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Theorem nvinvfval 21198
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 2283 . . . . 5  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
2 nvinvfval.4 . . . . 5  |-  S  =  ( .s OLD `  U
)
31, 2nvsf 21175 . . . 4  |-  ( U  e.  NrmCVec  ->  S : ( CC  X.  ( BaseSet `  U ) ) --> (
BaseSet `  U ) )
4 neg1cn 9813 . . . 4  |-  -u 1  e.  CC
5 nvinvfval.3 . . . . 5  |-  N  =  ( S  o.  `' ( 2nd  |`  ( { -u 1 }  X.  _V ) ) )
65curry1f 6212 . . . 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 5389 . . 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 21173 . . 3  |-  ( U  e.  NrmCVec  ->  G  e.  GrpOp )
121, 10bafval 21160 . . . 4  |-  ( BaseSet `  U )  =  ran  G
13 eqid 2283 . . . 4  |-  ( inv `  G )  =  ( inv `  G )
1412, 13grpoinvf 20907 . . 3  |-  ( G  e.  GrpOp  ->  ( inv `  G ) : (
BaseSet `  U ) -1-1-onto-> ( BaseSet `  U ) )
15 f1ofn 5473 . . 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 5389 . . . . . 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 6211 . . . 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 21197 . . 3  |-  ( ( U  e.  NrmCVec  /\  x  e.  ( BaseSet `  U )
)  ->  ( -u 1 S x )  =  ( ( inv `  G
) `  x )
)
2321, 22eqtrd 2315 . 2  |-  ( ( U  e.  NrmCVec  /\  x  e.  ( BaseSet `  U )
)  ->  ( N `  x )  =  ( ( inv `  G
) `  x )
)
249, 16, 23eqfnfvd 5625 1  |-  ( U  e.  NrmCVec  ->  N  =  ( inv `  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640    X. cxp 4687   `'ccnv 4688    |` cres 4691    o. ccom 4693    Fn wfn 5250   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   2ndc2nd 6121   CCcc 8735   1c1 8738   -ucneg 9038   GrpOpcgr 20853   invcgn 20855   NrmCVeccnv 21140   +vcpv 21141   BaseSetcba 21142   .s
OLDcns 21143
This theorem is referenced by:  hhssabloi  21839
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-sub 9039  df-neg 9040  df-grpo 20858  df-gid 20859  df-ginv 20860  df-ablo 20949  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-nmcv 21156
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