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Theorem nvinv 21190
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 2285 . . 3  |-  ( 1st `  U )  =  ( 1st `  U )
21nvvc 21164 . 2  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  e.  CVec OLD )
3 nvinv.2 . . . 4  |-  G  =  ( +v `  U
)
43vafval 21152 . . 3  |-  G  =  ( 1st `  ( 1st `  U ) )
5 nvinv.4 . . . 4  |-  S  =  ( .s OLD `  U
)
65smfval 21154 . . 3  |-  S  =  ( 2nd `  ( 1st `  U ) )
7 nvinv.1 . . . 4  |-  X  =  ( BaseSet `  U )
87, 3bafval 21153 . . 3  |-  X  =  ran  G
9 nvinv.5 . . 3  |-  M  =  ( inv `  G
)
104, 6, 8, 9vcm 21120 . 2  |-  ( ( ( 1st `  U
)  e.  CVec OLD  /\  A  e.  X )  ->  ( -u 1 S A )  =  ( M `  A ) )
112, 10sylan 459 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( -u 1 S A )  =  ( M `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1624    e. wcel 1685   ` cfv 5222  (class class class)co 5820   1stc1st 6082   1c1 8734   -ucneg 9034   invcgn 20848   CVec
OLDcvc 21094   NrmCVeccnv 21133   +vcpv 21134   BaseSetcba 21135   .s
OLDcns 21136
This theorem is referenced by:  nvinvfval  21191  nvmval  21193  nvmfval  21195  nvnegneg  21202  nvrinv  21204  nvlinv  21205
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  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-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-pnf 8865  df-mnf 8866  df-ltxr 8868  df-sub 9035  df-neg 9036  df-grpo 20851  df-gid 20852  df-ginv 20853  df-ablo 20942  df-vc 21095  df-nv 21141  df-va 21144  df-ba 21145  df-sm 21146  df-0v 21147  df-nmcv 21149
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