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Theorem blometi 21381
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 2283 . . . . . 6  |-  ( -v
`  U )  =  ( -v `  U
)
42, 3nvmcl 21205 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  P  e.  X  /\  Q  e.  X )  ->  ( P ( -v `  U ) Q )  e.  X )
51, 4mp3an1 1264 . . . 4  |-  ( ( P  e.  X  /\  Q  e.  X )  ->  ( P ( -v
`  U ) Q )  e.  X )
6 eqid 2283 . . . . 5  |-  ( normCV `  U )  =  (
normCV
`  U )
7 eqid 2283 . . . . 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 21378 . . . 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 460 . . 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 1147 . 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 21363 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  B )  ->  T : X --> Y )
161, 10, 15mp3an12 1267 . . . . . 6  |-  ( T  e.  B  ->  T : X --> Y )
17 ffvelrn 5663 . . . . . 6  |-  ( ( T : X --> Y  /\  P  e.  X )  ->  ( T `  P
)  e.  Y )
1816, 17sylan 457 . . . . 5  |-  ( ( T  e.  B  /\  P  e.  X )  ->  ( T `  P
)  e.  Y )
19183adant3 975 . . . 4  |-  ( ( T  e.  B  /\  P  e.  X  /\  Q  e.  X )  ->  ( T `  P
)  e.  Y )
20 ffvelrn 5663 . . . . . 6  |-  ( ( T : X --> Y  /\  Q  e.  X )  ->  ( T `  Q
)  e.  Y )
2116, 20sylan 457 . . . . 5  |-  ( ( T  e.  B  /\  Q  e.  X )  ->  ( T `  Q
)  e.  Y )
22213adant2 974 . . . 4  |-  ( ( T  e.  B  /\  P  e.  X  /\  Q  e.  X )  ->  ( T `  Q
)  e.  Y )
23 eqid 2283 . . . . . 6  |-  ( -v
`  W )  =  ( -v `  W
)
24 blometi.d . . . . . 6  |-  D  =  ( IndMet `  W )
2514, 23, 7, 24imsdval 21255 . . . . 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 ) ) ) )
2610, 25mp3an1 1264 . . . 4  |-  ( ( ( T `  P
)  e.  Y  /\  ( T `  Q )  e.  Y )  -> 
( ( T `  P ) D ( T `  Q ) )  =  ( (
normCV
`  W ) `  ( ( T `  P ) ( -v
`  W ) ( T `  Q ) ) ) )
2719, 22, 26syl2anc 642 . . 3  |-  ( ( T  e.  B  /\  P  e.  X  /\  Q  e.  X )  ->  ( ( T `  P ) D ( T `  Q ) )  =  ( (
normCV
`  W ) `  ( ( T `  P ) ( -v
`  W ) ( T `  Q ) ) ) )
28 eqid 2283 . . . . . . 7  |-  ( U 
LnOp  W )  =  ( U  LnOp  W )
2928, 9bloln 21362 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  B )  ->  T  e.  ( U  LnOp  W
) )
301, 10, 29mp3an12 1267 . . . . 5  |-  ( T  e.  B  ->  T  e.  ( U  LnOp  W
) )
312, 3, 23, 28lnosub 21337 . . . . . . . 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 ) ) )
321, 31mp3anl1 1271 . . . . . . 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
) ) )
3310, 32mpanl1 661 . . . . . 6  |-  ( ( T  e.  ( U 
LnOp  W )  /\  ( P  e.  X  /\  Q  e.  X )
)  ->  ( T `  ( P ( -v
`  U ) Q ) )  =  ( ( T `  P
) ( -v `  W ) ( T `
 Q ) ) )
34333impb 1147 . . . . 5  |-  ( ( T  e.  ( U 
LnOp  W )  /\  P  e.  X  /\  Q  e.  X )  ->  ( T `  ( P
( -v `  U
) Q ) )  =  ( ( T `
 P ) ( -v `  W ) ( T `  Q
) ) )
3530, 34syl3an1 1215 . . . 4  |-  ( ( T  e.  B  /\  P  e.  X  /\  Q  e.  X )  ->  ( T `  ( P ( -v `  U ) Q ) )  =  ( ( T `  P ) ( -v `  W
) ( T `  Q ) ) )
3635fveq2d 5529 . . 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 ) ) ) )
3727, 36eqtr4d 2318 . 2  |-  ( ( T  e.  B  /\  P  e.  X  /\  Q  e.  X )  ->  ( ( T `  P ) D ( T `  Q ) )  =  ( (
normCV
`  W ) `  ( T `  ( P ( -v `  U
) Q ) ) ) )
38 blometi.8 . . . . . 6  |-  C  =  ( IndMet `  U )
392, 3, 6, 38imsdval 21255 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  P  e.  X  /\  Q  e.  X )  ->  ( P C Q )  =  ( ( normCV `  U
) `  ( P
( -v `  U
) Q ) ) )
401, 39mp3an1 1264 . . . 4  |-  ( ( P  e.  X  /\  Q  e.  X )  ->  ( P C Q )  =  ( (
normCV
`  U ) `  ( P ( -v `  U ) Q ) ) )
41403adant1 973 . . 3  |-  ( ( T  e.  B  /\  P  e.  X  /\  Q  e.  X )  ->  ( P C Q )  =  ( (
normCV
`  U ) `  ( P ( -v `  U ) Q ) ) )
4241oveq2d 5874 . 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 ) ) ) )
4313, 37, 423brtr4d 4053 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   -->wf 5251   ` cfv 5255  (class class class)co 5858    x. cmul 8742    <_ cle 8868   NrmCVeccnv 21140   BaseSetcba 21142   -vcnsb 21145   normCVcnmcv 21146   IndMetcims 21147    LnOp clno 21318   normOp OLDcnmoo 21319    BLnOp cblo 21320
This theorem is referenced by:  blocni  21383
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ablo 20949  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-vs 21155  df-nmcv 21156  df-ims 21157  df-lno 21322  df-nmoo 21323  df-blo 21324  df-0o 21325
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