![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 24444 proved from this theorem and grpidcl 18891) or more generally monoids (see mndidcl 18678), or pointed sets). (Contributed by Mario Carneiro, 2-Oct-2015.) Extract this result from the proof of nmfval2 24444. (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 24441 | . 2 β’ π = (π₯ β π β¦ (π₯π· 0 )) |
6 | nmfval0.e | . . . . 5 β’ πΈ = (π· βΎ (π Γ π)) | |
7 | 6 | oveqi 7415 | . . . 4 β’ (π₯πΈ 0 ) = (π₯(π· βΎ (π Γ π)) 0 ) |
8 | ovres 7567 | . . . . 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 2777 | . . 3 β’ (( 0 β π β§ π₯ β π) β (π₯π· 0 ) = (π₯πΈ 0 )) |
11 | 10 | mpteq2dva 5239 | . 2 β’ ( 0 β π β (π₯ β π β¦ (π₯π· 0 )) = (π₯ β π β¦ (π₯πΈ 0 ))) |
12 | 5, 11 | eqtrid 2776 | 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 1533 β wcel 2098 β¦ cmpt 5222 Γ cxp 5665 βΎ cres 5669 βcfv 6534 (class class class)co 7402 Basecbs 17149 distcds 17211 0gc0g 17390 normcnm 24429 |
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-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-fv 6542 df-ov 7405 df-nm 24435 |
This theorem is referenced by: nmfval2 24444 |
Copyright terms: Public domain | W3C validator |