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Mirrors > Home > MPE Home > Th. List > nv1 | Structured version Visualization version GIF version |
Description: From any nonzero vector, construct a vector whose norm is one. (Contributed by NM, 6-Dec-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nv1.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nv1.4 | ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) |
nv1.5 | ⊢ 𝑍 = (0vec‘𝑈) |
nv1.6 | ⊢ 𝑁 = (normCV‘𝑈) |
Ref | Expression |
---|---|
nv1 | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → (𝑁‘((1 / (𝑁‘𝐴))𝑆𝐴)) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1132 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → 𝑈 ∈ NrmCVec) | |
2 | nv1.1 | . . . . . 6 ⊢ 𝑋 = (BaseSet‘𝑈) | |
3 | nv1.6 | . . . . . 6 ⊢ 𝑁 = (normCV‘𝑈) | |
4 | 2, 3 | nvcl 28441 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℝ) |
5 | 4 | 3adant3 1128 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → (𝑁‘𝐴) ∈ ℝ) |
6 | nv1.5 | . . . . . . 7 ⊢ 𝑍 = (0vec‘𝑈) | |
7 | 2, 6, 3 | nvz 28449 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴) = 0 ↔ 𝐴 = 𝑍)) |
8 | 7 | necon3bid 3063 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴) ≠ 0 ↔ 𝐴 ≠ 𝑍)) |
9 | 8 | biimp3ar 1466 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → (𝑁‘𝐴) ≠ 0) |
10 | 5, 9 | rereccld 11470 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → (1 / (𝑁‘𝐴)) ∈ ℝ) |
11 | 2, 6, 3 | nvgt0 28454 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴 ≠ 𝑍 ↔ 0 < (𝑁‘𝐴))) |
12 | 11 | biimp3a 1465 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → 0 < (𝑁‘𝐴)) |
13 | 1re 10644 | . . . . 5 ⊢ 1 ∈ ℝ | |
14 | 0le1 11166 | . . . . 5 ⊢ 0 ≤ 1 | |
15 | divge0 11512 | . . . . 5 ⊢ (((1 ∈ ℝ ∧ 0 ≤ 1) ∧ ((𝑁‘𝐴) ∈ ℝ ∧ 0 < (𝑁‘𝐴))) → 0 ≤ (1 / (𝑁‘𝐴))) | |
16 | 13, 14, 15 | mpanl12 700 | . . . 4 ⊢ (((𝑁‘𝐴) ∈ ℝ ∧ 0 < (𝑁‘𝐴)) → 0 ≤ (1 / (𝑁‘𝐴))) |
17 | 5, 12, 16 | syl2anc 586 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → 0 ≤ (1 / (𝑁‘𝐴))) |
18 | simp2 1133 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → 𝐴 ∈ 𝑋) | |
19 | nv1.4 | . . . 4 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
20 | 2, 19, 3 | nvsge0 28444 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ ((1 / (𝑁‘𝐴)) ∈ ℝ ∧ 0 ≤ (1 / (𝑁‘𝐴))) ∧ 𝐴 ∈ 𝑋) → (𝑁‘((1 / (𝑁‘𝐴))𝑆𝐴)) = ((1 / (𝑁‘𝐴)) · (𝑁‘𝐴))) |
21 | 1, 10, 17, 18, 20 | syl121anc 1371 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → (𝑁‘((1 / (𝑁‘𝐴))𝑆𝐴)) = ((1 / (𝑁‘𝐴)) · (𝑁‘𝐴))) |
22 | 4 | recnd 10672 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℂ) |
23 | 22 | 3adant3 1128 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → (𝑁‘𝐴) ∈ ℂ) |
24 | 23, 9 | recid2d 11415 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → ((1 / (𝑁‘𝐴)) · (𝑁‘𝐴)) = 1) |
25 | 21, 24 | eqtrd 2859 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → (𝑁‘((1 / (𝑁‘𝐴))𝑆𝐴)) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 ℂcc 10538 ℝcr 10539 0cc0 10540 1c1 10541 · cmul 10545 < clt 10678 ≤ cle 10679 / cdiv 11300 NrmCVeccnv 28364 BaseSetcba 28366 ·𝑠OLD cns 28367 0veccn0v 28368 normCVcnmcv 28370 |
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 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-sup 8909 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-seq 13373 df-exp 13433 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-grpo 28273 df-gid 28274 df-ginv 28275 df-ablo 28325 df-vc 28339 df-nv 28372 df-va 28375 df-ba 28376 df-sm 28377 df-0v 28378 df-nmcv 28380 |
This theorem is referenced by: nmlno0lem 28573 nmblolbii 28579 |
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