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Theorem nvge0 25977
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 25962 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  A )  e.  RR )
4 2re 10645 . . 3  |-  2  e.  RR
53, 4jctil 535 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
2  e.  RR  /\  ( N `  A )  e.  RR ) )
6 eqid 2402 . . . . . . . 8  |-  ( 0vec `  U )  =  (
0vec `  U )
76, 2nvz0 25971 . . . . . . 7  |-  ( U  e.  NrmCVec  ->  ( N `  ( 0vec `  U )
)  =  0 )
87adantr 463 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( 0vec `  U ) )  =  0 )
9 1pneg1e0 10684 . . . . . . . . . 10  |-  ( 1  +  -u 1 )  =  0
109oveq1i 6287 . . . . . . . . 9  |-  ( ( 1  +  -u 1
) ( .sOLD `  U ) A )  =  ( 0 ( .sOLD `  U
) A )
11 eqid 2402 . . . . . . . . . 10  |-  ( .sOLD `  U )  =  ( .sOLD `  U )
121, 11, 6nv0 25932 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
0 ( .sOLD `  U ) A )  =  ( 0vec `  U
) )
1310, 12syl5req 2456 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( 0vec `  U )  =  ( ( 1  + 
-u 1 ) ( .sOLD `  U
) A ) )
14 neg1cn 10679 . . . . . . . . 9  |-  -u 1  e.  CC
15 ax-1cn 9579 . . . . . . . . . 10  |-  1  e.  CC
16 eqid 2402 . . . . . . . . . . 11  |-  ( +v
`  U )  =  ( +v `  U
)
171, 16, 11nvdir 25926 . . . . . . . . . 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 1323 . . . . . . . . 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 681 . . . . . . . 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 25922 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
1 ( .sOLD `  U ) A )  =  A )
2120oveq1d 6292 . . . . . . . 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 2447 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( 0vec `  U )  =  ( A ( +v
`  U ) (
-u 1 ( .sOLD `  U ) A ) ) )
2322fveq2d 5852 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( 0vec `  U ) )  =  ( N `  ( A ( +v `  U ) ( -u
1 ( .sOLD `  U ) A ) ) ) )
248, 23eqtr3d 2445 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  0  =  ( N `  ( A ( +v `  U ) ( -u
1 ( .sOLD `  U ) A ) ) ) )
251, 11nvscl 25921 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  A  e.  X )  ->  ( -u 1 ( .sOLD `  U ) A )  e.  X )
2614, 25mp3an2 1314 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( -u 1 ( .sOLD `  U ) A )  e.  X )
271, 16, 2nvtri 25973 . . . . . 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 1327 . . . . 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 4414 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  0  <_  ( ( N `  A )  +  ( N `  ( -u
1 ( .sOLD `  U ) A ) ) ) )
301, 11, 2nvm1 25967 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( -u 1
( .sOLD `  U ) A ) )  =  ( N `
 A ) )
3130oveq2d 6293 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( N `  A
)  +  ( N `
 ( -u 1
( .sOLD `  U ) A ) ) )  =  ( ( N `  A
)  +  ( N `
 A ) ) )
323recnd 9651 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  A )  e.  CC )
33322timesd 10821 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
2  x.  ( N `
 A ) )  =  ( ( N `
 A )  +  ( N `  A
) ) )
3431, 33eqtr4d 2446 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( N `  A
)  +  ( N `
 ( -u 1
( .sOLD `  U ) A ) ) )  =  ( 2  x.  ( N `
 A ) ) )
3529, 34breqtrd 4418 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  0  <_  ( 2  x.  ( N `  A )
) )
36 2pos 10667 . . 3  |-  0  <  2
3735, 36jctil 535 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
0  <  2  /\  0  <_  ( 2  x.  ( N `  A
) ) ) )
38 prodge0 10429 . 2  |-  ( ( ( 2  e.  RR  /\  ( N `  A
)  e.  RR )  /\  ( 0  <  2  /\  0  <_ 
( 2  x.  ( N `  A )
) ) )  -> 
0  <_  ( N `  A ) )
395, 37, 38syl2anc 659 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  0  <_  ( N `  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   class class class wbr 4394   ` cfv 5568  (class class class)co 6277   CCcc 9519   RRcr 9520   0cc0 9521   1c1 9522    + caddc 9524    x. cmul 9526    < clt 9657    <_ cle 9658   -ucneg 9841   2c2 10625   NrmCVeccnv 25877   +vcpv 25878   BaseSetcba 25879   .sOLDcns 25880   0veccn0v 25881   normCVcnmcv 25883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-sup 7934  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-n0 10836  df-z 10905  df-uz 11127  df-rp 11265  df-seq 12150  df-exp 12209  df-cj 13079  df-re 13080  df-im 13081  df-sqrt 13215  df-abs 13216  df-grpo 25593  df-gid 25594  df-ginv 25595  df-ablo 25684  df-vc 25839  df-nv 25885  df-va 25888  df-ba 25889  df-sm 25890  df-0v 25891  df-nmcv 25893
This theorem is referenced by:  nvgt0  25978  smcnlem  26007  ipnm  26024  nmooge0  26082  nmoub3i  26088  siilem1  26166  siii  26168  ubthlem3  26188  minvecolem1  26190  minvecolem5  26197  minvecolem6  26198  htthlem  26234
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