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Theorem nvmtri2 22161
Description: Triangle inequality for the norm of a vector difference. (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvmtri.1  |-  X  =  ( BaseSet `  U )
nvmtri.3  |-  M  =  ( -v `  U
)
nvmtri.6  |-  N  =  ( normCV `  U )
Assertion
Ref Expression
nvmtri2  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( N `  ( A M C ) )  <_  (
( N `  ( A M B ) )  +  ( N `  ( B M C ) ) ) )

Proof of Theorem nvmtri2
StepHypRef Expression
1 eqid 2436 . . . . . 6  |-  ( +v
`  U )  =  ( +v `  U
)
21nvgrp 22096 . . . . 5  |-  ( U  e.  NrmCVec  ->  ( +v `  U )  e.  GrpOp )
3 nvmtri.1 . . . . . . 7  |-  X  =  ( BaseSet `  U )
43, 1bafval 22083 . . . . . 6  |-  X  =  ran  ( +v `  U )
5 nvmtri.3 . . . . . . 7  |-  M  =  ( -v `  U
)
61, 5vsfval 22114 . . . . . 6  |-  M  =  (  /g  `  ( +v `  U ) )
74, 6grponpncan 21843 . . . . 5  |-  ( ( ( +v `  U
)  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A M B ) ( +v `  U ) ( B M C ) )  =  ( A M C ) )
82, 7sylan 458 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A M B ) ( +v `  U ) ( B M C ) )  =  ( A M C ) )
98eqcomd 2441 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A M C )  =  ( ( A M B ) ( +v `  U ) ( B M C ) ) )
109fveq2d 5732 . 2  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( N `  ( A M C ) )  =  ( N `  ( ( A M B ) ( +v `  U
) ( B M C ) ) ) )
11 simpl 444 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  U  e.  NrmCVec )
123, 5nvmcl 22128 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  e.  X )
13123adant3r3 1164 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A M B )  e.  X
)
143, 5nvmcl 22128 . . . 4  |-  ( ( U  e.  NrmCVec  /\  B  e.  X  /\  C  e.  X )  ->  ( B M C )  e.  X )
15143adant3r1 1162 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B M C )  e.  X
)
16 nvmtri.6 . . . 4  |-  N  =  ( normCV `  U )
173, 1, 16nvtri 22159 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A M B )  e.  X  /\  ( B M C )  e.  X )  ->  ( N `  ( ( A M B ) ( +v `  U ) ( B M C ) ) )  <_ 
( ( N `  ( A M B ) )  +  ( N `
 ( B M C ) ) ) )
1811, 13, 15, 17syl3anc 1184 . 2  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( N `  ( ( A M B ) ( +v
`  U ) ( B M C ) ) )  <_  (
( N `  ( A M B ) )  +  ( N `  ( B M C ) ) ) )
1910, 18eqbrtrd 4232 1  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( N `  ( A M C ) )  <_  (
( N `  ( A M B ) )  +  ( N `  ( B M C ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4212   ` cfv 5454  (class class class)co 6081    + caddc 8993    <_ cle 9121   GrpOpcgr 21774   NrmCVeccnv 22063   +vcpv 22064   BaseSetcba 22065   -vcnsb 22068   normCVcnmcv 22069
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-ltxr 9125  df-sub 9293  df-neg 9294  df-grpo 21779  df-gid 21780  df-ginv 21781  df-gdiv 21782  df-ablo 21870  df-vc 22025  df-nv 22071  df-va 22074  df-ba 22075  df-sm 22076  df-0v 22077  df-vs 22078  df-nmcv 22079
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