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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 1054 | . . 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 26900 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℝ) |
5 | 4 | 3adant3 1074 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → (𝑁‘𝐴) ∈ ℝ) |
6 | nv1.5 | . . . . . . 7 ⊢ 𝑍 = (0vec‘𝑈) | |
7 | 2, 6, 3 | nvz 26908 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴) = 0 ↔ 𝐴 = 𝑍)) |
8 | 7 | necon3bid 2826 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴) ≠ 0 ↔ 𝐴 ≠ 𝑍)) |
9 | 8 | biimp3ar 1425 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → (𝑁‘𝐴) ≠ 0) |
10 | 5, 9 | rereccld 10731 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → (1 / (𝑁‘𝐴)) ∈ ℝ) |
11 | 2, 6, 3 | nvgt0 26913 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴 ≠ 𝑍 ↔ 0 < (𝑁‘𝐴))) |
12 | 11 | biimp3a 1424 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → 0 < (𝑁‘𝐴)) |
13 | 1re 9918 | . . . . 5 ⊢ 1 ∈ ℝ | |
14 | 0le1 10430 | . . . . 5 ⊢ 0 ≤ 1 | |
15 | divge0 10771 | . . . . 5 ⊢ (((1 ∈ ℝ ∧ 0 ≤ 1) ∧ ((𝑁‘𝐴) ∈ ℝ ∧ 0 < (𝑁‘𝐴))) → 0 ≤ (1 / (𝑁‘𝐴))) | |
16 | 13, 14, 15 | mpanl12 714 | . . . 4 ⊢ (((𝑁‘𝐴) ∈ ℝ ∧ 0 < (𝑁‘𝐴)) → 0 ≤ (1 / (𝑁‘𝐴))) |
17 | 5, 12, 16 | syl2anc 691 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → 0 ≤ (1 / (𝑁‘𝐴))) |
18 | simp2 1055 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → 𝐴 ∈ 𝑋) | |
19 | nv1.4 | . . . 4 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
20 | 2, 19, 3 | nvsge0 26903 | . . 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 1323 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → (𝑁‘((1 / (𝑁‘𝐴))𝑆𝐴)) = ((1 / (𝑁‘𝐴)) · (𝑁‘𝐴))) |
22 | 4 | recnd 9947 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℂ) |
23 | 22 | 3adant3 1074 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → (𝑁‘𝐴) ∈ ℂ) |
24 | 23, 9 | recid2d 10676 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → ((1 / (𝑁‘𝐴)) · (𝑁‘𝐴)) = 1) |
25 | 21, 24 | eqtrd 2644 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → (𝑁‘((1 / (𝑁‘𝐴))𝑆𝐴)) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 · cmul 9820 < clt 9953 ≤ cle 9954 / cdiv 10563 NrmCVeccnv 26823 BaseSetcba 26825 ·𝑠OLD cns 26826 0veccn0v 26827 normCVcnmcv 26829 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-grpo 26731 df-gid 26732 df-ginv 26733 df-ablo 26783 df-vc 26798 df-nv 26831 df-va 26834 df-ba 26835 df-sm 26836 df-0v 26837 df-nmcv 26839 |
This theorem is referenced by: nmlno0lem 27032 nmblolbii 27038 |
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