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Mirrors > Home > MPE Home > Th. List > nmf2 | Structured version Visualization version GIF version |
Description: The norm on a metric group is a function from the base set into the reals. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
nmf2.n | β’ π = (normβπ) |
nmf2.x | β’ π = (Baseβπ) |
nmf2.d | β’ π· = (distβπ) |
nmf2.e | β’ πΈ = (π· βΎ (π Γ π)) |
Ref | Expression |
---|---|
nmf2 | β’ ((π β Grp β§ πΈ β (Metβπ)) β π:πβΆβ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmf2.n | . . . 4 β’ π = (normβπ) | |
2 | nmf2.x | . . . 4 β’ π = (Baseβπ) | |
3 | eqid 2733 | . . . 4 β’ (0gβπ) = (0gβπ) | |
4 | nmf2.d | . . . 4 β’ π· = (distβπ) | |
5 | nmf2.e | . . . 4 β’ πΈ = (π· βΎ (π Γ π)) | |
6 | 1, 2, 3, 4, 5 | nmfval2 23970 | . . 3 β’ (π β Grp β π = (π₯ β π β¦ (π₯πΈ(0gβπ)))) |
7 | 6 | adantr 482 | . 2 β’ ((π β Grp β§ πΈ β (Metβπ)) β π = (π₯ β π β¦ (π₯πΈ(0gβπ)))) |
8 | 2, 3 | grpidcl 18786 | . . . 4 β’ (π β Grp β (0gβπ) β π) |
9 | metcl 23708 | . . . . 5 β’ ((πΈ β (Metβπ) β§ π₯ β π β§ (0gβπ) β π) β (π₯πΈ(0gβπ)) β β) | |
10 | 9 | 3comr 1126 | . . . 4 β’ (((0gβπ) β π β§ πΈ β (Metβπ) β§ π₯ β π) β (π₯πΈ(0gβπ)) β β) |
11 | 8, 10 | syl3an1 1164 | . . 3 β’ ((π β Grp β§ πΈ β (Metβπ) β§ π₯ β π) β (π₯πΈ(0gβπ)) β β) |
12 | 11 | 3expa 1119 | . 2 β’ (((π β Grp β§ πΈ β (Metβπ)) β§ π₯ β π) β (π₯πΈ(0gβπ)) β β) |
13 | 7, 12 | fmpt3d 7068 | 1 β’ ((π β Grp β§ πΈ β (Metβπ)) β π:πβΆβ) |
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 βΆwf 6496 βcfv 6500 (class class class)co 7361 βcr 11058 Basecbs 17091 distcds 17150 0gc0g 17329 Grpcgrp 18756 Metcmet 20805 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 ax-cnex 11115 ax-resscn 11116 |
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-oprab 7365 df-mpo 7366 df-map 8773 df-0g 17331 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-grp 18759 df-met 20813 df-nm 23961 |
This theorem is referenced by: isngp2 23976 isngp3 23977 nmf 23994 |
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