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Theorem nmoptrii 23589
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 23357 . . . . 5  |-  ( S  e.  BndLinOp  ->  S : ~H --> ~H )
31, 2ax-mp 8 . . . 4  |-  S : ~H
--> ~H
4 nmoptri.2 . . . . 5  |-  T  e.  BndLinOp
5 bdopf 23357 . . . . 5  |-  ( T  e.  BndLinOp  ->  T : ~H --> ~H )
64, 5ax-mp 8 . . . 4  |-  T : ~H
--> ~H
73, 6hoaddcli 23263 . . 3  |-  ( S 
+op  T ) : ~H --> ~H
8 nmopre 23365 . . . . . 6  |-  ( S  e.  BndLinOp  ->  ( normop `  S
)  e.  RR )
91, 8ax-mp 8 . . . . 5  |-  ( normop `  S )  e.  RR
10 nmopre 23365 . . . . . 6  |-  ( T  e.  BndLinOp  ->  ( normop `  T
)  e.  RR )
114, 10ax-mp 8 . . . . 5  |-  ( normop `  T )  e.  RR
129, 11readdcli 9095 . . . 4  |-  ( (
normop `  S )  +  ( normop `  T )
)  e.  RR
1312rexri 9129 . . 3  |-  ( (
normop `  S )  +  ( normop `  T )
)  e.  RR*
14 nmopub 23403 . . 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 654 . 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 23257 . . . . . 6  |-  ( x  e.  ~H  ->  (
( S  +op  T
) `  x )  e.  ~H )
17 normcl 22619 . . . . . 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 452 . . . 4  |-  ( ( x  e.  ~H  /\  ( normh `  x )  <_  1 )  ->  ( normh `  ( ( S 
+op  T ) `  x ) )  e.  RR )
203ffvelrni 5861 . . . . . . 7  |-  ( x  e.  ~H  ->  ( S `  x )  e.  ~H )
21 normcl 22619 . . . . . . 7  |-  ( ( S `  x )  e.  ~H  ->  ( normh `  ( S `  x ) )  e.  RR )
2220, 21syl 16 . . . . . 6  |-  ( x  e.  ~H  ->  ( normh `  ( S `  x ) )  e.  RR )
236ffvelrni 5861 . . . . . . 7  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
24 normcl 22619 . . . . . . 7  |-  ( ( T `  x )  e.  ~H  ->  ( normh `  ( T `  x ) )  e.  RR )
2523, 24syl 16 . . . . . 6  |-  ( x  e.  ~H  ->  ( normh `  ( T `  x ) )  e.  RR )
2622, 25readdcld 9107 . . . . 5  |-  ( x  e.  ~H  ->  (
( normh `  ( S `  x ) )  +  ( normh `  ( T `  x ) ) )  e.  RR )
2726adantr 452 . . . 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 23235 . . . . . . . 8  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( S  +op  T ) `  x )  =  ( ( S `
 x )  +h  ( T `  x
) ) )
303, 6, 29mp3an12 1269 . . . . . . 7  |-  ( x  e.  ~H  ->  (
( S  +op  T
) `  x )  =  ( ( S `
 x )  +h  ( T `  x
) ) )
3130fveq2d 5724 . . . . . 6  |-  ( x  e.  ~H  ->  ( normh `  ( ( S 
+op  T ) `  x ) )  =  ( normh `  ( ( S `  x )  +h  ( T `  x
) ) ) )
32 norm-ii 22632 . . . . . . 7  |-  ( ( ( S `  x
)  e.  ~H  /\  ( T `  x )  e.  ~H )  -> 
( normh `  ( ( S `  x )  +h  ( T `  x
) ) )  <_ 
( ( normh `  ( S `  x )
)  +  ( normh `  ( T `  x
) ) ) )
3320, 23, 32syl2anc 643 . . . . . 6  |-  ( x  e.  ~H  ->  ( normh `  ( ( S `
 x )  +h  ( T `  x
) ) )  <_ 
( ( normh `  ( S `  x )
)  +  ( normh `  ( T `  x
) ) ) )
3431, 33eqbrtrd 4224 . . . . 5  |-  ( x  e.  ~H  ->  ( normh `  ( ( S 
+op  T ) `  x ) )  <_ 
( ( normh `  ( S `  x )
)  +  ( normh `  ( T `  x
) ) ) )
3534adantr 452 . . . 4  |-  ( ( x  e.  ~H  /\  ( normh `  x )  <_  1 )  ->  ( normh `  ( ( S 
+op  T ) `  x ) )  <_ 
( ( normh `  ( S `  x )
)  +  ( normh `  ( T `  x
) ) ) )
36 nmoplb 23402 . . . . . 6  |-  ( ( S : ~H --> ~H  /\  x  e.  ~H  /\  ( normh `  x )  <_ 
1 )  ->  ( normh `  ( S `  x ) )  <_ 
( normop `  S )
)
373, 36mp3an1 1266 . . . . 5  |-  ( ( x  e.  ~H  /\  ( normh `  x )  <_  1 )  ->  ( normh `  ( S `  x ) )  <_ 
( normop `  S )
)
38 nmoplb 23402 . . . . . 6  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H  /\  ( normh `  x )  <_ 
1 )  ->  ( normh `  ( T `  x ) )  <_ 
( normop `  T )
)
396, 38mp3an1 1266 . . . . 5  |-  ( ( x  e.  ~H  /\  ( normh `  x )  <_  1 )  ->  ( normh `  ( T `  x ) )  <_ 
( normop `  T )
)
40 le2add 9502 . . . . . . . 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 667 . . . . . . 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 643 . . . . . 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 452 . . . . 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 661 . . . 4  |-  ( ( x  e.  ~H  /\  ( normh `  x )  <_  1 )  ->  (
( normh `  ( S `  x ) )  +  ( normh `  ( T `  x ) ) )  <_  ( ( normop `  S )  +  (
normop `  T ) ) )
4519, 27, 28, 35, 44letrd 9219 . . 3  |-  ( ( x  e.  ~H  /\  ( normh `  x )  <_  1 )  ->  ( normh `  ( ( S 
+op  T ) `  x ) )  <_ 
( ( normop `  S
)  +  ( normop `  T ) ) )
4645ex 424 . 2  |-  ( x  e.  ~H  ->  (
( normh `  x )  <_  1  ->  ( normh `  ( ( S  +op  T ) `  x ) )  <_  ( ( normop `  S )  +  (
normop `  T ) ) ) )
4715, 46mprgbir 2768 1  |-  ( normop `  ( S  +op  T
) )  <_  (
( normop `  S )  +  ( normop `  T
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   class class class wbr 4204   -->wf 5442   ` cfv 5446  (class class class)co 6073   RRcr 8981   1c1 8983    + caddc 8985   RR*cxr 9111    <_ cle 9113   ~Hchil 22414    +h cva 22415   normhcno 22418    +op chos 22433   normopcnop 22440   BndLinOpcbo 22443
This theorem is referenced by:  bdophsi  23591  nmoptri2i  23594  unierri  23599
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-hilex 22494  ax-hfvadd 22495  ax-hvcom 22496  ax-hvass 22497  ax-hv0cl 22498  ax-hvaddid 22499  ax-hfvmul 22500  ax-hvmulid 22501  ax-hvmulass 22502  ax-hvdistr1 22503  ax-hvdistr2 22504  ax-hvmul0 22505  ax-hfi 22573  ax-his1 22576  ax-his2 22577  ax-his3 22578  ax-his4 22579
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-grpo 21771  df-gid 21772  df-ablo 21862  df-vc 22017  df-nv 22063  df-va 22066  df-ba 22067  df-sm 22068  df-0v 22069  df-nmcv 22071  df-hnorm 22463  df-hba 22464  df-hvsub 22466  df-hosum 23225  df-nmop 23334  df-lnop 23336  df-bdop 23337
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