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Theorem nvge0 21232
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 21217 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  A )  e.  RR )
4 2re 9810 . . 3  |-  2  e.  RR
53, 4jctil 525 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
2  e.  RR  /\  ( N `  A )  e.  RR ) )
6 eqid 2284 . . . . . . . 8  |-  ( 0vec `  U )  =  (
0vec `  U )
76, 2nvz0 21226 . . . . . . 7  |-  ( U  e.  NrmCVec  ->  ( N `  ( 0vec `  U )
)  =  0 )
87adantr 453 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( 0vec `  U ) )  =  0 )
9 ax-1cn 8790 . . . . . . . . . . 11  |-  1  e.  CC
109negidi 9110 . . . . . . . . . 10  |-  ( 1  +  -u 1 )  =  0
1110oveq1i 5829 . . . . . . . . 9  |-  ( ( 1  +  -u 1
) ( .s OLD `  U ) A )  =  ( 0 ( .s OLD `  U
) A )
12 eqid 2284 . . . . . . . . . 10  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
131, 12, 6nv0 21187 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
0 ( .s OLD `  U ) A )  =  ( 0vec `  U
) )
1411, 13syl5req 2329 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( 0vec `  U )  =  ( ( 1  + 
-u 1 ) ( .s OLD `  U
) A ) )
15 neg1cn 9808 . . . . . . . . 9  |-  -u 1  e.  CC
16 eqid 2284 . . . . . . . . . . 11  |-  ( +v
`  U )  =  ( +v `  U
)
171, 16, 12nvdir 21181 . . . . . . . . . 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 666 . . . . . . . 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 21177 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
1 ( .s OLD `  U ) A )  =  A )
2120oveq1d 5834 . . . . . . . 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 2320 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( 0vec `  U )  =  ( A ( +v
`  U ) (
-u 1 ( .s
OLD `  U ) A ) ) )
2322fveq2d 5489 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( 0vec `  U ) )  =  ( N `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) A ) ) ) )
248, 23eqtr3d 2318 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  0  =  ( N `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) A ) ) ) )
251, 12nvscl 21176 . . . . . . 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 21228 . . . . . 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 4044 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  0  <_  ( ( N `  A )  +  ( N `  ( -u
1 ( .s OLD `  U ) A ) ) ) )
301, 12, 2nvm1 21222 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( -u 1
( .s OLD `  U
) A ) )  =  ( N `  A ) )
3130oveq2d 5835 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( N `  A
)  +  ( N `
 ( -u 1
( .s OLD `  U
) A ) ) )  =  ( ( N `  A )  +  ( N `  A ) ) )
323recnd 8856 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  A )  e.  CC )
33322timesd 9949 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
2  x.  ( N `
 A ) )  =  ( ( N `
 A )  +  ( N `  A
) ) )
3431, 33eqtr4d 2319 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( N `  A
)  +  ( N `
 ( -u 1
( .s OLD `  U
) A ) ) )  =  ( 2  x.  ( N `  A ) ) )
3529, 34breqtrd 4048 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  0  <_  ( 2  x.  ( N `  A )
) )
36 2pos 9823 . . 3  |-  0  <  2
3735, 36jctil 525 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
0  <  2  /\  0  <_  ( 2  x.  ( N `  A
) ) ) )
38 prodge0 9598 . 2  |-  ( ( ( 2  e.  RR  /\  ( N `  A
)  e.  RR )  /\  ( 0  <  2  /\  0  <_ 
( 2  x.  ( N `  A )
) ) )  -> 
0  <_  ( N `  A ) )
395, 37, 38syl2anc 644 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  0  <_  ( N `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1624    e. wcel 1685   class class class wbr 4024   ` cfv 5221  (class class class)co 5819   CCcc 8730   RRcr 8731   0cc0 8732   1c1 8733    + caddc 8735    x. cmul 8737    < clt 8862    <_ cle 8863   -ucneg 9033   2c2 9790   NrmCVeccnv 21132   +vcpv 21133   BaseSetcba 21134   .s
OLDcns 21135   0veccn0v 21136   normCVcnmcv 21138
This theorem is referenced by:  nvgt0  21233  smcnlem  21262  ipnm  21279  nmooge0  21337  nmoub3i  21343  siilem1  21421  siii  21423  ubthlem3  21443  minvecolem1  21445  minvecolem5  21452  minvecolem6  21453  htthlem  21489
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-sup 7189  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-2 9799  df-3 9800  df-n0 9961  df-z 10020  df-uz 10226  df-rp 10350  df-seq 11041  df-exp 11099  df-cj 11578  df-re 11579  df-im 11580  df-sqr 11714  df-abs 11715  df-grpo 20850  df-gid 20851  df-ginv 20852  df-ablo 20941  df-vc 21094  df-nv 21140  df-va 21143  df-ba 21144  df-sm 21145  df-0v 21146  df-nmcv 21148
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