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Theorem nmoptrii 23447
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 23215 . . . . 5  |-  ( S  e.  BndLinOp  ->  S : ~H --> ~H )
31, 2ax-mp 8 . . . 4  |-  S : ~H
--> ~H
4 nmoptri.2 . . . . 5  |-  T  e.  BndLinOp
5 bdopf 23215 . . . . 5  |-  ( T  e.  BndLinOp  ->  T : ~H --> ~H )
64, 5ax-mp 8 . . . 4  |-  T : ~H
--> ~H
73, 6hoaddcli 23121 . . 3  |-  ( S 
+op  T ) : ~H --> ~H
8 nmopre 23223 . . . . . 6  |-  ( S  e.  BndLinOp  ->  ( normop `  S
)  e.  RR )
91, 8ax-mp 8 . . . . 5  |-  ( normop `  S )  e.  RR
10 nmopre 23223 . . . . . 6  |-  ( T  e.  BndLinOp  ->  ( normop `  T
)  e.  RR )
114, 10ax-mp 8 . . . . 5  |-  ( normop `  T )  e.  RR
129, 11readdcli 9038 . . . 4  |-  ( (
normop `  S )  +  ( normop `  T )
)  e.  RR
1312rexri 9072 . . 3  |-  ( (
normop `  S )  +  ( normop `  T )
)  e.  RR*
14 nmopub 23261 . . 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 23115 . . . . . 6  |-  ( x  e.  ~H  ->  (
( S  +op  T
) `  x )  e.  ~H )
17 normcl 22477 . . . . . 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 5810 . . . . . . 7  |-  ( x  e.  ~H  ->  ( S `  x )  e.  ~H )
21 normcl 22477 . . . . . . 7  |-  ( ( S `  x )  e.  ~H  ->  ( normh `  ( S `  x ) )  e.  RR )
2220, 21syl 16 . . . . . 6  |-  ( x  e.  ~H  ->  ( normh `  ( S `  x ) )  e.  RR )
236ffvelrni 5810 . . . . . . 7  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
24 normcl 22477 . . . . . . 7  |-  ( ( T `  x )  e.  ~H  ->  ( normh `  ( T `  x ) )  e.  RR )
2523, 24syl 16 . . . . . 6  |-  ( x  e.  ~H  ->  ( normh `  ( T `  x ) )  e.  RR )
2622, 25readdcld 9050 . . . . 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 23093 . . . . . . . 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 5674 . . . . . 6  |-  ( x  e.  ~H  ->  ( normh `  ( ( S 
+op  T ) `  x ) )  =  ( normh `  ( ( S `  x )  +h  ( T `  x
) ) ) )
32 norm-ii 22490 . . . . . . 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 4175 . . . . 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 23260 . . . . . 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 23260 . . . . . 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 9444 . . . . . . . 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 9161 . . 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 2721 1  |-  ( normop `  ( S  +op  T
) )  <_  (
( normop `  S )  +  ( normop `  T
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2651   class class class wbr 4155   -->wf 5392   ` cfv 5396  (class class class)co 6022   RRcr 8924   1c1 8926    + caddc 8928   RR*cxr 9054    <_ cle 9056   ~Hchil 22272    +h cva 22273   normhcno 22276    +op chos 22291   normopcnop 22298   BndLinOpcbo 22301
This theorem is referenced by:  bdophsi  23449  nmoptri2i  23452  unierri  23457
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003  ax-hilex 22352  ax-hfvadd 22353  ax-hvcom 22354  ax-hvass 22355  ax-hv0cl 22356  ax-hvaddid 22357  ax-hfvmul 22358  ax-hvmulid 22359  ax-hvmulass 22360  ax-hvdistr1 22361  ax-hvdistr2 22362  ax-hvmul0 22363  ax-hfi 22431  ax-his1 22434  ax-his2 22435  ax-his3 22436  ax-his4 22437
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-map 6958  df-en 7048  df-dom 7049  df-sdom 7050  df-sup 7383  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-4 9994  df-n0 10156  df-z 10217  df-uz 10423  df-rp 10547  df-seq 11253  df-exp 11312  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970  df-grpo 21629  df-gid 21630  df-ablo 21720  df-vc 21875  df-nv 21921  df-va 21924  df-ba 21925  df-sm 21926  df-0v 21927  df-nmcv 21929  df-hnorm 22321  df-hba 22322  df-hvsub 22324  df-hosum 23083  df-nmop 23192  df-lnop 23194  df-bdop 23195
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