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Mirrors > Home > HSE Home > Th. List > bdophmi | Structured version Visualization version GIF version |
Description: The scalar product of a bounded linear operator is a bounded linear operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmophm.1 | โข ๐ โ BndLinOp |
Ref | Expression |
---|---|
bdophmi | โข (๐ด โ โ โ (๐ด ยทop ๐) โ BndLinOp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmophm.1 | . . . 4 โข ๐ โ BndLinOp | |
2 | bdopln 30852 | . . . 4 โข (๐ โ BndLinOp โ ๐ โ LinOp) | |
3 | 1, 2 | ax-mp 5 | . . 3 โข ๐ โ LinOp |
4 | 3 | lnopmi 30991 | . 2 โข (๐ด โ โ โ (๐ด ยทop ๐) โ LinOp) |
5 | 1 | nmophmi 31022 | . . 3 โข (๐ด โ โ โ (normopโ(๐ด ยทop ๐)) = ((absโ๐ด) ยท (normopโ๐))) |
6 | abscl 15172 | . . . 4 โข (๐ด โ โ โ (absโ๐ด) โ โ) | |
7 | nmopre 30861 | . . . . 5 โข (๐ โ BndLinOp โ (normopโ๐) โ โ) | |
8 | 1, 7 | ax-mp 5 | . . . 4 โข (normopโ๐) โ โ |
9 | remulcl 11144 | . . . 4 โข (((absโ๐ด) โ โ โง (normopโ๐) โ โ) โ ((absโ๐ด) ยท (normopโ๐)) โ โ) | |
10 | 6, 8, 9 | sylancl 587 | . . 3 โข (๐ด โ โ โ ((absโ๐ด) ยท (normopโ๐)) โ โ) |
11 | 5, 10 | eqeltrd 2834 | . 2 โข (๐ด โ โ โ (normopโ(๐ด ยทop ๐)) โ โ) |
12 | elbdop2 30862 | . 2 โข ((๐ด ยทop ๐) โ BndLinOp โ ((๐ด ยทop ๐) โ LinOp โง (normopโ(๐ด ยทop ๐)) โ โ)) | |
13 | 4, 11, 12 | sylanbrc 584 | 1 โข (๐ด โ โ โ (๐ด ยทop ๐) โ BndLinOp) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โ wcel 2107 โcfv 6500 (class class class)co 7361 โcc 11057 โcr 11058 ยท cmul 11064 abscabs 15128 ยทop chot 29930 normopcnop 29936 LinOpclo 29938 BndLinOpcbo 29939 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 ax-hilex 29990 ax-hfvadd 29991 ax-hvcom 29992 ax-hvass 29993 ax-hv0cl 29994 ax-hvaddid 29995 ax-hfvmul 29996 ax-hvmulid 29997 ax-hvmulass 29998 ax-hvdistr1 29999 ax-hvdistr2 30000 ax-hvmul0 30001 ax-hfi 30070 ax-his1 30073 ax-his2 30074 ax-his3 30075 ax-his4 30076 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-sup 9386 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-n0 12422 df-z 12508 df-uz 12772 df-rp 12924 df-seq 13916 df-exp 13977 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-grpo 29484 df-gid 29485 df-ablo 29536 df-vc 29550 df-nv 29583 df-va 29586 df-ba 29587 df-sm 29588 df-0v 29589 df-nmcv 29591 df-hnorm 29959 df-hba 29960 df-hvsub 29962 df-homul 30722 df-nmop 30830 df-lnop 30832 df-bdop 30833 |
This theorem is referenced by: bdophdi 31088 nmoptri2i 31090 |
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