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Theorem nvinv 22073
Description: Minus 1 times a vector is the underlying group's inverse element. Equation 2 of [Kreyszig] p. 51. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvinv.1  |-  X  =  ( BaseSet `  U )
nvinv.2  |-  G  =  ( +v `  U
)
nvinv.4  |-  S  =  ( .s OLD `  U
)
nvinv.5  |-  M  =  ( inv `  G
)
Assertion
Ref Expression
nvinv  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( -u 1 S A )  =  ( M `  A ) )

Proof of Theorem nvinv
StepHypRef Expression
1 eqid 2404 . . 3  |-  ( 1st `  U )  =  ( 1st `  U )
21nvvc 22047 . 2  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  e.  CVec OLD )
3 nvinv.2 . . . 4  |-  G  =  ( +v `  U
)
43vafval 22035 . . 3  |-  G  =  ( 1st `  ( 1st `  U ) )
5 nvinv.4 . . . 4  |-  S  =  ( .s OLD `  U
)
65smfval 22037 . . 3  |-  S  =  ( 2nd `  ( 1st `  U ) )
7 nvinv.1 . . . 4  |-  X  =  ( BaseSet `  U )
87, 3bafval 22036 . . 3  |-  X  =  ran  G
9 nvinv.5 . . 3  |-  M  =  ( inv `  G
)
104, 6, 8, 9vcm 22003 . 2  |-  ( ( ( 1st `  U
)  e.  CVec OLD  /\  A  e.  X )  ->  ( -u 1 S A )  =  ( M `  A ) )
112, 10sylan 458 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( -u 1 S A )  =  ( M `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   ` cfv 5413  (class class class)co 6040   1stc1st 6306   1c1 8947   -ucneg 9248   invcgn 21729   CVec
OLDcvc 21977   NrmCVeccnv 22016   +vcpv 22017   BaseSetcba 22018   .s
OLDcns 22019
This theorem is referenced by:  nvinvfval  22074  nvmval  22076  nvmfval  22078  nvnegneg  22085  nvrinv  22087  nvlinv  22088
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-ltxr 9081  df-sub 9249  df-neg 9250  df-grpo 21732  df-gid 21733  df-ginv 21734  df-ablo 21823  df-vc 21978  df-nv 22024  df-va 22027  df-ba 22028  df-sm 22029  df-0v 22030  df-nmcv 22032
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