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Theorem nmoptrii 22674
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 22442 . . . . 5  |-  ( S  e.  BndLinOp  ->  S : ~H --> ~H )
31, 2ax-mp 8 . . . 4  |-  S : ~H
--> ~H
4 nmoptri.2 . . . . 5  |-  T  e.  BndLinOp
5 bdopf 22442 . . . . 5  |-  ( T  e.  BndLinOp  ->  T : ~H --> ~H )
64, 5ax-mp 8 . . . 4  |-  T : ~H
--> ~H
73, 6hoaddcli 22348 . . 3  |-  ( S 
+op  T ) : ~H --> ~H
8 nmopre 22450 . . . . . 6  |-  ( S  e.  BndLinOp  ->  ( normop `  S
)  e.  RR )
91, 8ax-mp 8 . . . . 5  |-  ( normop `  S )  e.  RR
10 nmopre 22450 . . . . . 6  |-  ( T  e.  BndLinOp  ->  ( normop `  T
)  e.  RR )
114, 10ax-mp 8 . . . . 5  |-  ( normop `  T )  e.  RR
129, 11readdcli 8850 . . . 4  |-  ( (
normop `  S )  +  ( normop `  T )
)  e.  RR
13 rexr 8877 . . . 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 22488 . . 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 22342 . . . . . 6  |-  ( x  e.  ~H  ->  (
( S  +op  T
) `  x )  e.  ~H )
18 normcl 21704 . . . . . 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 5664 . . . . . . 7  |-  ( x  e.  ~H  ->  ( S `  x )  e.  ~H )
22 normcl 21704 . . . . . . 7  |-  ( ( S `  x )  e.  ~H  ->  ( normh `  ( S `  x ) )  e.  RR )
2321, 22syl 15 . . . . . 6  |-  ( x  e.  ~H  ->  ( normh `  ( S `  x ) )  e.  RR )
246ffvelrni 5664 . . . . . . 7  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
25 normcl 21704 . . . . . . 7  |-  ( ( T `  x )  e.  ~H  ->  ( normh `  ( T `  x ) )  e.  RR )
2624, 25syl 15 . . . . . 6  |-  ( x  e.  ~H  ->  ( normh `  ( T `  x ) )  e.  RR )
2723, 26readdcld 8862 . . . . 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 22320 . . . . . . . 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 5529 . . . . . 6  |-  ( x  e.  ~H  ->  ( normh `  ( ( S 
+op  T ) `  x ) )  =  ( normh `  ( ( S `  x )  +h  ( T `  x
) ) ) )
33 norm-ii 21717 . . . . . . 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 4043 . . . . 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 22487 . . . . . 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 22487 . . . . . 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 9256 . . . . . . . 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 8973 . . 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 2613 1  |-  ( normop `  ( S  +op  T
) )  <_  (
( normop `  S )  +  ( normop `  T
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   class class class wbr 4023   -->wf 5251   ` cfv 5255  (class class class)co 5858   RRcr 8736   1c1 8738    + caddc 8740   RR*cxr 8866    <_ cle 8868   ~Hchil 21499    +h cva 21500   normhcno 21503    +op chos 21518   normopcnop 21525   BndLinOpcbo 21528
This theorem is referenced by:  bdophsi  22676  nmoptri2i  22679  unierri  22684
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-hilex 21579  ax-hfvadd 21580  ax-hvcom 21581  ax-hvass 21582  ax-hv0cl 21583  ax-hvaddid 21584  ax-hfvmul 21585  ax-hvmulid 21586  ax-hvmulass 21587  ax-hvdistr1 21588  ax-hvdistr2 21589  ax-hvmul0 21590  ax-hfi 21658  ax-his1 21661  ax-his2 21662  ax-his3 21663  ax-his4 21664
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-grpo 20858  df-gid 20859  df-ablo 20949  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-nmcv 21156  df-hnorm 21548  df-hba 21549  df-hvsub 21551  df-hosum 22310  df-nmop 22419  df-lnop 22421  df-bdop 22422
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