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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 1134 | . . 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 30464 | . . . . 5 β’ ((π β NrmCVec β§ π΄ β π) β (πβπ΄) β β) |
5 | 4 | 3adant3 1130 | . . . 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 30472 | . . . . . 6 β’ ((π β NrmCVec β§ π΄ β π) β ((πβπ΄) = 0 β π΄ = π)) |
8 | 7 | necon3bid 2980 | . . . . 5 β’ ((π β NrmCVec β§ π΄ β π) β ((πβπ΄) β 0 β π΄ β π)) |
9 | 8 | biimp3ar 1467 | . . . 4 β’ ((π β NrmCVec β§ π΄ β π β§ π΄ β π) β (πβπ΄) β 0) |
10 | 5, 9 | rereccld 12065 | . . 3 β’ ((π β NrmCVec β§ π΄ β π β§ π΄ β π) β (1 / (πβπ΄)) β β) |
11 | 2, 6, 3 | nvgt0 30477 | . . . . 5 β’ ((π β NrmCVec β§ π΄ β π) β (π΄ β π β 0 < (πβπ΄))) |
12 | 11 | biimp3a 1466 | . . . 4 β’ ((π β NrmCVec β§ π΄ β π β§ π΄ β π) β 0 < (πβπ΄)) |
13 | 1re 11238 | . . . . 5 β’ 1 β β | |
14 | 0le1 11761 | . . . . 5 β’ 0 β€ 1 | |
15 | divge0 12107 | . . . . 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 701 | . . . 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 1135 | . . 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 30467 | . . 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 1373 | . 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 11266 | . . . 4 β’ ((π β NrmCVec β§ π΄ β π) β (πβπ΄) β β) |
23 | 22 | 3adant3 1130 | . . 3 β’ ((π β NrmCVec β§ π΄ β π β§ π΄ β π) β (πβπ΄) β β) |
24 | 23, 9 | recid2d 12010 | . 2 β’ ((π β NrmCVec β§ π΄ β π β§ π΄ β π) β ((1 / (πβπ΄)) Β· (πβπ΄)) = 1) |
25 | 21, 24 | eqtrd 2767 | 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 1085 = wceq 1534 β wcel 2099 β wne 2935 class class class wbr 5142 βcfv 6542 (class class class)co 7414 βcc 11130 βcr 11131 0cc0 11132 1c1 11133 Β· cmul 11137 < clt 11272 β€ cle 11273 / cdiv 11895 NrmCVeccnv 30387 BaseSetcba 30389 Β·π OLD cns 30390 0veccn0v 30391 normCVcnmcv 30393 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-sup 9459 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-n0 12497 df-z 12583 df-uz 12847 df-rp 13001 df-seq 13993 df-exp 14053 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-grpo 30296 df-gid 30297 df-ginv 30298 df-ablo 30348 df-vc 30362 df-nv 30395 df-va 30398 df-ba 30399 df-sm 30400 df-0v 30401 df-nmcv 30403 |
This theorem is referenced by: nmlno0lem 30596 nmblolbii 30602 |
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