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Theorem nvinvfval 21214
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  =  ( .s OLD `  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 2296 . . . . 5  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
2 nvinvfval.4 . . . . 5  |-  S  =  ( .s OLD `  U
)
31, 2nvsf 21191 . . . 4  |-  ( U  e.  NrmCVec  ->  S : ( CC  X.  ( BaseSet `  U ) ) --> (
BaseSet `  U ) )
4 neg1cn 9829 . . . 4  |-  -u 1  e.  CC
5 nvinvfval.3 . . . . 5  |-  N  =  ( S  o.  `' ( 2nd  |`  ( { -u 1 }  X.  _V ) ) )
65curry1f 6228 . . . 4  |-  ( ( S : ( CC 
X.  ( BaseSet `  U
) ) --> ( BaseSet `  U )  /\  -u 1  e.  CC )  ->  N : ( BaseSet `  U
) --> ( BaseSet `  U
) )
73, 4, 6sylancl 643 . . 3  |-  ( U  e.  NrmCVec  ->  N : (
BaseSet `  U ) --> (
BaseSet `  U ) )
8 ffn 5405 . . 3  |-  ( N : ( BaseSet `  U
) --> ( BaseSet `  U
)  ->  N  Fn  ( BaseSet `  U )
)
97, 8syl 15 . 2  |-  ( U  e.  NrmCVec  ->  N  Fn  ( BaseSet
`  U ) )
10 nvinvfval.2 . . . 4  |-  G  =  ( +v `  U
)
1110nvgrp 21189 . . 3  |-  ( U  e.  NrmCVec  ->  G  e.  GrpOp )
121, 10bafval 21176 . . . 4  |-  ( BaseSet `  U )  =  ran  G
13 eqid 2296 . . . 4  |-  ( inv `  G )  =  ( inv `  G )
1412, 13grpoinvf 20923 . . 3  |-  ( G  e.  GrpOp  ->  ( inv `  G ) : (
BaseSet `  U ) -1-1-onto-> ( BaseSet `  U ) )
15 f1ofn 5489 . . 3  |-  ( ( inv `  G ) : ( BaseSet `  U
)
-1-1-onto-> ( BaseSet `  U )  ->  ( inv `  G
)  Fn  ( BaseSet `  U ) )
1611, 14, 153syl 18 . 2  |-  ( U  e.  NrmCVec  ->  ( inv `  G
)  Fn  ( BaseSet `  U ) )
17 ffn 5405 . . . . . 6  |-  ( S : ( CC  X.  ( BaseSet `  U )
) --> ( BaseSet `  U
)  ->  S  Fn  ( CC  X.  ( BaseSet
`  U ) ) )
183, 17syl 15 . . . . 5  |-  ( U  e.  NrmCVec  ->  S  Fn  ( CC  X.  ( BaseSet `  U
) ) )
1918adantr 451 . . . 4  |-  ( ( U  e.  NrmCVec  /\  x  e.  ( BaseSet `  U )
)  ->  S  Fn  ( CC  X.  ( BaseSet
`  U ) ) )
205curry1val 6227 . . . 4  |-  ( ( S  Fn  ( CC 
X.  ( BaseSet `  U
) )  /\  -u 1  e.  CC )  ->  ( N `  x )  =  ( -u 1 S x ) )
2119, 4, 20sylancl 643 . . 3  |-  ( ( U  e.  NrmCVec  /\  x  e.  ( BaseSet `  U )
)  ->  ( N `  x )  =  (
-u 1 S x ) )
221, 10, 2, 13nvinv 21213 . . 3  |-  ( ( U  e.  NrmCVec  /\  x  e.  ( BaseSet `  U )
)  ->  ( -u 1 S x )  =  ( ( inv `  G
) `  x )
)
2321, 22eqtrd 2328 . 2  |-  ( ( U  e.  NrmCVec  /\  x  e.  ( BaseSet `  U )
)  ->  ( N `  x )  =  ( ( inv `  G
) `  x )
)
249, 16, 23eqfnfvd 5641 1  |-  ( U  e.  NrmCVec  ->  N  =  ( inv `  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   {csn 3653    X. cxp 4703   `'ccnv 4704    |` cres 4707    o. ccom 4709    Fn wfn 5266   -->wf 5267   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   2ndc2nd 6137   CCcc 8751   1c1 8754   -ucneg 9054   GrpOpcgr 20869   invcgn 20871   NrmCVeccnv 21156   +vcpv 21157   BaseSetcba 21158   .s
OLDcns 21159
This theorem is referenced by:  hhssabloi  21855
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888  df-sub 9055  df-neg 9056  df-grpo 20874  df-gid 20875  df-ginv 20876  df-ablo 20965  df-vc 21118  df-nv 21164  df-va 21167  df-ba 21168  df-sm 21169  df-0v 21170  df-nmcv 21172
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