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Theorem nvinv 21197
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 2283 . . 3  |-  ( 1st `  U )  =  ( 1st `  U )
21nvvc 21171 . 2  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  e.  CVec OLD )
3 nvinv.2 . . . 4  |-  G  =  ( +v `  U
)
43vafval 21159 . . 3  |-  G  =  ( 1st `  ( 1st `  U ) )
5 nvinv.4 . . . 4  |-  S  =  ( .s OLD `  U
)
65smfval 21161 . . 3  |-  S  =  ( 2nd `  ( 1st `  U ) )
7 nvinv.1 . . . 4  |-  X  =  ( BaseSet `  U )
87, 3bafval 21160 . . 3  |-  X  =  ran  G
9 nvinv.5 . . 3  |-  M  =  ( inv `  G
)
104, 6, 8, 9vcm 21127 . 2  |-  ( ( ( 1st `  U
)  e.  CVec OLD  /\  A  e.  X )  ->  ( -u 1 S A )  =  ( M `  A ) )
112, 10sylan 457 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( -u 1 S A )  =  ( M `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   1stc1st 6120   1c1 8738   -ucneg 9038   invcgn 20855   CVec
OLDcvc 21101   NrmCVeccnv 21140   +vcpv 21141   BaseSetcba 21142   .s
OLDcns 21143
This theorem is referenced by:  nvinvfval  21198  nvmval  21200  nvmfval  21202  nvnegneg  21209  nvrinv  21211  nvlinv  21212
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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