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Mirrors > Home > MPE Home > Th. List > nmfval0 | Structured version Visualization version GIF version |
Description: The value of the norm function on a structure containing a zero as the distance restricted to the elements of the base set to zero. Examples of structures containing a "zero" are groups (see nmfval2 23970 proved from this theorem and grpidcl 18786) or more generally monoids (see mndidcl 18579), or pointed sets). (Contributed by Mario Carneiro, 2-Oct-2015.) Extract this result from the proof of nmfval2 23970. (Revised by BJ, 27-Aug-2024.) |
Ref | Expression |
---|---|
nmfval0.n | β’ π = (normβπ) |
nmfval0.x | β’ π = (Baseβπ) |
nmfval0.z | β’ 0 = (0gβπ) |
nmfval0.d | β’ π· = (distβπ) |
nmfval0.e | β’ πΈ = (π· βΎ (π Γ π)) |
Ref | Expression |
---|---|
nmfval0 | β’ ( 0 β π β π = (π₯ β π β¦ (π₯πΈ 0 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmfval0.n | . . 3 β’ π = (normβπ) | |
2 | nmfval0.x | . . 3 β’ π = (Baseβπ) | |
3 | nmfval0.z | . . 3 β’ 0 = (0gβπ) | |
4 | nmfval0.d | . . 3 β’ π· = (distβπ) | |
5 | 1, 2, 3, 4 | nmfval 23967 | . 2 β’ π = (π₯ β π β¦ (π₯π· 0 )) |
6 | nmfval0.e | . . . . 5 β’ πΈ = (π· βΎ (π Γ π)) | |
7 | 6 | oveqi 7374 | . . . 4 β’ (π₯πΈ 0 ) = (π₯(π· βΎ (π Γ π)) 0 ) |
8 | ovres 7524 | . . . . 5 β’ ((π₯ β π β§ 0 β π) β (π₯(π· βΎ (π Γ π)) 0 ) = (π₯π· 0 )) | |
9 | 8 | ancoms 460 | . . . 4 β’ (( 0 β π β§ π₯ β π) β (π₯(π· βΎ (π Γ π)) 0 ) = (π₯π· 0 )) |
10 | 7, 9 | eqtr2id 2786 | . . 3 β’ (( 0 β π β§ π₯ β π) β (π₯π· 0 ) = (π₯πΈ 0 )) |
11 | 10 | mpteq2dva 5209 | . 2 β’ ( 0 β π β (π₯ β π β¦ (π₯π· 0 )) = (π₯ β π β¦ (π₯πΈ 0 ))) |
12 | 5, 11 | eqtrid 2785 | 1 β’ ( 0 β π β π = (π₯ β π β¦ (π₯πΈ 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β¦ cmpt 5192 Γ cxp 5635 βΎ cres 5639 βcfv 6500 (class class class)co 7361 Basecbs 17091 distcds 17150 0gc0g 17329 normcnm 23955 |
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-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-fv 6508 df-ov 7364 df-nm 23961 |
This theorem is referenced by: nmfval2 23970 |
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