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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 | ⊢ 𝑍 = (0vec‘𝑈) |
nvz0.6 | ⊢ 𝑁 = (normCV‘𝑈) |
Ref | Expression |
---|---|
nvz0 | ⊢ (𝑈 ∈ NrmCVec → (𝑁‘𝑍) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈) | |
2 | nvz0.5 | . . . 4 ⊢ 𝑍 = (0vec‘𝑈) | |
3 | 1, 2 | nvzcl 28405 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑍 ∈ (BaseSet‘𝑈)) |
4 | 0re 10637 | . . . . 5 ⊢ 0 ∈ ℝ | |
5 | 0le0 11732 | . . . . 5 ⊢ 0 ≤ 0 | |
6 | 4, 5 | pm3.2i 473 | . . . 4 ⊢ (0 ∈ ℝ ∧ 0 ≤ 0) |
7 | eqid 2821 | . . . . 5 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
8 | nvz0.6 | . . . . 5 ⊢ 𝑁 = (normCV‘𝑈) | |
9 | 1, 7, 8 | nvsge0 28435 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (0 ∈ ℝ ∧ 0 ≤ 0) ∧ 𝑍 ∈ (BaseSet‘𝑈)) → (𝑁‘(0( ·𝑠OLD ‘𝑈)𝑍)) = (0 · (𝑁‘𝑍))) |
10 | 6, 9 | mp3an2 1445 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑍 ∈ (BaseSet‘𝑈)) → (𝑁‘(0( ·𝑠OLD ‘𝑈)𝑍)) = (0 · (𝑁‘𝑍))) |
11 | 3, 10 | mpdan 685 | . 2 ⊢ (𝑈 ∈ NrmCVec → (𝑁‘(0( ·𝑠OLD ‘𝑈)𝑍)) = (0 · (𝑁‘𝑍))) |
12 | 1, 7, 2 | nv0 28408 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑍 ∈ (BaseSet‘𝑈)) → (0( ·𝑠OLD ‘𝑈)𝑍) = 𝑍) |
13 | 3, 12 | mpdan 685 | . . 3 ⊢ (𝑈 ∈ NrmCVec → (0( ·𝑠OLD ‘𝑈)𝑍) = 𝑍) |
14 | 13 | fveq2d 6668 | . 2 ⊢ (𝑈 ∈ NrmCVec → (𝑁‘(0( ·𝑠OLD ‘𝑈)𝑍)) = (𝑁‘𝑍)) |
15 | 1, 8 | nvcl 28432 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑍 ∈ (BaseSet‘𝑈)) → (𝑁‘𝑍) ∈ ℝ) |
16 | 15 | recnd 10663 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑍 ∈ (BaseSet‘𝑈)) → (𝑁‘𝑍) ∈ ℂ) |
17 | 3, 16 | mpdan 685 | . . 3 ⊢ (𝑈 ∈ NrmCVec → (𝑁‘𝑍) ∈ ℂ) |
18 | 17 | mul02d 10832 | . 2 ⊢ (𝑈 ∈ NrmCVec → (0 · (𝑁‘𝑍)) = 0) |
19 | 11, 14, 18 | 3eqtr3d 2864 | 1 ⊢ (𝑈 ∈ NrmCVec → (𝑁‘𝑍) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 ℝcr 10530 0cc0 10531 · cmul 10536 ≤ cle 10670 NrmCVeccnv 28355 BaseSetcba 28357 ·𝑠OLD cns 28358 0veccn0v 28359 normCVcnmcv 28361 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-grpo 28264 df-gid 28265 df-ginv 28266 df-ablo 28316 df-vc 28330 df-nv 28363 df-va 28366 df-ba 28367 df-sm 28368 df-0v 28369 df-nmcv 28371 |
This theorem is referenced by: nvz 28440 nvge0 28444 ipidsq 28481 nmosetn0 28536 nmoo0 28562 nmlnoubi 28567 nmblolbii 28570 blocnilem 28575 |
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