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Theorem blometi 22304
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 normOp OLD 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 2436 . . . . . 6  |-  ( -v
`  U )  =  ( -v `  U
)
42, 3nvmcl 22128 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  P  e.  X  /\  Q  e.  X )  ->  ( P ( -v `  U ) Q )  e.  X )
51, 4mp3an1 1266 . . . 4  |-  ( ( P  e.  X  /\  Q  e.  X )  ->  ( P ( -v
`  U ) Q )  e.  X )
6 eqid 2436 . . . . 5  |-  ( normCV `  U )  =  (
normCV
`  U )
7 eqid 2436 . . . . 5  |-  ( normCV `  W )  =  (
normCV
`  W )
8 blometi.6 . . . . 5  |-  N  =  ( U normOp OLD W
)
9 blometi.7 . . . . 5  |-  B  =  ( U  BLnOp  W )
10 blometi.w . . . . 5  |-  W  e.  NrmCVec
112, 6, 7, 8, 9, 1, 10nmblolbi 22301 . . . 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 461 . . 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 1149 . 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 22286 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  B )  ->  T : X --> Y )
161, 10, 15mp3an12 1269 . . . . . 6  |-  ( T  e.  B  ->  T : X --> Y )
1716ffvelrnda 5870 . . . . 5  |-  ( ( T  e.  B  /\  P  e.  X )  ->  ( T `  P
)  e.  Y )
18173adant3 977 . . . 4  |-  ( ( T  e.  B  /\  P  e.  X  /\  Q  e.  X )  ->  ( T `  P
)  e.  Y )
1916ffvelrnda 5870 . . . . 5  |-  ( ( T  e.  B  /\  Q  e.  X )  ->  ( T `  Q
)  e.  Y )
20193adant2 976 . . . 4  |-  ( ( T  e.  B  /\  P  e.  X  /\  Q  e.  X )  ->  ( T `  Q
)  e.  Y )
21 eqid 2436 . . . . . 6  |-  ( -v
`  W )  =  ( -v `  W
)
22 blometi.d . . . . . 6  |-  D  =  ( IndMet `  W )
2314, 21, 7, 22imsdval 22178 . . . . 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 1266 . . . 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 643 . . 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 2436 . . . . . . 7  |-  ( U 
LnOp  W )  =  ( U  LnOp  W )
2726, 9bloln 22285 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  B )  ->  T  e.  ( U  LnOp  W
) )
281, 10, 27mp3an12 1269 . . . . 5  |-  ( T  e.  B  ->  T  e.  ( U  LnOp  W
) )
292, 3, 21, 26lnosub 22260 . . . . . . . 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 1273 . . . . . . 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 662 . . . . . 6  |-  ( ( T  e.  ( U 
LnOp  W )  /\  ( P  e.  X  /\  Q  e.  X )
)  ->  ( T `  ( P ( -v
`  U ) Q ) )  =  ( ( T `  P
) ( -v `  W ) ( T `
 Q ) ) )
32313impb 1149 . . . . 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 1217 . . . 4  |-  ( ( T  e.  B  /\  P  e.  X  /\  Q  e.  X )  ->  ( T `  ( P ( -v `  U ) Q ) )  =  ( ( T `  P ) ( -v `  W
) ( T `  Q ) ) )
3433fveq2d 5732 . . 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 2471 . 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 22178 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  P  e.  X  /\  Q  e.  X )  ->  ( P C Q )  =  ( ( normCV `  U
) `  ( P
( -v `  U
) Q ) ) )
381, 37mp3an1 1266 . . . 4  |-  ( ( P  e.  X  /\  Q  e.  X )  ->  ( P C Q )  =  ( (
normCV
`  U ) `  ( P ( -v `  U ) Q ) ) )
39383adant1 975 . . 3  |-  ( ( T  e.  B  /\  P  e.  X  /\  Q  e.  X )  ->  ( P C Q )  =  ( (
normCV
`  U ) `  ( P ( -v `  U ) Q ) ) )
4039oveq2d 6097 . 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 4242 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4212   -->wf 5450   ` cfv 5454  (class class class)co 6081    x. cmul 8995    <_ cle 9121   NrmCVeccnv 22063   BaseSetcba 22065   -vcnsb 22068   normCVcnmcv 22069   IndMetcims 22070    LnOp clno 22241   normOp OLDcnmoo 22242    BLnOp cblo 22243
This theorem is referenced by:  blocni  22306
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-grpo 21779  df-gid 21780  df-ginv 21781  df-gdiv 21782  df-ablo 21870  df-vc 22025  df-nv 22071  df-va 22074  df-ba 22075  df-sm 22076  df-0v 22077  df-vs 22078  df-nmcv 22079  df-ims 22080  df-lno 22245  df-nmoo 22246  df-blo 22247  df-0o 22248
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