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Mirrors > Home > MPE Home > Th. List > nmval | Structured version Visualization version GIF version |
Description: The value of the norm as the distance to zero. Problem 1 of [Kreyszig] p. 63. (Contributed by NM, 4-Dec-2006.) (Revised by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
nmfval.n | β’ π = (normβπ) |
nmfval.x | β’ π = (Baseβπ) |
nmfval.z | β’ 0 = (0gβπ) |
nmfval.d | β’ π· = (distβπ) |
Ref | Expression |
---|---|
nmval | β’ (π΄ β π β (πβπ΄) = (π΄π· 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7369 | . 2 β’ (π₯ = π΄ β (π₯π· 0 ) = (π΄π· 0 )) | |
2 | nmfval.n | . . 3 β’ π = (normβπ) | |
3 | nmfval.x | . . 3 β’ π = (Baseβπ) | |
4 | nmfval.z | . . 3 β’ 0 = (0gβπ) | |
5 | nmfval.d | . . 3 β’ π· = (distβπ) | |
6 | 2, 3, 4, 5 | nmfval 23960 | . 2 β’ π = (π₯ β π β¦ (π₯π· 0 )) |
7 | ovex 7395 | . 2 β’ (π΄π· 0 ) β V | |
8 | 1, 6, 7 | fvmpt 6953 | 1 β’ (π΄ β π β (πβπ΄) = (π΄π· 0 )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 βcfv 6501 (class class class)co 7362 Basecbs 17090 distcds 17149 0gc0g 17328 normcnm 23948 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-fv 6509 df-ov 7365 df-nm 23954 |
This theorem is referenced by: nmval2 23964 ngpds2 23978 isngp4 23984 nmge0 23989 nmeq0 23990 nminv 23993 nmmtri 23994 nmrtri 23996 |
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