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Theorem nvrinv 26160
Description: A vector minus itself. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvrinv.1  |-  X  =  ( BaseSet `  U )
nvrinv.2  |-  G  =  ( +v `  U
)
nvrinv.4  |-  S  =  ( .sOLD `  U )
nvrinv.6  |-  Z  =  ( 0vec `  U
)
Assertion
Ref Expression
nvrinv  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G ( -u 1 S A ) )  =  Z )

Proof of Theorem nvrinv
StepHypRef Expression
1 nvrinv.2 . . . 4  |-  G  =  ( +v `  U
)
21nvgrp 26122 . . 3  |-  ( U  e.  NrmCVec  ->  G  e.  GrpOp )
3 nvrinv.1 . . . . 5  |-  X  =  ( BaseSet `  U )
43, 1bafval 26109 . . . 4  |-  X  =  ran  G
5 eqid 2420 . . . 4  |-  (GId `  G )  =  (GId
`  G )
6 eqid 2420 . . . 4  |-  ( inv `  G )  =  ( inv `  G )
74, 5, 6grporinv 25843 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G ( ( inv `  G ) `  A
) )  =  (GId
`  G ) )
82, 7sylan 473 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G ( ( inv `  G ) `  A
) )  =  (GId
`  G ) )
9 nvrinv.4 . . . 4  |-  S  =  ( .sOLD `  U )
103, 1, 9, 6nvinv 26146 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( -u 1 S A )  =  ( ( inv `  G ) `  A
) )
1110oveq2d 6312 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G ( -u 1 S A ) )  =  ( A G ( ( inv `  G
) `  A )
) )
12 nvrinv.6 . . . 4  |-  Z  =  ( 0vec `  U
)
131, 120vfval 26111 . . 3  |-  ( U  e.  NrmCVec  ->  Z  =  (GId
`  G ) )
1413adantr 466 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  Z  =  (GId `  G )
)
158, 11, 143eqtr4d 2471 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G ( -u 1 S A ) )  =  Z )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1867   ` cfv 5592  (class class class)co 6296   1c1 9529   -ucneg 9850   GrpOpcgr 25800  GIdcgi 25801   invcgn 25802   NrmCVeccnv 26089   +vcpv 26090   BaseSetcba 26091   .sOLDcns 26092   0veccn0v 26093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-po 4766  df-so 4767  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6798  df-2nd 6799  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-pnf 9666  df-mnf 9667  df-ltxr 9669  df-sub 9851  df-neg 9852  df-grpo 25805  df-gid 25806  df-ginv 25807  df-ablo 25896  df-vc 26051  df-nv 26097  df-va 26100  df-ba 26101  df-sm 26102  df-0v 26103  df-nmcv 26105
This theorem is referenced by:  nvsubadd  26162  nvpncan2  26163  nvnncan  26170  ipidsq  26235  ip2i  26355  ipdirilem  26356  ipasslem2  26359
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