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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 1137 | . . 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 29879 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℝ) |
5 | 4 | 3adant3 1133 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → (𝑁‘𝐴) ∈ ℝ) |
6 | nv1.5 | . . . . . . 7 ⊢ 𝑍 = (0vec‘𝑈) | |
7 | 2, 6, 3 | nvz 29887 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴) = 0 ↔ 𝐴 = 𝑍)) |
8 | 7 | necon3bid 2986 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴) ≠ 0 ↔ 𝐴 ≠ 𝑍)) |
9 | 8 | biimp3ar 1471 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → (𝑁‘𝐴) ≠ 0) |
10 | 5, 9 | rereccld 12028 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → (1 / (𝑁‘𝐴)) ∈ ℝ) |
11 | 2, 6, 3 | nvgt0 29892 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴 ≠ 𝑍 ↔ 0 < (𝑁‘𝐴))) |
12 | 11 | biimp3a 1470 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → 0 < (𝑁‘𝐴)) |
13 | 1re 11201 | . . . . 5 ⊢ 1 ∈ ℝ | |
14 | 0le1 11724 | . . . . 5 ⊢ 0 ≤ 1 | |
15 | divge0 12070 | . . . . 5 ⊢ (((1 ∈ ℝ ∧ 0 ≤ 1) ∧ ((𝑁‘𝐴) ∈ ℝ ∧ 0 < (𝑁‘𝐴))) → 0 ≤ (1 / (𝑁‘𝐴))) | |
16 | 13, 14, 15 | mpanl12 701 | . . . 4 ⊢ (((𝑁‘𝐴) ∈ ℝ ∧ 0 < (𝑁‘𝐴)) → 0 ≤ (1 / (𝑁‘𝐴))) |
17 | 5, 12, 16 | syl2anc 585 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → 0 ≤ (1 / (𝑁‘𝐴))) |
18 | simp2 1138 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → 𝐴 ∈ 𝑋) | |
19 | nv1.4 | . . . 4 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
20 | 2, 19, 3 | nvsge0 29882 | . . 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 1376 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → (𝑁‘((1 / (𝑁‘𝐴))𝑆𝐴)) = ((1 / (𝑁‘𝐴)) · (𝑁‘𝐴))) |
22 | 4 | recnd 11229 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℂ) |
23 | 22 | 3adant3 1133 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → (𝑁‘𝐴) ∈ ℂ) |
24 | 23, 9 | recid2d 11973 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → ((1 / (𝑁‘𝐴)) · (𝑁‘𝐴)) = 1) |
25 | 21, 24 | eqtrd 2773 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → (𝑁‘((1 / (𝑁‘𝐴))𝑆𝐴)) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 class class class wbr 5144 ‘cfv 6535 (class class class)co 7396 ℂcc 11095 ℝcr 11096 0cc0 11097 1c1 11098 · cmul 11102 < clt 11235 ≤ cle 11236 / cdiv 11858 NrmCVeccnv 29802 BaseSetcba 29804 ·𝑠OLD cns 29805 0veccn0v 29806 normCVcnmcv 29808 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 ax-pre-sup 11175 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-1st 7962 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-er 8691 df-en 8928 df-dom 8929 df-sdom 8930 df-sup 9424 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-div 11859 df-nn 12200 df-2 12262 df-3 12263 df-n0 12460 df-z 12546 df-uz 12810 df-rp 12962 df-seq 13954 df-exp 14015 df-cj 15033 df-re 15034 df-im 15035 df-sqrt 15169 df-abs 15170 df-grpo 29711 df-gid 29712 df-ginv 29713 df-ablo 29763 df-vc 29777 df-nv 29810 df-va 29813 df-ba 29814 df-sm 29815 df-0v 29816 df-nmcv 29818 |
This theorem is referenced by: nmlno0lem 30011 nmblolbii 30017 |
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