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Theorem blometi 24340
Description: Upper bound for the distance between the values of a bounded linear operator. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
blometi.1  |-  X  =  ( BaseSet `  U )
blometi.2  |-  Y  =  ( BaseSet `  W )
blometi.8  |-  C  =  ( IndMet `  U )
blometi.d  |-  D  =  ( IndMet `  W )
blometi.6  |-  N  =  ( U normOpOLD W
)
blometi.7  |-  B  =  ( U  BLnOp  W )
blometi.u  |-  U  e.  NrmCVec
blometi.w  |-  W  e.  NrmCVec
Assertion
Ref Expression
blometi  |-  ( ( T  e.  B  /\  P  e.  X  /\  Q  e.  X )  ->  ( ( T `  P ) D ( T `  Q ) )  <_  ( ( N `  T )  x.  ( P C Q ) ) )

Proof of Theorem blometi
StepHypRef Expression
1 blometi.u . . . . 5  |-  U  e.  NrmCVec
2 blometi.1 . . . . . 6  |-  X  =  ( BaseSet `  U )
3 eqid 2451 . . . . . 6  |-  ( -v
`  U )  =  ( -v `  U
)
42, 3nvmcl 24164 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  P  e.  X  /\  Q  e.  X )  ->  ( P ( -v `  U ) Q )  e.  X )
51, 4mp3an1 1302 . . . 4  |-  ( ( P  e.  X  /\  Q  e.  X )  ->  ( P ( -v
`  U ) Q )  e.  X )
6 eqid 2451 . . . . 5  |-  ( normCV `  U )  =  (
normCV
`  U )
7 eqid 2451 . . . . 5  |-  ( normCV `  W )  =  (
normCV
`  W )
8 blometi.6 . . . . 5  |-  N  =  ( U normOpOLD W
)
9 blometi.7 . . . . 5  |-  B  =  ( U  BLnOp  W )
10 blometi.w . . . . 5  |-  W  e.  NrmCVec
112, 6, 7, 8, 9, 1, 10nmblolbi 24337 . . . 4  |-  ( ( T  e.  B  /\  ( P ( -v `  U ) Q )  e.  X )  -> 
( ( normCV `  W
) `  ( T `  ( P ( -v
`  U ) Q ) ) )  <_ 
( ( N `  T )  x.  (
( normCV `  U ) `  ( P ( -v `  U ) Q ) ) ) )
125, 11sylan2 474 . . 3  |-  ( ( T  e.  B  /\  ( P  e.  X  /\  Q  e.  X
) )  ->  (
( normCV `  W ) `  ( T `  ( P ( -v `  U
) Q ) ) )  <_  ( ( N `  T )  x.  ( ( normCV `  U
) `  ( P
( -v `  U
) Q ) ) ) )
13123impb 1184 . 2  |-  ( ( T  e.  B  /\  P  e.  X  /\  Q  e.  X )  ->  ( ( normCV `  W
) `  ( T `  ( P ( -v
`  U ) Q ) ) )  <_ 
( ( N `  T )  x.  (
( normCV `  U ) `  ( P ( -v `  U ) Q ) ) ) )
14 blometi.2 . . . . . . . 8  |-  Y  =  ( BaseSet `  W )
152, 14, 9blof 24322 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  B )  ->  T : X --> Y )
161, 10, 15mp3an12 1305 . . . . . 6  |-  ( T  e.  B  ->  T : X --> Y )
1716ffvelrnda 5944 . . . . 5  |-  ( ( T  e.  B  /\  P  e.  X )  ->  ( T `  P
)  e.  Y )
18173adant3 1008 . . . 4  |-  ( ( T  e.  B  /\  P  e.  X  /\  Q  e.  X )  ->  ( T `  P
)  e.  Y )
1916ffvelrnda 5944 . . . . 5  |-  ( ( T  e.  B  /\  Q  e.  X )  ->  ( T `  Q
)  e.  Y )
20193adant2 1007 . . . 4  |-  ( ( T  e.  B  /\  P  e.  X  /\  Q  e.  X )  ->  ( T `  Q
)  e.  Y )
21 eqid 2451 . . . . . 6  |-  ( -v
`  W )  =  ( -v `  W
)
22 blometi.d . . . . . 6  |-  D  =  ( IndMet `  W )
2314, 21, 7, 22imsdval 24214 . . . . 5  |-  ( ( W  e.  NrmCVec  /\  ( T `  P )  e.  Y  /\  ( T `  Q )  e.  Y )  ->  (
( T `  P
) D ( T `
 Q ) )  =  ( ( normCV `  W ) `  (
( T `  P
) ( -v `  W ) ( T `
 Q ) ) ) )
2410, 23mp3an1 1302 . . . 4  |-  ( ( ( T `  P
)  e.  Y  /\  ( T `  Q )  e.  Y )  -> 
( ( T `  P ) D ( T `  Q ) )  =  ( (
normCV
`  W ) `  ( ( T `  P ) ( -v
`  W ) ( T `  Q ) ) ) )
2518, 20, 24syl2anc 661 . . 3  |-  ( ( T  e.  B  /\  P  e.  X  /\  Q  e.  X )  ->  ( ( T `  P ) D ( T `  Q ) )  =  ( (
normCV
`  W ) `  ( ( T `  P ) ( -v
`  W ) ( T `  Q ) ) ) )
26 eqid 2451 . . . . . . 7  |-  ( U 
LnOp  W )  =  ( U  LnOp  W )
2726, 9bloln 24321 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  B )  ->  T  e.  ( U  LnOp  W
) )
281, 10, 27mp3an12 1305 . . . . 5  |-  ( T  e.  B  ->  T  e.  ( U  LnOp  W
) )
292, 3, 21, 26lnosub 24296 . . . . . . . 8  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  ( U  LnOp  W
) )  /\  ( P  e.  X  /\  Q  e.  X )
)  ->  ( T `  ( P ( -v
`  U ) Q ) )  =  ( ( T `  P
) ( -v `  W ) ( T `
 Q ) ) )
301, 29mp3anl1 1309 . . . . . . 7  |-  ( ( ( W  e.  NrmCVec  /\  T  e.  ( U  LnOp  W ) )  /\  ( P  e.  X  /\  Q  e.  X
) )  ->  ( T `  ( P
( -v `  U
) Q ) )  =  ( ( T `
 P ) ( -v `  W ) ( T `  Q
) ) )
3110, 30mpanl1 680 . . . . . 6  |-  ( ( T  e.  ( U 
LnOp  W )  /\  ( P  e.  X  /\  Q  e.  X )
)  ->  ( T `  ( P ( -v
`  U ) Q ) )  =  ( ( T `  P
) ( -v `  W ) ( T `
 Q ) ) )
32313impb 1184 . . . . 5  |-  ( ( T  e.  ( U 
LnOp  W )  /\  P  e.  X  /\  Q  e.  X )  ->  ( T `  ( P
( -v `  U
) Q ) )  =  ( ( T `
 P ) ( -v `  W ) ( T `  Q
) ) )
3328, 32syl3an1 1252 . . . 4  |-  ( ( T  e.  B  /\  P  e.  X  /\  Q  e.  X )  ->  ( T `  ( P ( -v `  U ) Q ) )  =  ( ( T `  P ) ( -v `  W
) ( T `  Q ) ) )
3433fveq2d 5795 . . 3  |-  ( ( T  e.  B  /\  P  e.  X  /\  Q  e.  X )  ->  ( ( normCV `  W
) `  ( T `  ( P ( -v
`  U ) Q ) ) )  =  ( ( normCV `  W
) `  ( ( T `  P )
( -v `  W
) ( T `  Q ) ) ) )
3525, 34eqtr4d 2495 . 2  |-  ( ( T  e.  B  /\  P  e.  X  /\  Q  e.  X )  ->  ( ( T `  P ) D ( T `  Q ) )  =  ( (
normCV
`  W ) `  ( T `  ( P ( -v `  U
) Q ) ) ) )
36 blometi.8 . . . . . 6  |-  C  =  ( IndMet `  U )
372, 3, 6, 36imsdval 24214 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  P  e.  X  /\  Q  e.  X )  ->  ( P C Q )  =  ( ( normCV `  U
) `  ( P
( -v `  U
) Q ) ) )
381, 37mp3an1 1302 . . . 4  |-  ( ( P  e.  X  /\  Q  e.  X )  ->  ( P C Q )  =  ( (
normCV
`  U ) `  ( P ( -v `  U ) Q ) ) )
39383adant1 1006 . . 3  |-  ( ( T  e.  B  /\  P  e.  X  /\  Q  e.  X )  ->  ( P C Q )  =  ( (
normCV
`  U ) `  ( P ( -v `  U ) Q ) ) )
4039oveq2d 6208 . 2  |-  ( ( T  e.  B  /\  P  e.  X  /\  Q  e.  X )  ->  ( ( N `  T )  x.  ( P C Q ) )  =  ( ( N `
 T )  x.  ( ( normCV `  U
) `  ( P
( -v `  U
) Q ) ) ) )
4113, 35, 403brtr4d 4422 1  |-  ( ( T  e.  B  /\  P  e.  X  /\  Q  e.  X )  ->  ( ( T `  P ) D ( T `  Q ) )  <_  ( ( N `  T )  x.  ( P C Q ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   class class class wbr 4392   -->wf 5514   ` cfv 5518  (class class class)co 6192    x. cmul 9390    <_ cle 9522   NrmCVeccnv 24099   BaseSetcba 24101   -vcnsb 24104   normCVcnmcv 24105   IndMetcims 24106    LnOp clno 24277   normOpOLDcnmoo 24278    BLnOp cblo 24279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462  ax-pre-sup 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-er 7203  df-map 7318  df-en 7413  df-dom 7414  df-sdom 7415  df-sup 7794  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-div 10097  df-nn 10426  df-2 10483  df-3 10484  df-n0 10683  df-z 10750  df-uz 10965  df-rp 11095  df-seq 11910  df-exp 11969  df-cj 12692  df-re 12693  df-im 12694  df-sqr 12828  df-abs 12829  df-grpo 23815  df-gid 23816  df-ginv 23817  df-gdiv 23818  df-ablo 23906  df-vc 24061  df-nv 24107  df-va 24110  df-ba 24111  df-sm 24112  df-0v 24113  df-vs 24114  df-nmcv 24115  df-ims 24116  df-lno 24281  df-nmoo 24282  df-blo 24283  df-0o 24284
This theorem is referenced by:  blocni  24342
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