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Theorem nvmval 21216
Description: Value of vector subtraction on a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvmval.1  |-  X  =  ( BaseSet `  U )
nvmval.2  |-  G  =  ( +v `  U
)
nvmval.4  |-  S  =  ( .s OLD `  U
)
nvmval.3  |-  M  =  ( -v `  U
)
Assertion
Ref Expression
nvmval  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( A G (
-u 1 S B ) ) )

Proof of Theorem nvmval
StepHypRef Expression
1 nvmval.2 . . . 4  |-  G  =  ( +v `  U
)
21nvgrp 21189 . . 3  |-  ( U  e.  NrmCVec  ->  G  e.  GrpOp )
3 nvmval.1 . . . . 5  |-  X  =  ( BaseSet `  U )
43, 1bafval 21176 . . . 4  |-  X  =  ran  G
5 eqid 2296 . . . 4  |-  ( inv `  G )  =  ( inv `  G )
6 eqid 2296 . . . 4  |-  (  /g  `  G )  =  (  /g  `  G )
74, 5, 6grpodivval 20926 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A (  /g  `  G
) B )  =  ( A G ( ( inv `  G
) `  B )
) )
82, 7syl3an1 1215 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A (  /g  `  G
) B )  =  ( A G ( ( inv `  G
) `  B )
) )
9 nvmval.3 . . 3  |-  M  =  ( -v `  U
)
103, 1, 9, 6nvm 21215 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( A (  /g  `  G ) B ) )
11 nvmval.4 . . . . 5  |-  S  =  ( .s OLD `  U
)
123, 1, 11, 5nvinv 21213 . . . 4  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( -u 1 S B )  =  ( ( inv `  G ) `  B
) )
13123adant2 974 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 S B )  =  ( ( inv `  G ) `  B
) )
1413oveq2d 5890 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A G ( -u 1 S B ) )  =  ( A G ( ( inv `  G
) `  B )
) )
158, 10, 143eqtr4d 2338 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( A G (
-u 1 S B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   1c1 8754   -ucneg 9054   GrpOpcgr 20869   invcgn 20871    /g cgs 20872   NrmCVeccnv 21156   +vcpv 21157   BaseSetcba 21158   .s
OLDcns 21159   -vcnsb 21161
This theorem is referenced by:  nvmval2  21217  nvzs  21219  nvmdi  21224  nvsubadd  21229  nvpncan2  21230  nvaddsub4  21235  nvnncan  21237  nvsub  21249  nvmtri  21253  imsdval2  21272  nvnd  21273  ipval3  21298  sspmval  21325  isph  21416  dipsubdir  21442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888  df-sub 9055  df-neg 9056  df-grpo 20874  df-gid 20875  df-ginv 20876  df-gdiv 20877  df-ablo 20965  df-vc 21118  df-nv 21164  df-va 21167  df-ba 21168  df-sm 21169  df-0v 21170  df-vs 21171  df-nmcv 21172
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