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| Mirrors > Home > HSE Home > Th. List > nmopcoadji | Structured version Visualization version Unicode version | ||
| Description: The norm of an operator composed with its adjoint. Part of Theorem 3.11(vi) of [Beran] p. 106. (Contributed by NM, 8-Mar-2006.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| nmopcoadj.1 |
    |
| Ref | Expression |
|---|---|
| nmopcoadji |
         |
is distinct from all other
variables.| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nmopcoadj.1 |
. . . . . . 7
| |
| 2 | adjbdlnb 28943 |
. . . . . . 7
 | |
| 3 | 1, 2 | mpbi 220 |
. . . . . 6
|
| 4 | bdopf 28721 |
. . . . . 6
    | |
| 5 | 3, 4 | ax-mp 5 |
. . . . 5
|
| 6 | bdopf 28721 |
. . . . . 6
| |
| 7 | 1, 6 | ax-mp 5 |
. . . . 5
|
| 8 | 5, 7 | hocofi 28625 |
. . . 4
|
| 9 | nmopre 28729 |
. . . . . . 7
 | |
| 10 | 1, 9 | ax-mp 5 |
. . . . . 6
|
| 11 | 10 | resqcli 12949 |
. . . . 5
|
| 12 | rexr 10085 |
. . . . 5
 | |
| 13 | 11, 12 | ax-mp 5 |
. . . 4
|
| 14 | nmopub 28767 |
. . . 4
![/\](wedge.gif "/\"){kind=small}  
 | |
| 15 | 8, 13, 14 | mp2an 708 |
. . 3
|
| 16 | 5, 7 | hocoi 28623 |
. . . . . . . 8
|
| 17 | 16 | fveq2d 6195 |
. . . . . . 7
|
| 18 | 17 | adantr 481 |
. . . . . 6
|
| 19 | 7 | ffvelrni 6358 |
. . . . . . . . 9
|
| 20 | 5 | ffvelrni 6358 |
. . . . . . . . 9
|
| 21 | normcl 27982 |
. . . . . . . . 9
| |
| 22 | 19, 20, 21 | 3syl 18 |
. . . . . . . 8
|
| 23 | 22 | adantr 481 |
. . . . . . 7
|
| 24 | nmopre 28729 |
. . . . . . . . . 10
| |
| 25 | 3, 24 | ax-mp 5 |
. . . . . . . . 9
|
| 26 | normcl 27982 |
. . . . . . . . . 10
| |
| 27 | 19, 26 | syl 17 |
. . . . . . . . 9
|
| 28 | remulcl 10021 |
. . . . . . . . 9
 | |
| 29 | 25, 27, 28 | sylancr 695 |
. . . . . . . 8
|
| 30 | 29 | adantr 481 |
. . . . . . 7
|
| 31 | 25, 10 | remulcli 10054 |
. . . . . . . 8
|
| 32 | 31 | a1i 11 |
. . . . . . 7
|
| 33 | 3 | nmbdoplbi 28883 |
. . . . . . . . 9
|
| 34 | 19, 33 | syl 17 |
. . . . . . . 8
|
| 35 | 34 | adantr 481 |
. . . . . . 7
|
| 36 | 27 | adantr 481 |
. . . . . . . 8
|
| 37 | 10 | a1i 11 |
. . . . . . . 8
|
| 38 | normcl 27982 |
. . . . . . . . . . 11
| |
| 39 | remulcl 10021 |
. . . . . . . . . . 11
| |
| 40 | 10, 38, 39 | sylancr 695 |
. . . . . . . . . 10
|
| 41 | 40 | adantr 481 |
. . . . . . . . 9
|
| 42 | 1 | nmbdoplbi 28883 |
. . . . . . . . . 10
|
| 43 | 42 | adantr 481 |
. . . . . . . . 9
|
| 44 | 1re 10039 |
. . . . . . . . . . . 12
| |
| 45 | nmopge0 28770 |
. . . . . . . . . . . . . . 15
 | |
| 46 | 1, 6, 45 | mp2b 10 |
. . . . . . . . . . . . . 14
|
| 47 | 10, 46 | pm3.2i 471 |
. . . . . . . . . . . . 13
|
| 48 | lemul2a 10878 |
. . . . . . . . . . . . 13
| |
| 49 | 47, 48 | mp3anl3 1420 |
. . . . . . . . . . . 12
|
| 50 | 44, 49 | mpanl2 717 |
. . . . . . . . . . 11
|
| 51 | 38, 50 | sylan 488 |
. . . . . . . . . 10
|
| 52 | 10 | recni 10052 |
. . . . . . . . . . 11
 |
| 53 | 52 | mulid1i 10042 |
. . . . . . . . . 10
|
| 54 | 51, 53 | syl6breq 4694 |
. . . . . . . . 9
|
| 55 | 36, 41, 37, 43, 54 | letrd 10194 |
. . . . . . . 8
|
| 56 | nmopge0 28770 |
. . . . . . . . . . 11
| |
| 57 | 3, 4, 56 | mp2b 10 |
. . . . . . . . . 10
|
| 58 | 25, 57 | pm3.2i 471 |
. . . . . . . . 9
|
| 59 | lemul2a 10878 |
. . . . . . . . 9
| |
| 60 | 58, 59 | mp3anl3 1420 |
. . . . . . . 8
|
| 61 | 36, 37, 55, 60 | syl21anc 1325 |
. . . . . . 7
|
| 62 | 23, 30, 32, 35, 61 | letrd 10194 |
. . . . . 6
|
| 63 | 18, 62 | eqbrtrd 4675 |
. . . . 5
|
| 64 | 1 | nmopadji 28949 |
. . . . . . 7
|
| 65 | 64 | oveq1i 6660 |
. . . . . 6
|
| 66 | 52 | sqvali 12943 |
. . . . . 6
|
| 67 | 65, 66 | eqtr4i 2647 |
. . . . 5
|
| 68 | 63, 67 | syl6breq 4694 |
. . . 4
|
| 69 | 68 | ex 450 |
. . 3
|
| 70 | 15, 69 | mprgbir 2927 |
. 2
|
| 71 | nmopge0 28770 |
. . . . . . . 8
| |
| 72 | 8, 71 | ax-mp 5 |
. . . . . . 7
|
| 73 | 3, 1 | bdopcoi 28957 |
. . . . . . . . 9
|
| 74 | nmopre 28729 |
. . . . . . . . 9
| |
| 75 | 73, 74 | ax-mp 5 |
. . . . . . . 8
|
| 76 | 75 | sqrtcli 14111 |
. . . . . . 7
 |
| 77 | rexr 10085 |
. . . . . . 7
| |
| 78 | 72, 76, 77 | mp2b 10 |
. . . . . 6
|
| 79 | nmopub 28767 |
. . . . . 6
| |
| 80 | 7, 78, 79 | mp2an 708 |
. . . . 5
|
| 81 | 19, 20 | syl 17 |
. . . . . . . . . . . 12
|
| 82 | hicl 27937 |
. . . . . . . . . . . 12
 | |
| 83 | 81, 82 | mpancom 703 |
. . . . . . . . . . 11
|
| 84 | 83 | abscld 14175 |
. . . . . . . . . 10
 |
| 85 | 84 | adantr 481 |
. . . . . . . . 9
|
| 86 | 22, 38 | remulcld 10070 |
. . . . . . . . . 10
|
| 87 | 86 | adantr 481 |
. . . . . . . . 9
|
| 88 | 75 | a1i 11 |
. . . . . . . . 9
|
| 89 | bcs 28038 |
. . . . . . . . . . 11
| |
| 90 | 81, 89 | mpancom 703 |
. . . . . . . . . 10
|
| 91 | 90 | adantr 481 |
. . . . . . . . 9
|
| 92 | 5, 7 | hococli 28624 |
. . . . . . . . . . . 12
|
| 93 | normcl 27982 |
. . . . . . . . . . . 12
| |
| 94 | 92, 93 | syl 17 |
. . . . . . . . . . 11
|
| 95 | 94 | adantr 481 |
. . . . . . . . . 10
|
| 96 | 38 | adantr 481 |
. . . . . . . . . . . 12
|
| 97 | normge0 27983 |
. . . . . . . . . . . . . . 15
| |
| 98 | 19, 20, 97 | 3syl 18 |
. . . . . . . . . . . . . 14
|
| 99 | 22, 98 | jca 554 |
. . . . . . . . . . . . 13
|
| 100 | 99 | adantr 481 |
. . . . . . . . . . . 12
|
| 101 | simpr 477 |
. . . . . . . . . . . 12
| |
| 102 | lemul2a 10878 |
. . . . . . . . . . . . 13
| |
| 103 | 44, 102 | mp3anl2 1419 |
. . . . . . . . . . . 12
|
| 104 | 96, 100, 101, 103 | syl21anc 1325 |
. . . . . . . . . . 11
|
| 105 | 22 | recnd 10068 |
. . . . . . . . . . . . . 14
|
| 106 | 105 | mulid1d 10057 |
. . . . . . . . . . . . 13
|
| 107 | 106, 17 | eqtr4d 2659 |
. . . . . . . . . . . 12
|
| 108 | 107 | adantr 481 |
. . . . . . . . . . 11
|
| 109 | 104, 108 | breqtrd 4679 |
. . . . . . . . . 10
|
| 110 | remulcl 10021 |
. . . . . . . . . . . . 13
| |
| 111 | 75, 38, 110 | sylancr 695 |
. . . . . . . . . . . 12
|
| 112 | 111 | adantr 481 |
. . . . . . . . . . 11
|
| 113 | 73 | nmbdoplbi 28883 |
. . . . . . . . . . . 12
|
| 114 | 113 | adantr 481 |
. . . . . . . . . . 11
|
| 115 | 75, 72 | pm3.2i 471 |
. . . . . . . . . . . . . . 15
|
| 116 | lemul2a 10878 |
. . . . . . . . . . . . . . 15
| |
| 117 | 115, 116 | mp3anl3 1420 |
. . . . . . . . . . . . . 14
|
| 118 | 44, 117 | mpanl2 717 |
. . . . . . . . . . . . 13
|
| 119 | 38, 118 | sylan 488 |
. . . . . . . . . . . 12
|
| 120 | 75 | recni 10052 |
. . . . . . . . . . . . 13
|
| 121 | 120 | mulid1i 10042 |
. . . . . . . . . . . 12
|
| 122 | 119, 121 | syl6breq 4694 |
. . . . . . . . . . 11
|
| 123 | 95, 112, 88, 114, 122 | letrd 10194 |
. . . . . . . . . 10
|
| 124 | 87, 95, 88, 109, 123 | letrd 10194 |
. . . . . . . . 9
|
| 125 | 85, 87, 88, 91, 124 | letrd 10194 |
. . . . . . . 8
|
| 126 | resqcl 12931 |
. . . . . . . . . . . 12
| |
| 127 | sqge0 12940 |
. . . . . . . . . . . 12
| |
| 128 | 126, 127 | absidd 14161 |
. . . . . . . . . . 11
|
| 129 | 19, 26, 128 | 3syl 18 |
. . . . . . . . . 10
|
| 130 | normsq 27991 |
. . . . . . . . . . . . 13
| |
| 131 | 19, 130 | syl 17 |
. . . . . . . . . . . 12
|
| 132 | bdopadj 28941 |
. . . . . . . . . . . . . . . 16
 | |
| 133 | 3, 132 | ax-mp 5 |
. . . . . . . . . . . . . . 15
|
| 134 | adj2 28793 |
. . . . . . . . . . . . . . 15
| |
| 135 | 133, 134 | mp3an1 1411 |
. . . . . . . . . . . . . 14
|
| 136 | 19, 135 | mpancom 703 |
. . . . . . . . . . . . 13
|
| 137 | bdopadj 28941 |
. . . . . . . . . . . . . . . 16
| |
| 138 | adjadj 28795 |
. . . . . . . . . . . . . . . 16
| |
| 139 | 1, 137, 138 | mp2b 10 |
. . . . . . . . . . . . . . 15
|
| 140 | 139 | fveq1i 6192 |
. . . . . . . . . . . . . 14
|
| 141 | 140 | oveq2i 6661 |
. . . . . . . . . . . . 13
|
| 142 | 136, 141 | syl6req 2673 |
. . . . . . . . . . . 12
|
| 143 | 131, 142 | eqtrd 2656 |
. . . . . . . . . . 11
|
| 144 | 143 | fveq2d 6195 |
. . . . . . . . . 10
|
| 145 | 129, 144 | eqtr3d 2658 |
. . . . . . . . 9
|
| 146 | 145 | adantr 481 |
. . . . . . . 8
|
| 147 | 75 | sqsqrti 14115 |
. . . . . . . . . 10
|
| 148 | 8, 71, 147 | mp2b 10 |
. . . . . . . . 9
|
| 149 | 148 | a1i 11 |
. . . . . . . 8
|
| 150 | 125, 146, 149 | 3brtr4d 4685 |
. . . . . . 7
|
| 151 | normge0 27983 |
. . . . . . . . . 10
| |
| 152 | 19, 151 | syl 17 |
. . . . . . . . 9
|
| 153 | 8, 71, 76 | mp2b 10 |
. . . . . . . . . 10
|
| 154 | 75 | sqrtge0i 14116 |
. . . . . . . . . . 11
|
| 155 | 8, 71, 154 | mp2b 10 |
. . . . . . . . . 10
|
| 156 | le2sq 12938 |
. . . . . . . . . 10
| |
| 157 | 153, 155, 156 | mpanr12 721 |
. . . . . . . . 9
|
| 158 | 27, 152, 157 | syl2anc 693 |
. . . . . . . 8
|
| 159 | 158 | adantr 481 |
. . . . . . 7
|
| 160 | 150, 159 | mpbird 247 |
. . . . . 6
|
| 161 | 160 | ex 450 |
. . . . 5
|
| 162 | 80, 161 | mprgbir 2927 |
. . . 4
|
| 163 | 10, 153 | le2sqi 12953 |
. . . . 5
|
| 164 | 46, 155, 163 | mp2an 708 |
. . . 4
|
| 165 | 162, 164 | mpbi 220 |
. . 3
|
| 166 | 165, 148 | breqtri 4678 |
. 2
|
| 167 | 75, 11 | letri3i 10153 |
. 2
|
| 168 | 70, 166, 167 | mpbir2an 955 |
1
|
| Colors of variables: wff setvar class |
| Syntax hints: wi 4
wb 196 wa 384 wceq 1483 wcel 1990 wral 2912 class class class wbr 4653
cdm 5114
ccom 5118
wf 5884 cfv 5888 (class class class)co 6650
cc 9934
cr 9935
cc0 9936
c1 9937
cmul 9941
cxr 10073
cle 10075
c2 11070
cexp 12860
csqrt 13973 cabs 13974 chil 27776 csp 27779 cno 27780 cnop 27802
cbo 27805 cado 27812 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cc 9257 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 ax-hilex 27856 ax-hfvadd 27857 ax-hvcom 27858 ax-hvass 27859 ax-hv0cl 27860 ax-hvaddid 27861 ax-hfvmul 27862 ax-hvmulid 27863 ax-hvmulass 27864 ax-hvdistr1 27865 ax-hvdistr2 27866 ax-hvmul0 27867 ax-hfi 27936 ax-his1 27939 ax-his2 27940 ax-his3 27941 ax-his4 27942 ax-hcompl 28059 |
| This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-rlim 14220 df-sum 14417 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-cn 21031 df-cnp 21032 df-lm 21033 df-t1 21118 df-haus 21119 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cfil 23053 df-cau 23054 df-cmet 23055 df-grpo 27347 df-gid 27348 df-ginv 27349 df-gdiv 27350 df-ablo 27399 df-vc 27414 df-nv 27447 df-va 27450 df-ba 27451 df-sm 27452 df-0v 27453 df-vs 27454 df-nmcv 27455 df-ims 27456 df-dip 27556 df-ssp 27577 df-lno 27599 df-nmoo 27600 df-0o 27602 df-ph 27668 df-cbn 27719 df-hnorm 27825 df-hba 27826 df-hvsub 27828 df-hlim 27829 df-hcau 27830 df-sh 28064 df-ch 28078 df-oc 28109 df-ch0 28110 df-shs 28167 df-pjh 28254 df-h0op 28607 df-nmop 28698 df-cnop 28699 df-lnop 28700 df-bdop 28701 df-unop 28702 df-hmop 28703 df-nmfn 28704 df-nlfn 28705 df-cnfn 28706 df-lnfn 28707 df-adjh 28708 |
| This theorem is referenced by: nmopcoadj2i 28961 |
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