MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nvge0 Structured version   Unicode version

Theorem nvge0 24065
Description: The norm of a normed complex vector space is nonnegative. Second part of Problem 2 of [Kreyszig] p. 64. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvge0.1  |-  X  =  ( BaseSet `  U )
nvge0.6  |-  N  =  ( normCV `  U )
Assertion
Ref Expression
nvge0  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  0  <_  ( N `  A
) )

Proof of Theorem nvge0
StepHypRef Expression
1 nvge0.1 . . . 4  |-  X  =  ( BaseSet `  U )
2 nvge0.6 . . . 4  |-  N  =  ( normCV `  U )
31, 2nvcl 24050 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  A )  e.  RR )
4 2re 10394 . . 3  |-  2  e.  RR
53, 4jctil 537 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
2  e.  RR  /\  ( N `  A )  e.  RR ) )
6 eqid 2443 . . . . . . . 8  |-  ( 0vec `  U )  =  (
0vec `  U )
76, 2nvz0 24059 . . . . . . 7  |-  ( U  e.  NrmCVec  ->  ( N `  ( 0vec `  U )
)  =  0 )
87adantr 465 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( 0vec `  U ) )  =  0 )
9 1pneg1e0 10433 . . . . . . . . . 10  |-  ( 1  +  -u 1 )  =  0
109oveq1i 6104 . . . . . . . . 9  |-  ( ( 1  +  -u 1
) ( .sOLD `  U ) A )  =  ( 0 ( .sOLD `  U
) A )
11 eqid 2443 . . . . . . . . . 10  |-  ( .sOLD `  U )  =  ( .sOLD `  U )
121, 11, 6nv0 24020 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
0 ( .sOLD `  U ) A )  =  ( 0vec `  U
) )
1310, 12syl5req 2488 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( 0vec `  U )  =  ( ( 1  + 
-u 1 ) ( .sOLD `  U
) A ) )
14 neg1cn 10428 . . . . . . . . 9  |-  -u 1  e.  CC
15 ax-1cn 9343 . . . . . . . . . 10  |-  1  e.  CC
16 eqid 2443 . . . . . . . . . . 11  |-  ( +v
`  U )  =  ( +v `  U
)
171, 16, 11nvdir 24014 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  (
1  e.  CC  /\  -u 1  e.  CC  /\  A  e.  X )
)  ->  ( (
1  +  -u 1
) ( .sOLD `  U ) A )  =  ( ( 1 ( .sOLD `  U ) A ) ( +v `  U
) ( -u 1
( .sOLD `  U ) A ) ) )
1815, 17mp3anr1 1311 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  ( -u 1  e.  CC  /\  A  e.  X )
)  ->  ( (
1  +  -u 1
) ( .sOLD `  U ) A )  =  ( ( 1 ( .sOLD `  U ) A ) ( +v `  U
) ( -u 1
( .sOLD `  U ) A ) ) )
1914, 18mpanr1 683 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( 1  +  -u
1 ) ( .sOLD `  U ) A )  =  ( ( 1 ( .sOLD `  U ) A ) ( +v
`  U ) (
-u 1 ( .sOLD `  U ) A ) ) )
201, 11nvsid 24010 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
1 ( .sOLD `  U ) A )  =  A )
2120oveq1d 6109 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( 1 ( .sOLD `  U ) A ) ( +v
`  U ) (
-u 1 ( .sOLD `  U ) A ) )  =  ( A ( +v
`  U ) (
-u 1 ( .sOLD `  U ) A ) ) )
2213, 19, 213eqtrd 2479 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( 0vec `  U )  =  ( A ( +v
`  U ) (
-u 1 ( .sOLD `  U ) A ) ) )
2322fveq2d 5698 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( 0vec `  U ) )  =  ( N `  ( A ( +v `  U ) ( -u
1 ( .sOLD `  U ) A ) ) ) )
248, 23eqtr3d 2477 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  0  =  ( N `  ( A ( +v `  U ) ( -u
1 ( .sOLD `  U ) A ) ) ) )
251, 11nvscl 24009 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  A  e.  X )  ->  ( -u 1 ( .sOLD `  U ) A )  e.  X )
2614, 25mp3an2 1302 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( -u 1 ( .sOLD `  U ) A )  e.  X )
271, 16, 2nvtri 24061 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  ( -u 1 ( .sOLD `  U ) A )  e.  X )  -> 
( N `  ( A ( +v `  U ) ( -u
1 ( .sOLD `  U ) A ) ) )  <_  (
( N `  A
)  +  ( N `
 ( -u 1
( .sOLD `  U ) A ) ) ) )
2826, 27mpd3an3 1315 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( A
( +v `  U
) ( -u 1
( .sOLD `  U ) A ) ) )  <_  (
( N `  A
)  +  ( N `
 ( -u 1
( .sOLD `  U ) A ) ) ) )
2924, 28eqbrtrd 4315 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  0  <_  ( ( N `  A )  +  ( N `  ( -u
1 ( .sOLD `  U ) A ) ) ) )
301, 11, 2nvm1 24055 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( -u 1
( .sOLD `  U ) A ) )  =  ( N `
 A ) )
3130oveq2d 6110 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( N `  A
)  +  ( N `
 ( -u 1
( .sOLD `  U ) A ) ) )  =  ( ( N `  A
)  +  ( N `
 A ) ) )
323recnd 9415 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  A )  e.  CC )
33322timesd 10570 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
2  x.  ( N `
 A ) )  =  ( ( N `
 A )  +  ( N `  A
) ) )
3431, 33eqtr4d 2478 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( N `  A
)  +  ( N `
 ( -u 1
( .sOLD `  U ) A ) ) )  =  ( 2  x.  ( N `
 A ) ) )
3529, 34breqtrd 4319 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  0  <_  ( 2  x.  ( N `  A )
) )
36 2pos 10416 . . 3  |-  0  <  2
3735, 36jctil 537 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
0  <  2  /\  0  <_  ( 2  x.  ( N `  A
) ) ) )
38 prodge0 10179 . 2  |-  ( ( ( 2  e.  RR  /\  ( N `  A
)  e.  RR )  /\  ( 0  <  2  /\  0  <_ 
( 2  x.  ( N `  A )
) ) )  -> 
0  <_  ( N `  A ) )
395, 37, 38syl2anc 661 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  0  <_  ( N `  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   class class class wbr 4295   ` cfv 5421  (class class class)co 6094   CCcc 9283   RRcr 9284   0cc0 9285   1c1 9286    + caddc 9288    x. cmul 9290    < clt 9421    <_ cle 9422   -ucneg 9599   2c2 10374   NrmCVeccnv 23965   +vcpv 23966   BaseSetcba 23967   .sOLDcns 23968   0veccn0v 23969   normCVcnmcv 23971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362  ax-pre-sup 9363
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-om 6480  df-1st 6580  df-2nd 6581  df-recs 6835  df-rdg 6869  df-er 7104  df-en 7314  df-dom 7315  df-sdom 7316  df-sup 7694  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-div 9997  df-nn 10326  df-2 10383  df-3 10384  df-n0 10583  df-z 10650  df-uz 10865  df-rp 10995  df-seq 11810  df-exp 11869  df-cj 12591  df-re 12592  df-im 12593  df-sqr 12727  df-abs 12728  df-grpo 23681  df-gid 23682  df-ginv 23683  df-ablo 23772  df-vc 23927  df-nv 23973  df-va 23976  df-ba 23977  df-sm 23978  df-0v 23979  df-nmcv 23981
This theorem is referenced by:  nvgt0  24066  smcnlem  24095  ipnm  24112  nmooge0  24170  nmoub3i  24176  siilem1  24254  siii  24256  ubthlem3  24276  minvecolem1  24278  minvecolem5  24285  minvecolem6  24286  htthlem  24322
  Copyright terms: Public domain W3C validator