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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 2732 | . . . 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 24099 | . . 3 β’ (π β Grp β π = (π₯ β π β¦ (π₯πΈ(0gβπ)))) |
| 7 | 6 | adantr 481 | . 2 β’ ((π β Grp β§ πΈ β (Metβπ)) β π = (π₯ β π β¦ (π₯πΈ(0gβπ)))) |
| 8 | 2, 3 | grpidcl 18849 | . . . 4 β’ (π β Grp β (0gβπ) β π) |
| 9 | metcl 23837 | . . . . 5 β’ ((πΈ β (Metβπ) β§ π₯ β π β§ (0gβπ) β π) β (π₯πΈ(0gβπ)) β β) | |
| 10 | 9 | 3comr 1125 | . . . 4 β’ (((0gβπ) β π β§ πΈ β (Metβπ) β§ π₯ β π) β (π₯πΈ(0gβπ)) β β) |
| 11 | 8, 10 | syl3an1 1163 | . . 3 β’ ((π β Grp β§ πΈ β (Metβπ) β§ π₯ β π) β (π₯πΈ(0gβπ)) β β) |
| 12 | 11 | 3expa 1118 | . 2 β’ (((π β Grp β§ πΈ β (Metβπ)) β§ π₯ β π) β (π₯πΈ(0gβπ)) β β) |
| 13 | 7, 12 | fmpt3d 7115 | 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 396 = wceq 1541 β wcel 2106 β¦ cmpt 5231 Γ cxp 5674 βΎ cres 5678 βΆwf 6539 βcfv 6543 (class class class)co 7408 βcr 11108 Basecbs 17143 distcds 17205 0gc0g 17384 Grpcgrp 18818 Metcmet 20929 normcnm 24084 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-map 8821 df-0g 17386 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-grp 18821 df-met 20937 df-nm 24090 |
| This theorem is referenced by: isngp2 24105 isngp3 24106 nmf 24123 |
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