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Theorem nvmtri2 25692
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 2382 . . . . . 6  |-  ( +v
`  U )  =  ( +v `  U
)
21nvgrp 25627 . . . . 5  |-  ( U  e.  NrmCVec  ->  ( +v `  U )  e.  GrpOp )
3 nvmtri.1 . . . . . . 7  |-  X  =  ( BaseSet `  U )
43, 1bafval 25614 . . . . . 6  |-  X  =  ran  ( +v `  U )
5 nvmtri.3 . . . . . . 7  |-  M  =  ( -v `  U
)
61, 5vsfval 25645 . . . . . 6  |-  M  =  (  /g  `  ( +v `  U ) )
74, 6grponpncan 25374 . . . . 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 469 . . . 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 2390 . . 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 5778 . 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 455 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  U  e.  NrmCVec )
123, 5nvmcl 25659 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  e.  X )
13123adant3r3 1205 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A M B )  e.  X
)
143, 5nvmcl 25659 . . . 4  |-  ( ( U  e.  NrmCVec  /\  B  e.  X  /\  C  e.  X )  ->  ( B M C )  e.  X )
15143adant3r1 1203 . . 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 25690 . . 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 1226 . 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 4387 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 setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826   class class class wbr 4367   ` cfv 5496  (class class class)co 6196    + caddc 9406    <_ cle 9540   GrpOpcgr 25305   NrmCVeccnv 25594   +vcpv 25595   BaseSetcba 25596   -vcnsb 25599   normCVcnmcv 25600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-po 4714  df-so 4715  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-ltxr 9544  df-sub 9720  df-neg 9721  df-grpo 25310  df-gid 25311  df-ginv 25312  df-gdiv 25313  df-ablo 25401  df-vc 25556  df-nv 25602  df-va 25605  df-ba 25606  df-sm 25607  df-0v 25608  df-vs 25609  df-nmcv 25610
This theorem is referenced by: (None)
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