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Theorem blometi 25916
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 2454 . . . . . 6  |-  ( -v
`  U )  =  ( -v `  U
)
42, 3nvmcl 25740 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  P  e.  X  /\  Q  e.  X )  ->  ( P ( -v `  U ) Q )  e.  X )
51, 4mp3an1 1309 . . . 4  |-  ( ( P  e.  X  /\  Q  e.  X )  ->  ( P ( -v
`  U ) Q )  e.  X )
6 eqid 2454 . . . . 5  |-  ( normCV `  U )  =  (
normCV
`  U )
7 eqid 2454 . . . . 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 25913 . . . 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 472 . . 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 1190 . 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 25898 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  B )  ->  T : X --> Y )
161, 10, 15mp3an12 1312 . . . . . 6  |-  ( T  e.  B  ->  T : X --> Y )
1716ffvelrnda 6007 . . . . 5  |-  ( ( T  e.  B  /\  P  e.  X )  ->  ( T `  P
)  e.  Y )
18173adant3 1014 . . . 4  |-  ( ( T  e.  B  /\  P  e.  X  /\  Q  e.  X )  ->  ( T `  P
)  e.  Y )
1916ffvelrnda 6007 . . . . 5  |-  ( ( T  e.  B  /\  Q  e.  X )  ->  ( T `  Q
)  e.  Y )
20193adant2 1013 . . . 4  |-  ( ( T  e.  B  /\  P  e.  X  /\  Q  e.  X )  ->  ( T `  Q
)  e.  Y )
21 eqid 2454 . . . . . 6  |-  ( -v
`  W )  =  ( -v `  W
)
22 blometi.d . . . . . 6  |-  D  =  ( IndMet `  W )
2314, 21, 7, 22imsdval 25790 . . . . 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 1309 . . . 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 659 . . 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 2454 . . . . . . 7  |-  ( U 
LnOp  W )  =  ( U  LnOp  W )
2726, 9bloln 25897 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  B )  ->  T  e.  ( U  LnOp  W
) )
281, 10, 27mp3an12 1312 . . . . 5  |-  ( T  e.  B  ->  T  e.  ( U  LnOp  W
) )
292, 3, 21, 26lnosub 25872 . . . . . . . 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 1316 . . . . . . 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 678 . . . . . 6  |-  ( ( T  e.  ( U 
LnOp  W )  /\  ( P  e.  X  /\  Q  e.  X )
)  ->  ( T `  ( P ( -v
`  U ) Q ) )  =  ( ( T `  P
) ( -v `  W ) ( T `
 Q ) ) )
32313impb 1190 . . . . 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 1259 . . . 4  |-  ( ( T  e.  B  /\  P  e.  X  /\  Q  e.  X )  ->  ( T `  ( P ( -v `  U ) Q ) )  =  ( ( T `  P ) ( -v `  W
) ( T `  Q ) ) )
3433fveq2d 5852 . . 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 2498 . 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 25790 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  P  e.  X  /\  Q  e.  X )  ->  ( P C Q )  =  ( ( normCV `  U
) `  ( P
( -v `  U
) Q ) ) )
381, 37mp3an1 1309 . . . 4  |-  ( ( P  e.  X  /\  Q  e.  X )  ->  ( P C Q )  =  ( (
normCV
`  U ) `  ( P ( -v `  U ) Q ) ) )
39383adant1 1012 . . 3  |-  ( ( T  e.  B  /\  P  e.  X  /\  Q  e.  X )  ->  ( P C Q )  =  ( (
normCV
`  U ) `  ( P ( -v `  U ) Q ) ) )
4039oveq2d 6286 . 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 4469 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   class class class wbr 4439   -->wf 5566   ` cfv 5570  (class class class)co 6270    x. cmul 9486    <_ cle 9618   NrmCVeccnv 25675   BaseSetcba 25677   -vcnsb 25680   normCVcnmcv 25681   IndMetcims 25682    LnOp clno 25853   normOpOLDcnmoo 25854    BLnOp cblo 25855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-seq 12090  df-exp 12149  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-grpo 25391  df-gid 25392  df-ginv 25393  df-gdiv 25394  df-ablo 25482  df-vc 25637  df-nv 25683  df-va 25686  df-ba 25687  df-sm 25688  df-0v 25689  df-vs 25690  df-nmcv 25691  df-ims 25692  df-lno 25857  df-nmoo 25858  df-blo 25859  df-0o 25860
This theorem is referenced by:  blocni  25918
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