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Theorem nvnegneg 22124
Description: Double negative of a vector. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvnegneg.1  |-  X  =  ( BaseSet `  U )
nvnegneg.4  |-  S  =  ( .s OLD `  U
)
Assertion
Ref Expression
nvnegneg  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( -u 1 S ( -u
1 S A ) )  =  A )

Proof of Theorem nvnegneg
StepHypRef Expression
1 neg1cn 10059 . . . 4  |-  -u 1  e.  CC
2 nvnegneg.1 . . . . 5  |-  X  =  ( BaseSet `  U )
3 nvnegneg.4 . . . . 5  |-  S  =  ( .s OLD `  U
)
42, 3nvscl 22099 . . . 4  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  A  e.  X )  ->  ( -u 1 S A )  e.  X )
51, 4mp3an2 1267 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( -u 1 S A )  e.  X )
6 eqid 2435 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
7 eqid 2435 . . . 4  |-  ( inv `  ( +v `  U
) )  =  ( inv `  ( +v
`  U ) )
82, 6, 3, 7nvinv 22112 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( -u 1 S A )  e.  X )  -> 
( -u 1 S (
-u 1 S A ) )  =  ( ( inv `  ( +v `  U ) ) `
 ( -u 1 S A ) ) )
95, 8syldan 457 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( -u 1 S ( -u
1 S A ) )  =  ( ( inv `  ( +v
`  U ) ) `
 ( -u 1 S A ) ) )
102, 6, 3, 7nvinv 22112 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( -u 1 S A )  =  ( ( inv `  ( +v `  U
) ) `  A
) )
1110fveq2d 5724 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( inv `  ( +v `  U ) ) `
 ( -u 1 S A ) )  =  ( ( inv `  ( +v `  U ) ) `
 ( ( inv `  ( +v `  U
) ) `  A
) ) )
126nvgrp 22088 . . 3  |-  ( U  e.  NrmCVec  ->  ( +v `  U )  e.  GrpOp )
132, 6bafval 22075 . . . 4  |-  X  =  ran  ( +v `  U )
1413, 7grpo2inv 21819 . . 3  |-  ( ( ( +v `  U
)  e.  GrpOp  /\  A  e.  X )  ->  (
( inv `  ( +v `  U ) ) `
 ( ( inv `  ( +v `  U
) ) `  A
) )  =  A )
1512, 14sylan 458 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( inv `  ( +v `  U ) ) `
 ( ( inv `  ( +v `  U
) ) `  A
) )  =  A )
169, 11, 153eqtrd 2471 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( -u 1 S ( -u
1 S A ) )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   CCcc 8980   1c1 8983   -ucneg 9284   GrpOpcgr 21766   invcgn 21768   NrmCVeccnv 22055   +vcpv 22056   BaseSetcba 22057   .s
OLDcns 22058
This theorem is referenced by:  nvdif  22146
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-ltxr 9117  df-sub 9285  df-neg 9286  df-grpo 21771  df-gid 21772  df-ginv 21773  df-ablo 21862  df-vc 22017  df-nv 22063  df-va 22066  df-ba 22067  df-sm 22068  df-0v 22069  df-nmcv 22071
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