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Theorem nvge0 23997
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 23982 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  A )  e.  RR )
4 2re 10387 . . 3  |-  2  e.  RR
53, 4jctil 534 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
2  e.  RR  /\  ( N `  A )  e.  RR ) )
6 eqid 2441 . . . . . . . 8  |-  ( 0vec `  U )  =  (
0vec `  U )
76, 2nvz0 23991 . . . . . . 7  |-  ( U  e.  NrmCVec  ->  ( N `  ( 0vec `  U )
)  =  0 )
87adantr 462 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( 0vec `  U ) )  =  0 )
9 1pneg1e0 10426 . . . . . . . . . 10  |-  ( 1  +  -u 1 )  =  0
109oveq1i 6100 . . . . . . . . 9  |-  ( ( 1  +  -u 1
) ( .sOLD `  U ) A )  =  ( 0 ( .sOLD `  U
) A )
11 eqid 2441 . . . . . . . . . 10  |-  ( .sOLD `  U )  =  ( .sOLD `  U )
121, 11, 6nv0 23952 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
0 ( .sOLD `  U ) A )  =  ( 0vec `  U
) )
1310, 12syl5req 2486 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( 0vec `  U )  =  ( ( 1  + 
-u 1 ) ( .sOLD `  U
) A ) )
14 neg1cn 10421 . . . . . . . . 9  |-  -u 1  e.  CC
15 ax-1cn 9336 . . . . . . . . . 10  |-  1  e.  CC
16 eqid 2441 . . . . . . . . . . 11  |-  ( +v
`  U )  =  ( +v `  U
)
171, 16, 11nvdir 23946 . . . . . . . . . 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 1306 . . . . . . . . 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 678 . . . . . . . 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 23942 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
1 ( .sOLD `  U ) A )  =  A )
2120oveq1d 6105 . . . . . . . 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 2477 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( 0vec `  U )  =  ( A ( +v
`  U ) (
-u 1 ( .sOLD `  U ) A ) ) )
2322fveq2d 5692 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( 0vec `  U ) )  =  ( N `  ( A ( +v `  U ) ( -u
1 ( .sOLD `  U ) A ) ) ) )
248, 23eqtr3d 2475 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  0  =  ( N `  ( A ( +v `  U ) ( -u
1 ( .sOLD `  U ) A ) ) ) )
251, 11nvscl 23941 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  A  e.  X )  ->  ( -u 1 ( .sOLD `  U ) A )  e.  X )
2614, 25mp3an2 1297 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( -u 1 ( .sOLD `  U ) A )  e.  X )
271, 16, 2nvtri 23993 . . . . . 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 1310 . . . . 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 4309 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  0  <_  ( ( N `  A )  +  ( N `  ( -u
1 ( .sOLD `  U ) A ) ) ) )
301, 11, 2nvm1 23987 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( -u 1
( .sOLD `  U ) A ) )  =  ( N `
 A ) )
3130oveq2d 6106 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( N `  A
)  +  ( N `
 ( -u 1
( .sOLD `  U ) A ) ) )  =  ( ( N `  A
)  +  ( N `
 A ) ) )
323recnd 9408 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  A )  e.  CC )
33322timesd 10563 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
2  x.  ( N `
 A ) )  =  ( ( N `
 A )  +  ( N `  A
) ) )
3431, 33eqtr4d 2476 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( N `  A
)  +  ( N `
 ( -u 1
( .sOLD `  U ) A ) ) )  =  ( 2  x.  ( N `
 A ) ) )
3529, 34breqtrd 4313 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  0  <_  ( 2  x.  ( N `  A )
) )
36 2pos 10409 . . 3  |-  0  <  2
3735, 36jctil 534 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
0  <  2  /\  0  <_  ( 2  x.  ( N `  A
) ) ) )
38 prodge0 10172 . 2  |-  ( ( ( 2  e.  RR  /\  ( N `  A
)  e.  RR )  /\  ( 0  <  2  /\  0  <_ 
( 2  x.  ( N `  A )
) ) )  -> 
0  <_  ( N `  A ) )
395, 37, 38syl2anc 656 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  0  <_  ( N `  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279    + caddc 9281    x. cmul 9283    < clt 9414    <_ cle 9415   -ucneg 9592   2c2 10367   NrmCVeccnv 23897   +vcpv 23898   BaseSetcba 23899   .sOLDcns 23900   0veccn0v 23901   normCVcnmcv 23903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-seq 11803  df-exp 11862  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-grpo 23613  df-gid 23614  df-ginv 23615  df-ablo 23704  df-vc 23859  df-nv 23905  df-va 23908  df-ba 23909  df-sm 23910  df-0v 23911  df-nmcv 23913
This theorem is referenced by:  nvgt0  23998  smcnlem  24027  ipnm  24044  nmooge0  24102  nmoub3i  24108  siilem1  24186  siii  24188  ubthlem3  24208  minvecolem1  24210  minvecolem5  24217  minvecolem6  24218  htthlem  24254
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