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Mirrors > Home > MPE Home > Th. List > tngnrg | Structured version Visualization version GIF version |
Description: Given any absolute value over a ring, augmenting the ring with the absolute value produces a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
tngnrg.t | β’ π = (π toNrmGrp πΉ) |
tngnrg.a | β’ π΄ = (AbsValβπ ) |
Ref | Expression |
---|---|
tngnrg | β’ (πΉ β π΄ β π β NrmRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tngnrg.a | . . . . 5 β’ π΄ = (AbsValβπ ) | |
2 | 1 | abvrcl 20323 | . . . 4 β’ (πΉ β π΄ β π β Ring) |
3 | ringgrp 19977 | . . . 4 β’ (π β Ring β π β Grp) | |
4 | 2, 3 | syl 17 | . . 3 β’ (πΉ β π΄ β π β Grp) |
5 | tngnrg.t | . . . . 5 β’ π = (π toNrmGrp πΉ) | |
6 | eqid 2733 | . . . . 5 β’ (-gβπ ) = (-gβπ ) | |
7 | 5, 6 | tngds 24034 | . . . 4 β’ (πΉ β π΄ β (πΉ β (-gβπ )) = (distβπ)) |
8 | eqid 2733 | . . . . 5 β’ (Baseβπ ) = (Baseβπ ) | |
9 | 8, 1, 6 | abvmet 23954 | . . . 4 β’ (πΉ β π΄ β (πΉ β (-gβπ )) β (Metβ(Baseβπ ))) |
10 | 7, 9 | eqeltrrd 2835 | . . 3 β’ (πΉ β π΄ β (distβπ) β (Metβ(Baseβπ ))) |
11 | 1, 8 | abvf 20325 | . . . 4 β’ (πΉ β π΄ β πΉ:(Baseβπ )βΆβ) |
12 | eqid 2733 | . . . . 5 β’ (distβπ) = (distβπ) | |
13 | 5, 8, 12 | tngngp2 24039 | . . . 4 β’ (πΉ:(Baseβπ )βΆβ β (π β NrmGrp β (π β Grp β§ (distβπ) β (Metβ(Baseβπ ))))) |
14 | 11, 13 | syl 17 | . . 3 β’ (πΉ β π΄ β (π β NrmGrp β (π β Grp β§ (distβπ) β (Metβ(Baseβπ ))))) |
15 | 4, 10, 14 | mpbir2and 712 | . 2 β’ (πΉ β π΄ β π β NrmGrp) |
16 | reex 11150 | . . . . . 6 β’ β β V | |
17 | 5, 8, 16 | tngnm 24038 | . . . . 5 β’ ((π β Grp β§ πΉ:(Baseβπ )βΆβ) β πΉ = (normβπ)) |
18 | 4, 11, 17 | syl2anc 585 | . . . 4 β’ (πΉ β π΄ β πΉ = (normβπ)) |
19 | eqidd 2734 | . . . . . 6 β’ (πΉ β π΄ β (Baseβπ ) = (Baseβπ )) | |
20 | 5, 8 | tngbas 24021 | . . . . . 6 β’ (πΉ β π΄ β (Baseβπ ) = (Baseβπ)) |
21 | eqid 2733 | . . . . . . . 8 β’ (+gβπ ) = (+gβπ ) | |
22 | 5, 21 | tngplusg 24023 | . . . . . . 7 β’ (πΉ β π΄ β (+gβπ ) = (+gβπ)) |
23 | 22 | oveqdr 7389 | . . . . . 6 β’ ((πΉ β π΄ β§ (π₯ β (Baseβπ ) β§ π¦ β (Baseβπ ))) β (π₯(+gβπ )π¦) = (π₯(+gβπ)π¦)) |
24 | eqid 2733 | . . . . . . . 8 β’ (.rβπ ) = (.rβπ ) | |
25 | 5, 24 | tngmulr 24026 | . . . . . . 7 β’ (πΉ β π΄ β (.rβπ ) = (.rβπ)) |
26 | 25 | oveqdr 7389 | . . . . . 6 β’ ((πΉ β π΄ β§ (π₯ β (Baseβπ ) β§ π¦ β (Baseβπ ))) β (π₯(.rβπ )π¦) = (π₯(.rβπ)π¦)) |
27 | 19, 20, 23, 26 | abvpropd 20344 | . . . . 5 β’ (πΉ β π΄ β (AbsValβπ ) = (AbsValβπ)) |
28 | 1, 27 | eqtrid 2785 | . . . 4 β’ (πΉ β π΄ β π΄ = (AbsValβπ)) |
29 | 18, 28 | eleq12d 2828 | . . 3 β’ (πΉ β π΄ β (πΉ β π΄ β (normβπ) β (AbsValβπ))) |
30 | 29 | ibi 267 | . 2 β’ (πΉ β π΄ β (normβπ) β (AbsValβπ)) |
31 | eqid 2733 | . . 3 β’ (normβπ) = (normβπ) | |
32 | eqid 2733 | . . 3 β’ (AbsValβπ) = (AbsValβπ) | |
33 | 31, 32 | isnrg 24047 | . 2 β’ (π β NrmRing β (π β NrmGrp β§ (normβπ) β (AbsValβπ))) |
34 | 15, 30, 33 | sylanbrc 584 | 1 β’ (πΉ β π΄ β π β NrmRing) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 β ccom 5641 βΆwf 6496 βcfv 6500 (class class class)co 7361 βcr 11058 Basecbs 17091 +gcplusg 17141 .rcmulr 17142 distcds 17150 Grpcgrp 18756 -gcsg 18758 Ringcrg 19972 AbsValcabv 20318 Metcmet 20805 normcnm 23955 NrmGrpcngp 23956 toNrmGrp ctng 23957 NrmRingcnrg 23958 |
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-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 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-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-sup 9386 df-inf 9387 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-uz 12772 df-q 12882 df-rp 12924 df-xneg 13041 df-xadd 13042 df-xmul 13043 df-ico 13279 df-seq 13916 df-exp 13977 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-plusg 17154 df-mulr 17155 df-tset 17160 df-ds 17163 df-rest 17312 df-topn 17313 df-0g 17331 df-topgen 17333 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-grp 18759 df-minusg 18760 df-sbg 18761 df-mgp 19905 df-ur 19922 df-ring 19974 df-abv 20319 df-psmet 20811 df-xmet 20812 df-met 20813 df-bl 20814 df-mopn 20815 df-top 22266 df-topon 22283 df-topsp 22305 df-bases 22319 df-xms 23696 df-ms 23697 df-nm 23961 df-ngp 23962 df-tng 23963 df-nrg 23964 |
This theorem is referenced by: (None) |
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