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Theorem blometi 21397
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 2296 . . . . . 6  |-  ( -v
`  U )  =  ( -v `  U
)
42, 3nvmcl 21221 . . . . 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 2296 . . . . 5  |-  ( normCV `  U )  =  (
normCV
`  U )
7 eqid 2296 . . . . 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 21394 . . . 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 21379 . . . . . . 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 5679 . . . . . 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 5679 . . . . . 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 2296 . . . . . 6  |-  ( -v
`  W )  =  ( -v `  W
)
24 blometi.d . . . . . 6  |-  D  =  ( IndMet `  W )
2514, 23, 7, 24imsdval 21271 . . . . 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 2296 . . . . . . 7  |-  ( U 
LnOp  W )  =  ( U  LnOp  W )
2928, 9bloln 21378 . . . . . 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 21353 . . . . . . . 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 5545 . . 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 2331 . 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 21271 . . . . 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 5890 . 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 4069 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 1632    e. wcel 1696   class class class wbr 4039   -->wf 5267   ` cfv 5271  (class class class)co 5874    x. cmul 8758    <_ cle 8884   NrmCVeccnv 21156   BaseSetcba 21158   -vcnsb 21161   normCVcnmcv 21162   IndMetcims 21163    LnOp clno 21334   normOp OLDcnmoo 21335    BLnOp cblo 21336
This theorem is referenced by:  blocni  21399
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-grpo 20874  df-gid 20875  df-ginv 20876  df-gdiv 20877  df-ablo 20965  df-vc 21118  df-nv 21164  df-va 21167  df-ba 21168  df-sm 21169  df-0v 21170  df-vs 21171  df-nmcv 21172  df-ims 21173  df-lno 21338  df-nmoo 21339  df-blo 21340  df-0o 21341
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