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| Mirrors > Home > MPE Home > Th. List > nmlno0i | Structured version Visualization version GIF version | ||
| Description: The norm of a linear operator is zero iff the operator is zero. (Contributed by NM, 6-Dec-2007.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| nmlno0.3 | β’ π = (π normOpOLD π) |
| nmlno0.0 | β’ π = (π 0op π) |
| nmlno0.7 | β’ πΏ = (π LnOp π) |
| nmlno0i.u | β’ π β NrmCVec |
| nmlno0i.w | β’ π β NrmCVec |
| Ref | Expression |
|---|---|
| nmlno0i | β’ (π β πΏ β ((πβπ) = 0 β π = π)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fveqeq2 6899 | . . 3 β’ (π = if(π β πΏ, π, π) β ((πβπ) = 0 β (πβif(π β πΏ, π, π)) = 0)) | |
| 2 | eqeq1 2734 | . . 3 β’ (π = if(π β πΏ, π, π) β (π = π β if(π β πΏ, π, π) = π)) | |
| 3 | 1, 2 | bibi12d 344 | . 2 β’ (π = if(π β πΏ, π, π) β (((πβπ) = 0 β π = π) β ((πβif(π β πΏ, π, π)) = 0 β if(π β πΏ, π, π) = π))) |
| 4 | nmlno0.3 | . . 3 β’ π = (π normOpOLD π) | |
| 5 | nmlno0.0 | . . 3 β’ π = (π 0op π) | |
| 6 | nmlno0.7 | . . 3 β’ πΏ = (π LnOp π) | |
| 7 | nmlno0i.u | . . 3 β’ π β NrmCVec | |
| 8 | nmlno0i.w | . . 3 β’ π β NrmCVec | |
| 9 | 5, 6 | 0lno 30310 | . . . . 5 β’ ((π β NrmCVec β§ π β NrmCVec) β π β πΏ) |
| 10 | 7, 8, 9 | mp2an 688 | . . . 4 β’ π β πΏ |
| 11 | 10 | elimel 4596 | . . 3 β’ if(π β πΏ, π, π) β πΏ |
| 12 | eqid 2730 | . . 3 β’ (BaseSetβπ) = (BaseSetβπ) | |
| 13 | eqid 2730 | . . 3 β’ (BaseSetβπ) = (BaseSetβπ) | |
| 14 | eqid 2730 | . . 3 β’ ( Β·π OLD βπ) = ( Β·π OLD βπ) | |
| 15 | eqid 2730 | . . 3 β’ ( Β·π OLD βπ) = ( Β·π OLD βπ) | |
| 16 | eqid 2730 | . . 3 β’ (0vecβπ) = (0vecβπ) | |
| 17 | eqid 2730 | . . 3 β’ (0vecβπ) = (0vecβπ) | |
| 18 | eqid 2730 | . . 3 β’ (normCVβπ) = (normCVβπ) | |
| 19 | eqid 2730 | . . 3 β’ (normCVβπ) = (normCVβπ) | |
| 20 | 4, 5, 6, 7, 8, 11, 12, 13, 14, 15, 16, 17, 18, 19 | nmlno0lem 30313 | . 2 β’ ((πβif(π β πΏ, π, π)) = 0 β if(π β πΏ, π, π) = π) |
| 21 | 3, 20 | dedth 4585 | 1 β’ (π β πΏ β ((πβπ) = 0 β π = π)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 = wceq 1539 β wcel 2104 ifcif 4527 βcfv 6542 (class class class)co 7411 0cc0 11112 NrmCVeccnv 30104 BaseSetcba 30106 Β·π OLD cns 30107 0veccn0v 30108 normCVcnmcv 30110 LnOp clno 30260 normOpOLD cnmoo 30261 0op c0o 30263 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12979 df-seq 13971 df-exp 14032 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-grpo 30013 df-gid 30014 df-ginv 30015 df-ablo 30065 df-vc 30079 df-nv 30112 df-va 30115 df-ba 30116 df-sm 30117 df-0v 30118 df-nmcv 30120 df-lno 30264 df-nmoo 30265 df-0o 30267 |
| This theorem is referenced by: nmlno0 30315 nmlnop0iHIL 31516 |
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