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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 24499 proved from this theorem and grpidcl 18921) or more generally monoids (see mndidcl 18708), or pointed sets). (Contributed by Mario Carneiro, 2-Oct-2015.) Extract this result from the proof of nmfval2 24499. (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 24496 | . 2 β’ π = (π₯ β π β¦ (π₯π· 0 )) |
6 | nmfval0.e | . . . . 5 β’ πΈ = (π· βΎ (π Γ π)) | |
7 | 6 | oveqi 7433 | . . . 4 β’ (π₯πΈ 0 ) = (π₯(π· βΎ (π Γ π)) 0 ) |
8 | ovres 7587 | . . . . 5 β’ ((π₯ β π β§ 0 β π) β (π₯(π· βΎ (π Γ π)) 0 ) = (π₯π· 0 )) | |
9 | 8 | ancoms 458 | . . . 4 β’ (( 0 β π β§ π₯ β π) β (π₯(π· βΎ (π Γ π)) 0 ) = (π₯π· 0 )) |
10 | 7, 9 | eqtr2id 2781 | . . 3 β’ (( 0 β π β§ π₯ β π) β (π₯π· 0 ) = (π₯πΈ 0 )) |
11 | 10 | mpteq2dva 5248 | . 2 β’ ( 0 β π β (π₯ β π β¦ (π₯π· 0 )) = (π₯ β π β¦ (π₯πΈ 0 ))) |
12 | 5, 11 | eqtrid 2780 | 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 395 = wceq 1534 β wcel 2099 β¦ cmpt 5231 Γ cxp 5676 βΎ cres 5680 βcfv 6548 (class class class)co 7420 Basecbs 17179 distcds 17241 0gc0g 17420 normcnm 24484 |
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 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fv 6556 df-ov 7423 df-nm 24490 |
This theorem is referenced by: nmfval2 24499 |
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