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Mirrors > Home > HSE Home > Th. List > nmopun | Structured version Visualization version Unicode version |
Description: Norm of a unitary Hilbert space operator. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmopun |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unoplin 28779 | . . . . 5 | |
2 | lnopf 28718 | . . . . 5 | |
3 | 1, 2 | syl 17 | . . . 4 |
4 | nmopval 28715 | . . . 4 | |
5 | 3, 4 | syl 17 | . . 3 |
6 | 5 | adantl 482 | . 2 |
7 | nmopsetretHIL 28723 | . . . . . . 7 | |
8 | ressxr 10083 | . . . . . . 7 | |
9 | 7, 8 | syl6ss 3615 | . . . . . 6 |
10 | 3, 9 | syl 17 | . . . . 5 |
11 | 10 | adantl 482 | . . . 4 |
12 | 1re 10039 | . . . . 5 | |
13 | 12 | rexri 10097 | . . . 4 |
14 | 11, 13 | jctir 561 | . . 3 |
15 | vex 3203 | . . . . . . 7 | |
16 | eqeq1 2626 | . . . . . . . . 9 | |
17 | 16 | anbi2d 740 | . . . . . . . 8 |
18 | 17 | rexbidv 3052 | . . . . . . 7 |
19 | 15, 18 | elab 3350 | . . . . . 6 |
20 | unopnorm 28776 | . . . . . . . . . . 11 | |
21 | 20 | eqeq2d 2632 | . . . . . . . . . 10 |
22 | 21 | anbi2d 740 | . . . . . . . . 9 |
23 | breq1 4656 | . . . . . . . . . 10 | |
24 | 23 | biimparc 504 | . . . . . . . . 9 |
25 | 22, 24 | syl6bi 243 | . . . . . . . 8 |
26 | 25 | rexlimdva 3031 | . . . . . . 7 |
27 | 26 | imp 445 | . . . . . 6 |
28 | 19, 27 | sylan2b 492 | . . . . 5 |
29 | 28 | ralrimiva 2966 | . . . 4 |
30 | 29 | adantl 482 | . . 3 |
31 | hne0 28406 | . . . . . . . . . . 11 | |
32 | norm1hex 28108 | . . . . . . . . . . 11 | |
33 | 31, 32 | sylbb 209 | . . . . . . . . . 10 |
34 | 33 | adantr 481 | . . . . . . . . 9 |
35 | 1le1 10655 | . . . . . . . . . . . . . 14 | |
36 | breq1 4656 | . . . . . . . . . . . . . 14 | |
37 | 35, 36 | mpbiri 248 | . . . . . . . . . . . . 13 |
38 | 37 | a1i 11 | . . . . . . . . . . . 12 |
39 | 20 | adantr 481 | . . . . . . . . . . . . . . 15 |
40 | eqeq2 2633 | . . . . . . . . . . . . . . . 16 | |
41 | 40 | adantl 482 | . . . . . . . . . . . . . . 15 |
42 | 39, 41 | mpbid 222 | . . . . . . . . . . . . . 14 |
43 | 42 | eqcomd 2628 | . . . . . . . . . . . . 13 |
44 | 43 | ex 450 | . . . . . . . . . . . 12 |
45 | 38, 44 | jcad 555 | . . . . . . . . . . 11 |
46 | 45 | adantll 750 | . . . . . . . . . 10 |
47 | 46 | reximdva 3017 | . . . . . . . . 9 |
48 | 34, 47 | mpd 15 | . . . . . . . 8 |
49 | 1ex 10035 | . . . . . . . . 9 | |
50 | eqeq1 2626 | . . . . . . . . . . 11 | |
51 | 50 | anbi2d 740 | . . . . . . . . . 10 |
52 | 51 | rexbidv 3052 | . . . . . . . . 9 |
53 | 49, 52 | elab 3350 | . . . . . . . 8 |
54 | 48, 53 | sylibr 224 | . . . . . . 7 |
55 | 54 | adantr 481 | . . . . . 6 |
56 | breq2 4657 | . . . . . . 7 | |
57 | 56 | rspcev 3309 | . . . . . 6 |
58 | 55, 57 | sylan 488 | . . . . 5 |
59 | 58 | ex 450 | . . . 4 |
60 | 59 | ralrimiva 2966 | . . 3 |
61 | supxr2 12144 | . . 3 | |
62 | 14, 30, 60, 61 | syl12anc 1324 | . 2 |
63 | 6, 62 | eqtrd 2656 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 cab 2608 wne 2794 wral 2912 wrex 2913 wss 3574 class class class wbr 4653 wf 5884 cfv 5888 csup 8346 cr 9935 c1 9937 cxr 10073 clt 10074 cle 10075 chil 27776 cno 27780 c0v 27781 c0h 27792 cnop 27802 clo 27804 cuo 27806 |
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-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-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 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 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-iun 4522 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-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-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 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-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-grpo 27347 df-gid 27348 df-ablo 27399 df-vc 27414 df-nv 27447 df-va 27450 df-ba 27451 df-sm 27452 df-0v 27453 df-nmcv 27455 df-hnorm 27825 df-hba 27826 df-hvsub 27828 df-hlim 27829 df-sh 28064 df-ch 28078 df-ch0 28110 df-nmop 28698 df-lnop 28700 df-unop 28702 |
This theorem is referenced by: unopbd 28874 unierri 28963 |
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