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Theorem nvinvfval 25935
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 2402 . . . . 5  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
2 nvinvfval.4 . . . . 5  |-  S  =  ( .sOLD `  U )
31, 2nvsf 25912 . . . 4  |-  ( U  e.  NrmCVec  ->  S : ( CC  X.  ( BaseSet `  U ) ) --> (
BaseSet `  U ) )
4 neg1cn 10679 . . . 4  |-  -u 1  e.  CC
5 nvinvfval.3 . . . . 5  |-  N  =  ( S  o.  `' ( 2nd  |`  ( { -u 1 }  X.  _V ) ) )
65curry1f 6877 . . . 4  |-  ( ( S : ( CC 
X.  ( BaseSet `  U
) ) --> ( BaseSet `  U )  /\  -u 1  e.  CC )  ->  N : ( BaseSet `  U
) --> ( BaseSet `  U
) )
73, 4, 6sylancl 660 . . 3  |-  ( U  e.  NrmCVec  ->  N : (
BaseSet `  U ) --> (
BaseSet `  U ) )
8 ffn 5713 . . 3  |-  ( N : ( BaseSet `  U
) --> ( BaseSet `  U
)  ->  N  Fn  ( BaseSet `  U )
)
97, 8syl 17 . 2  |-  ( U  e.  NrmCVec  ->  N  Fn  ( BaseSet
`  U ) )
10 nvinvfval.2 . . . 4  |-  G  =  ( +v `  U
)
1110nvgrp 25910 . . 3  |-  ( U  e.  NrmCVec  ->  G  e.  GrpOp )
121, 10bafval 25897 . . . 4  |-  ( BaseSet `  U )  =  ran  G
13 eqid 2402 . . . 4  |-  ( inv `  G )  =  ( inv `  G )
1412, 13grpoinvf 25642 . . 3  |-  ( G  e.  GrpOp  ->  ( inv `  G ) : (
BaseSet `  U ) -1-1-onto-> ( BaseSet `  U ) )
15 f1ofn 5799 . . 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 5713 . . . . . 6  |-  ( S : ( CC  X.  ( BaseSet `  U )
) --> ( BaseSet `  U
)  ->  S  Fn  ( CC  X.  ( BaseSet
`  U ) ) )
183, 17syl 17 . . . . 5  |-  ( U  e.  NrmCVec  ->  S  Fn  ( CC  X.  ( BaseSet `  U
) ) )
1918adantr 463 . . . 4  |-  ( ( U  e.  NrmCVec  /\  x  e.  ( BaseSet `  U )
)  ->  S  Fn  ( CC  X.  ( BaseSet
`  U ) ) )
205curry1val 6876 . . . 4  |-  ( ( S  Fn  ( CC 
X.  ( BaseSet `  U
) )  /\  -u 1  e.  CC )  ->  ( N `  x )  =  ( -u 1 S x ) )
2119, 4, 20sylancl 660 . . 3  |-  ( ( U  e.  NrmCVec  /\  x  e.  ( BaseSet `  U )
)  ->  ( N `  x )  =  (
-u 1 S x ) )
221, 10, 2, 13nvinv 25934 . . 3  |-  ( ( U  e.  NrmCVec  /\  x  e.  ( BaseSet `  U )
)  ->  ( -u 1 S x )  =  ( ( inv `  G
) `  x )
)
2321, 22eqtrd 2443 . 2  |-  ( ( U  e.  NrmCVec  /\  x  e.  ( BaseSet `  U )
)  ->  ( N `  x )  =  ( ( inv `  G
) `  x )
)
249, 16, 23eqfnfvd 5961 1  |-  ( U  e.  NrmCVec  ->  N  =  ( inv `  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   _Vcvv 3058   {csn 3971    X. cxp 4820   `'ccnv 4821    |` cres 4824    o. ccom 4826    Fn wfn 5563   -->wf 5564   -1-1-onto->wf1o 5567   ` cfv 5568  (class class class)co 6277   2ndc2nd 6782   CCcc 9519   1c1 9522   -ucneg 9841   GrpOpcgr 25588   invcgn 25590   NrmCVeccnv 25877   +vcpv 25878   BaseSetcba 25879   .sOLDcns 25880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-po 4743  df-so 4744  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-pnf 9659  df-mnf 9660  df-ltxr 9662  df-sub 9842  df-neg 9843  df-grpo 25593  df-gid 25594  df-ginv 25595  df-ablo 25684  df-vc 25839  df-nv 25885  df-va 25888  df-ba 25889  df-sm 25890  df-0v 25891  df-nmcv 25893
This theorem is referenced by:  hhssabloi  26578
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