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Theorem nvge0 22116
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 22101 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  A )  e.  RR )
4 2re 10025 . . 3  |-  2  e.  RR
53, 4jctil 524 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
2  e.  RR  /\  ( N `  A )  e.  RR ) )
6 eqid 2404 . . . . . . . 8  |-  ( 0vec `  U )  =  (
0vec `  U )
76, 2nvz0 22110 . . . . . . 7  |-  ( U  e.  NrmCVec  ->  ( N `  ( 0vec `  U )
)  =  0 )
87adantr 452 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( 0vec `  U ) )  =  0 )
9 ax-1cn 9004 . . . . . . . . . . 11  |-  1  e.  CC
109negidi 9325 . . . . . . . . . 10  |-  ( 1  +  -u 1 )  =  0
1110oveq1i 6050 . . . . . . . . 9  |-  ( ( 1  +  -u 1
) ( .s OLD `  U ) A )  =  ( 0 ( .s OLD `  U
) A )
12 eqid 2404 . . . . . . . . . 10  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
131, 12, 6nv0 22071 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
0 ( .s OLD `  U ) A )  =  ( 0vec `  U
) )
1411, 13syl5req 2449 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( 0vec `  U )  =  ( ( 1  + 
-u 1 ) ( .s OLD `  U
) A ) )
15 neg1cn 10023 . . . . . . . . 9  |-  -u 1  e.  CC
16 eqid 2404 . . . . . . . . . . 11  |-  ( +v
`  U )  =  ( +v `  U
)
171, 16, 12nvdir 22065 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  (
1  e.  CC  /\  -u 1  e.  CC  /\  A  e.  X )
)  ->  ( (
1  +  -u 1
) ( .s OLD `  U ) A )  =  ( ( 1 ( .s OLD `  U
) A ) ( +v `  U ) ( -u 1 ( .s OLD `  U
) A ) ) )
189, 17mp3anr1 1276 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  ( -u 1  e.  CC  /\  A  e.  X )
)  ->  ( (
1  +  -u 1
) ( .s OLD `  U ) A )  =  ( ( 1 ( .s OLD `  U
) A ) ( +v `  U ) ( -u 1 ( .s OLD `  U
) A ) ) )
1915, 18mpanr1 665 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( 1  +  -u
1 ) ( .s
OLD `  U ) A )  =  ( ( 1 ( .s
OLD `  U ) A ) ( +v
`  U ) (
-u 1 ( .s
OLD `  U ) A ) ) )
201, 12nvsid 22061 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
1 ( .s OLD `  U ) A )  =  A )
2120oveq1d 6055 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( 1 ( .s
OLD `  U ) A ) ( +v
`  U ) (
-u 1 ( .s
OLD `  U ) A ) )  =  ( A ( +v
`  U ) (
-u 1 ( .s
OLD `  U ) A ) ) )
2214, 19, 213eqtrd 2440 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( 0vec `  U )  =  ( A ( +v
`  U ) (
-u 1 ( .s
OLD `  U ) A ) ) )
2322fveq2d 5691 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( 0vec `  U ) )  =  ( N `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) A ) ) ) )
248, 23eqtr3d 2438 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  0  =  ( N `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) A ) ) ) )
251, 12nvscl 22060 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  A  e.  X )  ->  ( -u 1 ( .s OLD `  U ) A )  e.  X )
2615, 25mp3an2 1267 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( -u 1 ( .s OLD `  U ) A )  e.  X )
271, 16, 2nvtri 22112 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  ( -u 1 ( .s OLD `  U ) A )  e.  X )  -> 
( N `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) A ) ) )  <_  (
( N `  A
)  +  ( N `
 ( -u 1
( .s OLD `  U
) A ) ) ) )
2826, 27mpd3an3 1280 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( A
( +v `  U
) ( -u 1
( .s OLD `  U
) A ) ) )  <_  ( ( N `  A )  +  ( N `  ( -u 1 ( .s
OLD `  U ) A ) ) ) )
2924, 28eqbrtrd 4192 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  0  <_  ( ( N `  A )  +  ( N `  ( -u
1 ( .s OLD `  U ) A ) ) ) )
301, 12, 2nvm1 22106 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( -u 1
( .s OLD `  U
) A ) )  =  ( N `  A ) )
3130oveq2d 6056 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( N `  A
)  +  ( N `
 ( -u 1
( .s OLD `  U
) A ) ) )  =  ( ( N `  A )  +  ( N `  A ) ) )
323recnd 9070 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  A )  e.  CC )
33322timesd 10166 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
2  x.  ( N `
 A ) )  =  ( ( N `
 A )  +  ( N `  A
) ) )
3431, 33eqtr4d 2439 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( N `  A
)  +  ( N `
 ( -u 1
( .s OLD `  U
) A ) ) )  =  ( 2  x.  ( N `  A ) ) )
3529, 34breqtrd 4196 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  0  <_  ( 2  x.  ( N `  A )
) )
36 2pos 10038 . . 3  |-  0  <  2
3735, 36jctil 524 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
0  <  2  /\  0  <_  ( 2  x.  ( N `  A
) ) ) )
38 prodge0 9813 . 2  |-  ( ( ( 2  e.  RR  /\  ( N `  A
)  e.  RR )  /\  ( 0  <  2  /\  0  <_ 
( 2  x.  ( N `  A )
) ) )  -> 
0  <_  ( N `  A ) )
395, 37, 38syl2anc 643 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  0  <_  ( N `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    < clt 9076    <_ cle 9077   -ucneg 9248   2c2 10005   NrmCVeccnv 22016   +vcpv 22017   BaseSetcba 22018   .s
OLDcns 22019   0veccn0v 22020   normCVcnmcv 22022
This theorem is referenced by:  nvgt0  22117  smcnlem  22146  ipnm  22163  nmooge0  22221  nmoub3i  22227  siilem1  22305  siii  22307  ubthlem3  22327  minvecolem1  22329  minvecolem5  22336  minvecolem6  22337  htthlem  22373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-grpo 21732  df-gid 21733  df-ginv 21734  df-ablo 21823  df-vc 21978  df-nv 22024  df-va 22027  df-ba 22028  df-sm 22029  df-0v 22030  df-nmcv 22032
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