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Theorem nvge0 26303
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 26288 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  A )  e.  RR )
4 2re 10679 . . 3  |-  2  e.  RR
53, 4jctil 540 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
2  e.  RR  /\  ( N `  A )  e.  RR ) )
6 eqid 2451 . . . . . . . 8  |-  ( 0vec `  U )  =  (
0vec `  U )
76, 2nvz0 26297 . . . . . . 7  |-  ( U  e.  NrmCVec  ->  ( N `  ( 0vec `  U )
)  =  0 )
87adantr 467 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( 0vec `  U ) )  =  0 )
9 1pneg1e0 10718 . . . . . . . . . 10  |-  ( 1  +  -u 1 )  =  0
109oveq1i 6300 . . . . . . . . 9  |-  ( ( 1  +  -u 1
) ( .sOLD `  U ) A )  =  ( 0 ( .sOLD `  U
) A )
11 eqid 2451 . . . . . . . . . 10  |-  ( .sOLD `  U )  =  ( .sOLD `  U )
121, 11, 6nv0 26258 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
0 ( .sOLD `  U ) A )  =  ( 0vec `  U
) )
1310, 12syl5req 2498 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( 0vec `  U )  =  ( ( 1  + 
-u 1 ) ( .sOLD `  U
) A ) )
14 neg1cn 10713 . . . . . . . . 9  |-  -u 1  e.  CC
15 ax-1cn 9597 . . . . . . . . . 10  |-  1  e.  CC
16 eqid 2451 . . . . . . . . . . 11  |-  ( +v
`  U )  =  ( +v `  U
)
171, 16, 11nvdir 26252 . . . . . . . . . 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 1361 . . . . . . . . 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 689 . . . . . . . 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 26248 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
1 ( .sOLD `  U ) A )  =  A )
2120oveq1d 6305 . . . . . . . 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 2489 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( 0vec `  U )  =  ( A ( +v
`  U ) (
-u 1 ( .sOLD `  U ) A ) ) )
2322fveq2d 5869 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( 0vec `  U ) )  =  ( N `  ( A ( +v `  U ) ( -u
1 ( .sOLD `  U ) A ) ) ) )
248, 23eqtr3d 2487 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  0  =  ( N `  ( A ( +v `  U ) ( -u
1 ( .sOLD `  U ) A ) ) ) )
251, 11nvscl 26247 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  A  e.  X )  ->  ( -u 1 ( .sOLD `  U ) A )  e.  X )
2614, 25mp3an2 1352 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( -u 1 ( .sOLD `  U ) A )  e.  X )
271, 16, 2nvtri 26299 . . . . . 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 1365 . . . . 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 4423 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  0  <_  ( ( N `  A )  +  ( N `  ( -u
1 ( .sOLD `  U ) A ) ) ) )
301, 11, 2nvm1 26293 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  ( -u 1
( .sOLD `  U ) A ) )  =  ( N `
 A ) )
3130oveq2d 6306 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( N `  A
)  +  ( N `
 ( -u 1
( .sOLD `  U ) A ) ) )  =  ( ( N `  A
)  +  ( N `
 A ) ) )
323recnd 9669 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  A )  e.  CC )
33322timesd 10855 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
2  x.  ( N `
 A ) )  =  ( ( N `
 A )  +  ( N `  A
) ) )
3431, 33eqtr4d 2488 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( N `  A
)  +  ( N `
 ( -u 1
( .sOLD `  U ) A ) ) )  =  ( 2  x.  ( N `
 A ) ) )
3529, 34breqtrd 4427 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  0  <_  ( 2  x.  ( N `  A )
) )
36 2pos 10701 . . 3  |-  0  <  2
3735, 36jctil 540 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
0  <  2  /\  0  <_  ( 2  x.  ( N `  A
) ) ) )
38 prodge0 10452 . 2  |-  ( ( ( 2  e.  RR  /\  ( N `  A
)  e.  RR )  /\  ( 0  <  2  /\  0  <_ 
( 2  x.  ( N `  A )
) ) )  -> 
0  <_  ( N `  A ) )
395, 37, 38syl2anc 667 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  0  <_  ( N `  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887   class class class wbr 4402   ` cfv 5582  (class class class)co 6290   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542    x. cmul 9544    < clt 9675    <_ cle 9676   -ucneg 9861   2c2 10659   NrmCVeccnv 26203   +vcpv 26204   BaseSetcba 26205   .sOLDcns 26206   0veccn0v 26207   normCVcnmcv 26209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-sup 7956  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-seq 12214  df-exp 12273  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-grpo 25919  df-gid 25920  df-ginv 25921  df-ablo 26010  df-vc 26165  df-nv 26211  df-va 26214  df-ba 26215  df-sm 26216  df-0v 26217  df-nmcv 26219
This theorem is referenced by:  nvgt0  26304  smcnlem  26333  ipnm  26350  nmooge0  26408  nmoub3i  26414  siilem1  26492  siii  26494  ubthlem3  26514  minvecolem1  26516  minvecolem5  26523  minvecolem6  26524  minvecolem5OLD  26533  minvecolem6OLD  26534  htthlem  26570
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