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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 | β’ π = (BaseSetβπ) |
nv1.4 | β’ π = ( Β·π OLD βπ) |
nv1.5 | β’ π = (0vecβπ) |
nv1.6 | β’ π = (normCVβπ) |
Ref | Expression |
---|---|
nv1 | β’ ((π β NrmCVec β§ π΄ β π β§ π΄ β π) β (πβ((1 / (πβπ΄))ππ΄)) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1133 | . . 3 β’ ((π β NrmCVec β§ π΄ β π β§ π΄ β π) β π β NrmCVec) | |
2 | nv1.1 | . . . . . 6 β’ π = (BaseSetβπ) | |
3 | nv1.6 | . . . . . 6 β’ π = (normCVβπ) | |
4 | 2, 3 | nvcl 30383 | . . . . 5 β’ ((π β NrmCVec β§ π΄ β π) β (πβπ΄) β β) |
5 | 4 | 3adant3 1129 | . . . 4 β’ ((π β NrmCVec β§ π΄ β π β§ π΄ β π) β (πβπ΄) β β) |
6 | nv1.5 | . . . . . . 7 β’ π = (0vecβπ) | |
7 | 2, 6, 3 | nvz 30391 | . . . . . 6 β’ ((π β NrmCVec β§ π΄ β π) β ((πβπ΄) = 0 β π΄ = π)) |
8 | 7 | necon3bid 2977 | . . . . 5 β’ ((π β NrmCVec β§ π΄ β π) β ((πβπ΄) β 0 β π΄ β π)) |
9 | 8 | biimp3ar 1466 | . . . 4 β’ ((π β NrmCVec β§ π΄ β π β§ π΄ β π) β (πβπ΄) β 0) |
10 | 5, 9 | rereccld 12038 | . . 3 β’ ((π β NrmCVec β§ π΄ β π β§ π΄ β π) β (1 / (πβπ΄)) β β) |
11 | 2, 6, 3 | nvgt0 30396 | . . . . 5 β’ ((π β NrmCVec β§ π΄ β π) β (π΄ β π β 0 < (πβπ΄))) |
12 | 11 | biimp3a 1465 | . . . 4 β’ ((π β NrmCVec β§ π΄ β π β§ π΄ β π) β 0 < (πβπ΄)) |
13 | 1re 11211 | . . . . 5 β’ 1 β β | |
14 | 0le1 11734 | . . . . 5 β’ 0 β€ 1 | |
15 | divge0 12080 | . . . . 5 β’ (((1 β β β§ 0 β€ 1) β§ ((πβπ΄) β β β§ 0 < (πβπ΄))) β 0 β€ (1 / (πβπ΄))) | |
16 | 13, 14, 15 | mpanl12 699 | . . . 4 β’ (((πβπ΄) β β β§ 0 < (πβπ΄)) β 0 β€ (1 / (πβπ΄))) |
17 | 5, 12, 16 | syl2anc 583 | . . 3 β’ ((π β NrmCVec β§ π΄ β π β§ π΄ β π) β 0 β€ (1 / (πβπ΄))) |
18 | simp2 1134 | . . 3 β’ ((π β NrmCVec β§ π΄ β π β§ π΄ β π) β π΄ β π) | |
19 | nv1.4 | . . . 4 β’ π = ( Β·π OLD βπ) | |
20 | 2, 19, 3 | nvsge0 30386 | . . 3 β’ ((π β NrmCVec β§ ((1 / (πβπ΄)) β β β§ 0 β€ (1 / (πβπ΄))) β§ π΄ β π) β (πβ((1 / (πβπ΄))ππ΄)) = ((1 / (πβπ΄)) Β· (πβπ΄))) |
21 | 1, 10, 17, 18, 20 | syl121anc 1372 | . 2 β’ ((π β NrmCVec β§ π΄ β π β§ π΄ β π) β (πβ((1 / (πβπ΄))ππ΄)) = ((1 / (πβπ΄)) Β· (πβπ΄))) |
22 | 4 | recnd 11239 | . . . 4 β’ ((π β NrmCVec β§ π΄ β π) β (πβπ΄) β β) |
23 | 22 | 3adant3 1129 | . . 3 β’ ((π β NrmCVec β§ π΄ β π β§ π΄ β π) β (πβπ΄) β β) |
24 | 23, 9 | recid2d 11983 | . 2 β’ ((π β NrmCVec β§ π΄ β π β§ π΄ β π) β ((1 / (πβπ΄)) Β· (πβπ΄)) = 1) |
25 | 21, 24 | eqtrd 2764 | 1 β’ ((π β NrmCVec β§ π΄ β π β§ π΄ β π) β (πβ((1 / (πβπ΄))ππ΄)) = 1) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2932 class class class wbr 5138 βcfv 6533 (class class class)co 7401 βcc 11104 βcr 11105 0cc0 11106 1c1 11107 Β· cmul 11111 < clt 11245 β€ cle 11246 / cdiv 11868 NrmCVeccnv 30306 BaseSetcba 30308 Β·π OLD cns 30309 0veccn0v 30310 normCVcnmcv 30312 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-n0 12470 df-z 12556 df-uz 12820 df-rp 12972 df-seq 13964 df-exp 14025 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-grpo 30215 df-gid 30216 df-ginv 30217 df-ablo 30267 df-vc 30281 df-nv 30314 df-va 30317 df-ba 30318 df-sm 30319 df-0v 30320 df-nmcv 30322 |
This theorem is referenced by: nmlno0lem 30515 nmblolbii 30521 |
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