HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  nmoptrii Structured version   Unicode version

Theorem nmoptrii 25513
Description: Triangle inequality for the norms of bounded linear operators. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
nmoptri.1  |-  S  e.  BndLinOp
nmoptri.2  |-  T  e.  BndLinOp
Assertion
Ref Expression
nmoptrii  |-  ( normop `  ( S  +op  T
) )  <_  (
( normop `  S )  +  ( normop `  T
) )

Proof of Theorem nmoptrii
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 nmoptri.1 . . . . 5  |-  S  e.  BndLinOp
2 bdopf 25281 . . . . 5  |-  ( S  e.  BndLinOp  ->  S : ~H --> ~H )
31, 2ax-mp 5 . . . 4  |-  S : ~H
--> ~H
4 nmoptri.2 . . . . 5  |-  T  e.  BndLinOp
5 bdopf 25281 . . . . 5  |-  ( T  e.  BndLinOp  ->  T : ~H --> ~H )
64, 5ax-mp 5 . . . 4  |-  T : ~H
--> ~H
73, 6hoaddcli 25187 . . 3  |-  ( S 
+op  T ) : ~H --> ~H
8 nmopre 25289 . . . . . 6  |-  ( S  e.  BndLinOp  ->  ( normop `  S
)  e.  RR )
91, 8ax-mp 5 . . . . 5  |-  ( normop `  S )  e.  RR
10 nmopre 25289 . . . . . 6  |-  ( T  e.  BndLinOp  ->  ( normop `  T
)  e.  RR )
114, 10ax-mp 5 . . . . 5  |-  ( normop `  T )  e.  RR
129, 11readdcli 9414 . . . 4  |-  ( (
normop `  S )  +  ( normop `  T )
)  e.  RR
1312rexri 9451 . . 3  |-  ( (
normop `  S )  +  ( normop `  T )
)  e.  RR*
14 nmopub 25327 . . 3  |-  ( ( ( S  +op  T
) : ~H --> ~H  /\  ( ( normop `  S
)  +  ( normop `  T ) )  e. 
RR* )  ->  (
( normop `  ( S  +op  T ) )  <_ 
( ( normop `  S
)  +  ( normop `  T ) )  <->  A. x  e.  ~H  ( ( normh `  x )  <_  1  ->  ( normh `  ( ( S  +op  T ) `  x ) )  <_ 
( ( normop `  S
)  +  ( normop `  T ) ) ) ) )
157, 13, 14mp2an 672 . 2  |-  ( (
normop `  ( S  +op  T ) )  <_  (
( normop `  S )  +  ( normop `  T
) )  <->  A. x  e.  ~H  ( ( normh `  x )  <_  1  ->  ( normh `  ( ( S  +op  T ) `  x ) )  <_ 
( ( normop `  S
)  +  ( normop `  T ) ) ) )
163, 6hoscli 25181 . . . . . 6  |-  ( x  e.  ~H  ->  (
( S  +op  T
) `  x )  e.  ~H )
17 normcl 24542 . . . . . 6  |-  ( ( ( S  +op  T
) `  x )  e.  ~H  ->  ( normh `  ( ( S  +op  T ) `  x ) )  e.  RR )
1816, 17syl 16 . . . . 5  |-  ( x  e.  ~H  ->  ( normh `  ( ( S 
+op  T ) `  x ) )  e.  RR )
1918adantr 465 . . . 4  |-  ( ( x  e.  ~H  /\  ( normh `  x )  <_  1 )  ->  ( normh `  ( ( S 
+op  T ) `  x ) )  e.  RR )
203ffvelrni 5857 . . . . . . 7  |-  ( x  e.  ~H  ->  ( S `  x )  e.  ~H )
21 normcl 24542 . . . . . . 7  |-  ( ( S `  x )  e.  ~H  ->  ( normh `  ( S `  x ) )  e.  RR )
2220, 21syl 16 . . . . . 6  |-  ( x  e.  ~H  ->  ( normh `  ( S `  x ) )  e.  RR )
236ffvelrni 5857 . . . . . . 7  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
24 normcl 24542 . . . . . . 7  |-  ( ( T `  x )  e.  ~H  ->  ( normh `  ( T `  x ) )  e.  RR )
2523, 24syl 16 . . . . . 6  |-  ( x  e.  ~H  ->  ( normh `  ( T `  x ) )  e.  RR )
2622, 25readdcld 9428 . . . . 5  |-  ( x  e.  ~H  ->  (
( normh `  ( S `  x ) )  +  ( normh `  ( T `  x ) ) )  e.  RR )
2726adantr 465 . . . 4  |-  ( ( x  e.  ~H  /\  ( normh `  x )  <_  1 )  ->  (
( normh `  ( S `  x ) )  +  ( normh `  ( T `  x ) ) )  e.  RR )
2812a1i 11 . . . 4  |-  ( ( x  e.  ~H  /\  ( normh `  x )  <_  1 )  ->  (
( normop `  S )  +  ( normop `  T
) )  e.  RR )
29 hosval 25159 . . . . . . . 8  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( S  +op  T ) `  x )  =  ( ( S `
 x )  +h  ( T `  x
) ) )
303, 6, 29mp3an12 1304 . . . . . . 7  |-  ( x  e.  ~H  ->  (
( S  +op  T
) `  x )  =  ( ( S `
 x )  +h  ( T `  x
) ) )
3130fveq2d 5710 . . . . . 6  |-  ( x  e.  ~H  ->  ( normh `  ( ( S 
+op  T ) `  x ) )  =  ( normh `  ( ( S `  x )  +h  ( T `  x
) ) ) )
32 norm-ii 24555 . . . . . . 7  |-  ( ( ( S `  x
)  e.  ~H  /\  ( T `  x )  e.  ~H )  -> 
( normh `  ( ( S `  x )  +h  ( T `  x
) ) )  <_ 
( ( normh `  ( S `  x )
)  +  ( normh `  ( T `  x
) ) ) )
3320, 23, 32syl2anc 661 . . . . . 6  |-  ( x  e.  ~H  ->  ( normh `  ( ( S `
 x )  +h  ( T `  x
) ) )  <_ 
( ( normh `  ( S `  x )
)  +  ( normh `  ( T `  x
) ) ) )
3431, 33eqbrtrd 4327 . . . . 5  |-  ( x  e.  ~H  ->  ( normh `  ( ( S 
+op  T ) `  x ) )  <_ 
( ( normh `  ( S `  x )
)  +  ( normh `  ( T `  x
) ) ) )
3534adantr 465 . . . 4  |-  ( ( x  e.  ~H  /\  ( normh `  x )  <_  1 )  ->  ( normh `  ( ( S 
+op  T ) `  x ) )  <_ 
( ( normh `  ( S `  x )
)  +  ( normh `  ( T `  x
) ) ) )
36 nmoplb 25326 . . . . . 6  |-  ( ( S : ~H --> ~H  /\  x  e.  ~H  /\  ( normh `  x )  <_ 
1 )  ->  ( normh `  ( S `  x ) )  <_ 
( normop `  S )
)
373, 36mp3an1 1301 . . . . 5  |-  ( ( x  e.  ~H  /\  ( normh `  x )  <_  1 )  ->  ( normh `  ( S `  x ) )  <_ 
( normop `  S )
)
38 nmoplb 25326 . . . . . 6  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H  /\  ( normh `  x )  <_ 
1 )  ->  ( normh `  ( T `  x ) )  <_ 
( normop `  T )
)
396, 38mp3an1 1301 . . . . 5  |-  ( ( x  e.  ~H  /\  ( normh `  x )  <_  1 )  ->  ( normh `  ( T `  x ) )  <_ 
( normop `  T )
)
40 le2add 9836 . . . . . . . 8  |-  ( ( ( ( normh `  ( S `  x )
)  e.  RR  /\  ( normh `  ( T `  x ) )  e.  RR )  /\  (
( normop `  S )  e.  RR  /\  ( normop `  T )  e.  RR ) )  ->  (
( ( normh `  ( S `  x )
)  <_  ( normop `  S
)  /\  ( normh `  ( T `  x
) )  <_  ( normop `  T ) )  -> 
( ( normh `  ( S `  x )
)  +  ( normh `  ( T `  x
) ) )  <_ 
( ( normop `  S
)  +  ( normop `  T ) ) ) )
419, 11, 40mpanr12 685 . . . . . . 7  |-  ( ( ( normh `  ( S `  x ) )  e.  RR  /\  ( normh `  ( T `  x
) )  e.  RR )  ->  ( ( (
normh `  ( S `  x ) )  <_ 
( normop `  S )  /\  ( normh `  ( T `  x ) )  <_ 
( normop `  T )
)  ->  ( ( normh `  ( S `  x ) )  +  ( normh `  ( T `  x ) ) )  <_  ( ( normop `  S )  +  (
normop `  T ) ) ) )
4222, 25, 41syl2anc 661 . . . . . 6  |-  ( x  e.  ~H  ->  (
( ( normh `  ( S `  x )
)  <_  ( normop `  S
)  /\  ( normh `  ( T `  x
) )  <_  ( normop `  T ) )  -> 
( ( normh `  ( S `  x )
)  +  ( normh `  ( T `  x
) ) )  <_ 
( ( normop `  S
)  +  ( normop `  T ) ) ) )
4342adantr 465 . . . . 5  |-  ( ( x  e.  ~H  /\  ( normh `  x )  <_  1 )  ->  (
( ( normh `  ( S `  x )
)  <_  ( normop `  S
)  /\  ( normh `  ( T `  x
) )  <_  ( normop `  T ) )  -> 
( ( normh `  ( S `  x )
)  +  ( normh `  ( T `  x
) ) )  <_ 
( ( normop `  S
)  +  ( normop `  T ) ) ) )
4437, 39, 43mp2and 679 . . . 4  |-  ( ( x  e.  ~H  /\  ( normh `  x )  <_  1 )  ->  (
( normh `  ( S `  x ) )  +  ( normh `  ( T `  x ) ) )  <_  ( ( normop `  S )  +  (
normop `  T ) ) )
4519, 27, 28, 35, 44letrd 9543 . . 3  |-  ( ( x  e.  ~H  /\  ( normh `  x )  <_  1 )  ->  ( normh `  ( ( S 
+op  T ) `  x ) )  <_ 
( ( normop `  S
)  +  ( normop `  T ) ) )
4645ex 434 . 2  |-  ( x  e.  ~H  ->  (
( normh `  x )  <_  1  ->  ( normh `  ( ( S  +op  T ) `  x ) )  <_  ( ( normop `  S )  +  (
normop `  T ) ) ) )
4715, 46mprgbir 2801 1  |-  ( normop `  ( S  +op  T
) )  <_  (
( normop `  S )  +  ( normop `  T
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2730   class class class wbr 4307   -->wf 5429   ` cfv 5433  (class class class)co 6106   RRcr 9296   1c1 9298    + caddc 9300   RR*cxr 9432    <_ cle 9434   ~Hchil 24336    +h cva 24337   normhcno 24340    +op chos 24355   normopcnop 24362   BndLinOpcbo 24365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374  ax-pre-sup 9375  ax-hilex 24416  ax-hfvadd 24417  ax-hvcom 24418  ax-hvass 24419  ax-hv0cl 24420  ax-hvaddid 24421  ax-hfvmul 24422  ax-hvmulid 24423  ax-hvmulass 24424  ax-hvdistr1 24425  ax-hvdistr2 24426  ax-hvmul0 24427  ax-hfi 24496  ax-his1 24499  ax-his2 24500  ax-his3 24501  ax-his4 24502
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-om 6492  df-1st 6592  df-2nd 6593  df-recs 6847  df-rdg 6881  df-er 7116  df-map 7231  df-en 7326  df-dom 7327  df-sdom 7328  df-sup 7706  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-div 10009  df-nn 10338  df-2 10395  df-3 10396  df-4 10397  df-n0 10595  df-z 10662  df-uz 10877  df-rp 11007  df-seq 11822  df-exp 11881  df-cj 12603  df-re 12604  df-im 12605  df-sqr 12739  df-abs 12740  df-grpo 23693  df-gid 23694  df-ablo 23784  df-vc 23939  df-nv 23985  df-va 23988  df-ba 23989  df-sm 23990  df-0v 23991  df-nmcv 23993  df-hnorm 24385  df-hba 24386  df-hvsub 24388  df-hosum 25149  df-nmop 25258  df-lnop 25260  df-bdop 25261
This theorem is referenced by:  bdophsi  25515  nmoptri2i  25518  unierri  25523
  Copyright terms: Public domain W3C validator