ) )
221, 10, 2, 13nvinv 25198 . . 3  |-  ( ( U  e.  NrmCVec  /\  x  e.  ( BaseSet `  U )
)  ->  ( -u 1 S x )  =  ( ( inv `  G
) `  x )
)
2321, 22eqtrd 2503 . 2  |-  ( ( U  e.  NrmCVec  /\  x  e.  ( BaseSet `  U )
)  ->  ( N `  x )  =  ( ( inv `  G
) `  x )
)
249, 16, 23eqfnfvd 5971 1  |-  ( U  e.  NrmCVec  ->  N  =  ( inv `  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   _Vcvv 3108   {csn 4022    X. cxp 4992   `'ccnv 4993    |` cres 4996    o. ccom 4998    Fn wfn 5576   -->wf 5577   -1-1-onto->wf1o 5580   ` cfv 5581  (class class class)co 6277   2ndc2nd 6775   CCcc 9481   1c1 9484   -ucneg 9797   GrpOpcgr 24852   invcgn 24854   NrmCVeccnv 25141   +vcpv 25142   BaseSetcba 25143   .sOLDcns 25144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-po 4795  df-so 4796  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-ltxr 9624  df-sub 9798  df-neg 9799  df-grpo 24857  df-gid 24858  df-ginv 24859  df-ablo 24948  df-vc 25103  df-nv 25149  df-va 25152  df-ba 25153  df-sm 25154  df-0v 25155  df-nmcv 25157
This theorem is referenced by:  hhssabloi  25842
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