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Mirrors > Home > MPE Home > Th. List > nvz0 | Structured version Visualization version GIF version |
Description: The norm of a zero vector is zero. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvz0.5 | β’ π = (0vecβπ) |
nvz0.6 | β’ π = (normCVβπ) |
Ref | Expression |
---|---|
nvz0 | β’ (π β NrmCVec β (πβπ) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2731 | . . . 4 β’ (BaseSetβπ) = (BaseSetβπ) | |
2 | nvz0.5 | . . . 4 β’ π = (0vecβπ) | |
3 | 1, 2 | nvzcl 30155 | . . 3 β’ (π β NrmCVec β π β (BaseSetβπ)) |
4 | 0re 11221 | . . . . 5 β’ 0 β β | |
5 | 0le0 12318 | . . . . 5 β’ 0 β€ 0 | |
6 | 4, 5 | pm3.2i 470 | . . . 4 β’ (0 β β β§ 0 β€ 0) |
7 | eqid 2731 | . . . . 5 β’ ( Β·π OLD βπ) = ( Β·π OLD βπ) | |
8 | nvz0.6 | . . . . 5 β’ π = (normCVβπ) | |
9 | 1, 7, 8 | nvsge0 30185 | . . . 4 β’ ((π β NrmCVec β§ (0 β β β§ 0 β€ 0) β§ π β (BaseSetβπ)) β (πβ(0( Β·π OLD βπ)π)) = (0 Β· (πβπ))) |
10 | 6, 9 | mp3an2 1448 | . . 3 β’ ((π β NrmCVec β§ π β (BaseSetβπ)) β (πβ(0( Β·π OLD βπ)π)) = (0 Β· (πβπ))) |
11 | 3, 10 | mpdan 684 | . 2 β’ (π β NrmCVec β (πβ(0( Β·π OLD βπ)π)) = (0 Β· (πβπ))) |
12 | 1, 7, 2 | nv0 30158 | . . . 4 β’ ((π β NrmCVec β§ π β (BaseSetβπ)) β (0( Β·π OLD βπ)π) = π) |
13 | 3, 12 | mpdan 684 | . . 3 β’ (π β NrmCVec β (0( Β·π OLD βπ)π) = π) |
14 | 13 | fveq2d 6895 | . 2 β’ (π β NrmCVec β (πβ(0( Β·π OLD βπ)π)) = (πβπ)) |
15 | 1, 8 | nvcl 30182 | . . . . 5 β’ ((π β NrmCVec β§ π β (BaseSetβπ)) β (πβπ) β β) |
16 | 15 | recnd 11247 | . . . 4 β’ ((π β NrmCVec β§ π β (BaseSetβπ)) β (πβπ) β β) |
17 | 3, 16 | mpdan 684 | . . 3 β’ (π β NrmCVec β (πβπ) β β) |
18 | 17 | mul02d 11417 | . 2 β’ (π β NrmCVec β (0 Β· (πβπ)) = 0) |
19 | 11, 14, 18 | 3eqtr3d 2779 | 1 β’ (π β NrmCVec β (πβπ) = 0) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1540 β wcel 2105 class class class wbr 5148 βcfv 6543 (class class class)co 7412 βcc 11112 βcr 11113 0cc0 11114 Β· cmul 11119 β€ cle 11254 NrmCVeccnv 30105 BaseSetcba 30107 Β·π OLD cns 30108 0veccn0v 30109 normCVcnmcv 30111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-sup 9441 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-n0 12478 df-z 12564 df-uz 12828 df-rp 12980 df-seq 13972 df-exp 14033 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-grpo 30014 df-gid 30015 df-ginv 30016 df-ablo 30066 df-vc 30080 df-nv 30113 df-va 30116 df-ba 30117 df-sm 30118 df-0v 30119 df-nmcv 30121 |
This theorem is referenced by: nvz 30190 nvge0 30194 ipidsq 30231 nmosetn0 30286 nmoo0 30312 nmlnoubi 30317 nmblolbii 30320 blocnilem 30325 |
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