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Theorem nvmval 21964
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 21937 . . 3  |-  ( U  e.  NrmCVec  ->  G  e.  GrpOp )
3 nvmval.1 . . . . 5  |-  X  =  ( BaseSet `  U )
43, 1bafval 21924 . . . 4  |-  X  =  ran  G
5 eqid 2380 . . . 4  |-  ( inv `  G )  =  ( inv `  G )
6 eqid 2380 . . . 4  |-  (  /g  `  G )  =  (  /g  `  G )
74, 5, 6grpodivval 21672 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A (  /g  `  G
) B )  =  ( A G ( ( inv `  G
) `  B )
) )
82, 7syl3an1 1217 . 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 21963 . 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 21961 . . . 4  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( -u 1 S B )  =  ( ( inv `  G ) `  B
) )
13123adant2 976 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 S B )  =  ( ( inv `  G ) `  B
) )
1413oveq2d 6029 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A G ( -u 1 S B ) )  =  ( A G ( ( inv `  G
) `  B )
) )
158, 10, 143eqtr4d 2422 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 936    = wceq 1649    e. wcel 1717   ` cfv 5387  (class class class)co 6013   1c1 8917   -ucneg 9217   GrpOpcgr 21615   invcgn 21617    /g cgs 21618   NrmCVeccnv 21904   +vcpv 21905   BaseSetcba 21906   .s
OLDcns 21907   -vcnsb 21909
This theorem is referenced by:  nvmval2  21965  nvzs  21967  nvmdi  21972  nvsubadd  21977  nvpncan2  21978  nvaddsub4  21983  nvnncan  21985  nvsub  21997  nvmtri  22001  imsdval2  22020  nvnd  22021  ipval3  22046  sspmval  22073  isph  22164  dipsubdir  22190
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-po 4437  df-so 4438  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-ltxr 9051  df-sub 9218  df-neg 9219  df-grpo 21620  df-gid 21621  df-ginv 21622  df-gdiv 21623  df-ablo 21711  df-vc 21866  df-nv 21912  df-va 21915  df-ba 21916  df-sm 21917  df-0v 21918  df-vs 21919  df-nmcv 21920
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