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Mirrors > Home > HSE Home > Th. List > nmlnop0iHIL | Structured version Visualization version GIF version |
Description: A linear operator with a zero norm is identically zero. (Contributed by NM, 18-Jan-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmlnop0.1 | ⊢ 𝑇 ∈ LinOp |
Ref | Expression |
---|---|
nmlnop0iHIL | ⊢ ((normop‘𝑇) = 0 ↔ 𝑇 = 0hop ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmlnop0.1 | . 2 ⊢ 𝑇 ∈ LinOp | |
2 | eqid 2772 | . . . 4 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
3 | eqid 2772 | . . . 4 ⊢ (〈〈 +ℎ , ·ℎ 〉, normℎ〉 normOpOLD 〈〈 +ℎ , ·ℎ 〉, normℎ〉) = (〈〈 +ℎ , ·ℎ 〉, normℎ〉 normOpOLD 〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
4 | 2, 3 | hhnmoi 29453 | . . 3 ⊢ normop = (〈〈 +ℎ , ·ℎ 〉, normℎ〉 normOpOLD 〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
5 | eqid 2772 | . . . 4 ⊢ (〈〈 +ℎ , ·ℎ 〉, normℎ〉 0op 〈〈 +ℎ , ·ℎ 〉, normℎ〉) = (〈〈 +ℎ , ·ℎ 〉, normℎ〉 0op 〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
6 | 2, 5 | hh0oi 29455 | . . 3 ⊢ 0hop = (〈〈 +ℎ , ·ℎ 〉, normℎ〉 0op 〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
7 | eqid 2772 | . . . 4 ⊢ (〈〈 +ℎ , ·ℎ 〉, normℎ〉 LnOp 〈〈 +ℎ , ·ℎ 〉, normℎ〉) = (〈〈 +ℎ , ·ℎ 〉, normℎ〉 LnOp 〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
8 | 2, 7 | hhlnoi 29452 | . . 3 ⊢ LinOp = (〈〈 +ℎ , ·ℎ 〉, normℎ〉 LnOp 〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
9 | 2 | hhnv 28715 | . . 3 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ NrmCVec |
10 | 4, 6, 8, 9, 9 | nmlno0i 28342 | . 2 ⊢ (𝑇 ∈ LinOp → ((normop‘𝑇) = 0 ↔ 𝑇 = 0hop )) |
11 | 1, 10 | ax-mp 5 | 1 ⊢ ((normop‘𝑇) = 0 ↔ 𝑇 = 0hop ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1507 ∈ wcel 2050 〈cop 4441 ‘cfv 6182 (class class class)co 6970 0cc0 10329 LnOp clno 28288 normOpOLD cnmoo 28289 0op c0o 28291 +ℎ cva 28470 ·ℎ csm 28471 normℎcno 28473 0hop ch0o 28493 normopcnop 28495 LinOpclo 28497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-inf2 8892 ax-cc 9649 ax-cnex 10385 ax-resscn 10386 ax-1cn 10387 ax-icn 10388 ax-addcl 10389 ax-addrcl 10390 ax-mulcl 10391 ax-mulrcl 10392 ax-mulcom 10393 ax-addass 10394 ax-mulass 10395 ax-distr 10396 ax-i2m1 10397 ax-1ne0 10398 ax-1rid 10399 ax-rnegex 10400 ax-rrecex 10401 ax-cnre 10402 ax-pre-lttri 10403 ax-pre-lttrn 10404 ax-pre-ltadd 10405 ax-pre-mulgt0 10406 ax-pre-sup 10407 ax-addf 10408 ax-mulf 10409 ax-hilex 28549 ax-hfvadd 28550 ax-hvcom 28551 ax-hvass 28552 ax-hv0cl 28553 ax-hvaddid 28554 ax-hfvmul 28555 ax-hvmulid 28556 ax-hvmulass 28557 ax-hvdistr1 28558 ax-hvdistr2 28559 ax-hvmul0 28560 ax-hfi 28629 ax-his1 28632 ax-his2 28633 ax-his3 28634 ax-his4 28635 ax-hcompl 28752 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-iin 4789 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-se 5361 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-isom 6191 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-of 7221 df-om 7391 df-1st 7495 df-2nd 7496 df-supp 7628 df-wrecs 7744 df-recs 7806 df-rdg 7844 df-1o 7899 df-2o 7900 df-oadd 7903 df-omul 7904 df-er 8083 df-map 8202 df-pm 8203 df-ixp 8254 df-en 8301 df-dom 8302 df-sdom 8303 df-fin 8304 df-fsupp 8623 df-fi 8664 df-sup 8695 df-inf 8696 df-oi 8763 df-card 9156 df-acn 9159 df-cda 9382 df-pnf 10470 df-mnf 10471 df-xr 10472 df-ltxr 10473 df-le 10474 df-sub 10666 df-neg 10667 df-div 11093 df-nn 11434 df-2 11497 df-3 11498 df-4 11499 df-5 11500 df-6 11501 df-7 11502 df-8 11503 df-9 11504 df-n0 11702 df-z 11788 df-dec 11906 df-uz 12053 df-q 12157 df-rp 12199 df-xneg 12318 df-xadd 12319 df-xmul 12320 df-ioo 12552 df-ico 12554 df-icc 12555 df-fz 12703 df-fzo 12844 df-fl 12971 df-seq 13179 df-exp 13239 df-hash 13500 df-cj 14313 df-re 14314 df-im 14315 df-sqrt 14449 df-abs 14450 df-clim 14700 df-rlim 14701 df-sum 14898 df-struct 16335 df-ndx 16336 df-slot 16337 df-base 16339 df-sets 16340 df-ress 16341 df-plusg 16428 df-mulr 16429 df-starv 16430 df-sca 16431 df-vsca 16432 df-ip 16433 df-tset 16434 df-ple 16435 df-ds 16437 df-unif 16438 df-hom 16439 df-cco 16440 df-rest 16546 df-topn 16547 df-0g 16565 df-gsum 16566 df-topgen 16567 df-pt 16568 df-prds 16571 df-xrs 16625 df-qtop 16630 df-imas 16631 df-xps 16633 df-mre 16709 df-mrc 16710 df-acs 16712 df-mgm 17704 df-sgrp 17746 df-mnd 17757 df-submnd 17798 df-mulg 18006 df-cntz 18212 df-cmn 18662 df-psmet 20233 df-xmet 20234 df-met 20235 df-bl 20236 df-mopn 20237 df-fbas 20238 df-fg 20239 df-cnfld 20242 df-top 21200 df-topon 21217 df-topsp 21239 df-bases 21252 df-cld 21325 df-ntr 21326 df-cls 21327 df-nei 21404 df-cn 21533 df-cnp 21534 df-lm 21535 df-haus 21621 df-tx 21868 df-hmeo 22061 df-fil 22152 df-fm 22244 df-flim 22245 df-flf 22246 df-xms 22627 df-ms 22628 df-tms 22629 df-cfil 23555 df-cau 23556 df-cmet 23557 df-grpo 28041 df-gid 28042 df-ginv 28043 df-gdiv 28044 df-ablo 28093 df-vc 28107 df-nv 28140 df-va 28143 df-ba 28144 df-sm 28145 df-0v 28146 df-vs 28147 df-nmcv 28148 df-ims 28149 df-dip 28249 df-ssp 28270 df-lno 28292 df-nmoo 28293 df-0o 28295 df-ph 28361 df-cbn 28412 df-hnorm 28518 df-hba 28519 df-hvsub 28521 df-hlim 28522 df-hcau 28523 df-sh 28757 df-ch 28771 df-oc 28802 df-ch0 28803 df-shs 28860 df-pjh 28947 df-h0op 29300 df-nmop 29391 df-lnop 29393 |
This theorem is referenced by: nmlnopgt0i 29549 nmlnop0 29550 lnopco0i 29556 nmopcoi 29647 nmopcoadj0i 29655 |
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