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Mirrors > Home > MPE Home > Th. List > nmval2 | Structured version Visualization version GIF version |
Description: The value of the norm on a group as the distance restricted to the elements of the base set to zero. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
nmfval2.n | β’ π = (normβπ) |
nmfval2.x | β’ π = (Baseβπ) |
nmfval2.z | β’ 0 = (0gβπ) |
nmfval2.d | β’ π· = (distβπ) |
nmfval2.e | β’ πΈ = (π· βΎ (π Γ π)) |
Ref | Expression |
---|---|
nmval2 | β’ ((π β Grp β§ π΄ β π) β (πβπ΄) = (π΄πΈ 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmfval2.n | . . . 4 β’ π = (normβπ) | |
2 | nmfval2.x | . . . 4 β’ π = (Baseβπ) | |
3 | nmfval2.z | . . . 4 β’ 0 = (0gβπ) | |
4 | nmfval2.d | . . . 4 β’ π· = (distβπ) | |
5 | 1, 2, 3, 4 | nmval 23968 | . . 3 β’ (π΄ β π β (πβπ΄) = (π΄π· 0 )) |
6 | 5 | adantl 483 | . 2 β’ ((π β Grp β§ π΄ β π) β (πβπ΄) = (π΄π· 0 )) |
7 | nmfval2.e | . . . 4 β’ πΈ = (π· βΎ (π Γ π)) | |
8 | 7 | oveqi 7374 | . . 3 β’ (π΄πΈ 0 ) = (π΄(π· βΎ (π Γ π)) 0 ) |
9 | id 22 | . . . 4 β’ (π΄ β π β π΄ β π) | |
10 | 2, 3 | grpidcl 18786 | . . . 4 β’ (π β Grp β 0 β π) |
11 | ovres 7524 | . . . 4 β’ ((π΄ β π β§ 0 β π) β (π΄(π· βΎ (π Γ π)) 0 ) = (π΄π· 0 )) | |
12 | 9, 10, 11 | syl2anr 598 | . . 3 β’ ((π β Grp β§ π΄ β π) β (π΄(π· βΎ (π Γ π)) 0 ) = (π΄π· 0 )) |
13 | 8, 12 | eqtr2id 2786 | . 2 β’ ((π β Grp β§ π΄ β π) β (π΄π· 0 ) = (π΄πΈ 0 )) |
14 | 6, 13 | eqtrd 2773 | 1 β’ ((π β Grp β§ π΄ β π) β (πβπ΄) = (π΄πΈ 0 )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 Γ cxp 5635 βΎ cres 5639 βcfv 6500 (class class class)co 7361 Basecbs 17091 distcds 17150 0gc0g 17329 Grpcgrp 18756 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-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 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-riota 7317 df-ov 7364 df-0g 17331 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-grp 18759 df-nm 23961 |
This theorem is referenced by: nmhmcn 24506 nglmle 24689 |
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