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Theorem nvmval 22113
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 22086 . . 3  |-  ( U  e.  NrmCVec  ->  G  e.  GrpOp )
3 nvmval.1 . . . . 5  |-  X  =  ( BaseSet `  U )
43, 1bafval 22073 . . . 4  |-  X  =  ran  G
5 eqid 2435 . . . 4  |-  ( inv `  G )  =  ( inv `  G )
6 eqid 2435 . . . 4  |-  (  /g  `  G )  =  (  /g  `  G )
74, 5, 6grpodivval 21821 . . 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 22112 . 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 22110 . . . 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 6089 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A G ( -u 1 S B ) )  =  ( A G ( ( inv `  G
) `  B )
) )
158, 10, 143eqtr4d 2477 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 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   1c1 8981   -ucneg 9282   GrpOpcgr 21764   invcgn 21766    /g cgs 21767   NrmCVeccnv 22053   +vcpv 22054   BaseSetcba 22055   .s
OLDcns 22056   -vcnsb 22058
This theorem is referenced by:  nvmval2  22114  nvzs  22116  nvmdi  22121  nvsubadd  22126  nvpncan2  22127  nvaddsub4  22132  nvnncan  22134  nvsub  22146  nvmtri  22150  imsdval2  22169  nvnd  22170  ipval3  22195  sspmval  22222  isph  22313  dipsubdir  22339
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9112  df-mnf 9113  df-ltxr 9115  df-sub 9283  df-neg 9284  df-grpo 21769  df-gid 21770  df-ginv 21771  df-gdiv 21772  df-ablo 21860  df-vc 22015  df-nv 22061  df-va 22064  df-ba 22065  df-sm 22066  df-0v 22067  df-vs 22068  df-nmcv 22069
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