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Theorem nmoptrii 22690
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 22458 . . . . 5  |-  ( S  e.  BndLinOp  ->  S : ~H --> ~H )
31, 2ax-mp 8 . . . 4  |-  S : ~H
--> ~H
4 nmoptri.2 . . . . 5  |-  T  e.  BndLinOp
5 bdopf 22458 . . . . 5  |-  ( T  e.  BndLinOp  ->  T : ~H --> ~H )
64, 5ax-mp 8 . . . 4  |-  T : ~H
--> ~H
73, 6hoaddcli 22364 . . 3  |-  ( S 
+op  T ) : ~H --> ~H
8 nmopre 22466 . . . . . 6  |-  ( S  e.  BndLinOp  ->  ( normop `  S
)  e.  RR )
91, 8ax-mp 8 . . . . 5  |-  ( normop `  S )  e.  RR
10 nmopre 22466 . . . . . 6  |-  ( T  e.  BndLinOp  ->  ( normop `  T
)  e.  RR )
114, 10ax-mp 8 . . . . 5  |-  ( normop `  T )  e.  RR
129, 11readdcli 8866 . . . 4  |-  ( (
normop `  S )  +  ( normop `  T )
)  e.  RR
13 rexr 8893 . . . 4  |-  ( ( ( normop `  S )  +  ( normop `  T
) )  e.  RR  ->  ( ( normop `  S
)  +  ( normop `  T ) )  e. 
RR* )
1412, 13ax-mp 8 . . 3  |-  ( (
normop `  S )  +  ( normop `  T )
)  e.  RR*
15 nmopub 22504 . . 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 ) ) ) ) )
167, 14, 15mp2an 653 . 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 ) ) ) )
173, 6hoscli 22358 . . . . . 6  |-  ( x  e.  ~H  ->  (
( S  +op  T
) `  x )  e.  ~H )
18 normcl 21720 . . . . . 6  |-  ( ( ( S  +op  T
) `  x )  e.  ~H  ->  ( normh `  ( ( S  +op  T ) `  x ) )  e.  RR )
1917, 18syl 15 . . . . 5  |-  ( x  e.  ~H  ->  ( normh `  ( ( S 
+op  T ) `  x ) )  e.  RR )
2019adantr 451 . . . 4  |-  ( ( x  e.  ~H  /\  ( normh `  x )  <_  1 )  ->  ( normh `  ( ( S 
+op  T ) `  x ) )  e.  RR )
213ffvelrni 5680 . . . . . . 7  |-  ( x  e.  ~H  ->  ( S `  x )  e.  ~H )
22 normcl 21720 . . . . . . 7  |-  ( ( S `  x )  e.  ~H  ->  ( normh `  ( S `  x ) )  e.  RR )
2321, 22syl 15 . . . . . 6  |-  ( x  e.  ~H  ->  ( normh `  ( S `  x ) )  e.  RR )
246ffvelrni 5680 . . . . . . 7  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
25 normcl 21720 . . . . . . 7  |-  ( ( T `  x )  e.  ~H  ->  ( normh `  ( T `  x ) )  e.  RR )
2624, 25syl 15 . . . . . 6  |-  ( x  e.  ~H  ->  ( normh `  ( T `  x ) )  e.  RR )
2723, 26readdcld 8878 . . . . 5  |-  ( x  e.  ~H  ->  (
( normh `  ( S `  x ) )  +  ( normh `  ( T `  x ) ) )  e.  RR )
2827adantr 451 . . . 4  |-  ( ( x  e.  ~H  /\  ( normh `  x )  <_  1 )  ->  (
( normh `  ( S `  x ) )  +  ( normh `  ( T `  x ) ) )  e.  RR )
2912a1i 10 . . . 4  |-  ( ( x  e.  ~H  /\  ( normh `  x )  <_  1 )  ->  (
( normop `  S )  +  ( normop `  T
) )  e.  RR )
30 hosval 22336 . . . . . . . 8  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( S  +op  T ) `  x )  =  ( ( S `
 x )  +h  ( T `  x
) ) )
313, 6, 30mp3an12 1267 . . . . . . 7  |-  ( x  e.  ~H  ->  (
( S  +op  T
) `  x )  =  ( ( S `
 x )  +h  ( T `  x
) ) )
3231fveq2d 5545 . . . . . 6  |-  ( x  e.  ~H  ->  ( normh `  ( ( S 
+op  T ) `  x ) )  =  ( normh `  ( ( S `  x )  +h  ( T `  x
) ) ) )
33 norm-ii 21733 . . . . . . 7  |-  ( ( ( S `  x
)  e.  ~H  /\  ( T `  x )  e.  ~H )  -> 
( normh `  ( ( S `  x )  +h  ( T `  x
) ) )  <_ 
( ( normh `  ( S `  x )
)  +  ( normh `  ( T `  x
) ) ) )
3421, 24, 33syl2anc 642 . . . . . 6  |-  ( x  e.  ~H  ->  ( normh `  ( ( S `
 x )  +h  ( T `  x
) ) )  <_ 
( ( normh `  ( S `  x )
)  +  ( normh `  ( T `  x
) ) ) )
3532, 34eqbrtrd 4059 . . . . 5  |-  ( x  e.  ~H  ->  ( normh `  ( ( S 
+op  T ) `  x ) )  <_ 
( ( normh `  ( S `  x )
)  +  ( normh `  ( T `  x
) ) ) )
3635adantr 451 . . . 4  |-  ( ( x  e.  ~H  /\  ( normh `  x )  <_  1 )  ->  ( normh `  ( ( S 
+op  T ) `  x ) )  <_ 
( ( normh `  ( S `  x )
)  +  ( normh `  ( T `  x
) ) ) )
37 nmoplb 22503 . . . . . 6  |-  ( ( S : ~H --> ~H  /\  x  e.  ~H  /\  ( normh `  x )  <_ 
1 )  ->  ( normh `  ( S `  x ) )  <_ 
( normop `  S )
)
383, 37mp3an1 1264 . . . . 5  |-  ( ( x  e.  ~H  /\  ( normh `  x )  <_  1 )  ->  ( normh `  ( S `  x ) )  <_ 
( normop `  S )
)
39 nmoplb 22503 . . . . . 6  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H  /\  ( normh `  x )  <_ 
1 )  ->  ( normh `  ( T `  x ) )  <_ 
( normop `  T )
)
406, 39mp3an1 1264 . . . . 5  |-  ( ( x  e.  ~H  /\  ( normh `  x )  <_  1 )  ->  ( normh `  ( T `  x ) )  <_ 
( normop `  T )
)
41 le2add 9272 . . . . . . . 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 ) ) ) )
429, 11, 41mpanr12 666 . . . . . . 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 ) ) ) )
4323, 26, 42syl2anc 642 . . . . . 6  |-  ( x  e.  ~H  ->  (
( ( normh `  ( S `  x )
)  <_  ( normop `  S
)  /\  ( normh `  ( T `  x
) )  <_  ( normop `  T ) )  -> 
( ( normh `  ( S `  x )
)  +  ( normh `  ( T `  x
) ) )  <_ 
( ( normop `  S
)  +  ( normop `  T ) ) ) )
4443adantr 451 . . . . 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 ) ) ) )
4538, 40, 44mp2and 660 . . . 4  |-  ( ( x  e.  ~H  /\  ( normh `  x )  <_  1 )  ->  (
( normh `  ( S `  x ) )  +  ( normh `  ( T `  x ) ) )  <_  ( ( normop `  S )  +  (
normop `  T ) ) )
4620, 28, 29, 36, 45letrd 8989 . . 3  |-  ( ( x  e.  ~H  /\  ( normh `  x )  <_  1 )  ->  ( normh `  ( ( S 
+op  T ) `  x ) )  <_ 
( ( normop `  S
)  +  ( normop `  T ) ) )
4746ex 423 . 2  |-  ( x  e.  ~H  ->  (
( normh `  x )  <_  1  ->  ( normh `  ( ( S  +op  T ) `  x ) )  <_  ( ( normop `  S )  +  (
normop `  T ) ) ) )
4816, 47mprgbir 2626 1  |-  ( normop `  ( S  +op  T
) )  <_  (
( normop `  S )  +  ( normop `  T
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   class class class wbr 4039   -->wf 5267   ` cfv 5271  (class class class)co 5874   RRcr 8752   1c1 8754    + caddc 8756   RR*cxr 8882    <_ cle 8884   ~Hchil 21515    +h cva 21516   normhcno 21519    +op chos 21534   normopcnop 21541   BndLinOpcbo 21544
This theorem is referenced by:  bdophsi  22692  nmoptri2i  22695  unierri  22700
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  ax-hilex 21595  ax-hfvadd 21596  ax-hvcom 21597  ax-hvass 21598  ax-hv0cl 21599  ax-hvaddid 21600  ax-hfvmul 21601  ax-hvmulid 21602  ax-hvmulass 21603  ax-hvdistr1 21604  ax-hvdistr2 21605  ax-hvmul0 21606  ax-hfi 21674  ax-his1 21677  ax-his2 21678  ax-his3 21679  ax-his4 21680
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-4 9822  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-ablo 20965  df-vc 21118  df-nv 21164  df-va 21167  df-ba 21168  df-sm 21169  df-0v 21170  df-nmcv 21172  df-hnorm 21564  df-hba 21565  df-hvsub 21567  df-hosum 22326  df-nmop 22435  df-lnop 22437  df-bdop 22438
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