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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 20701 | . . . 4 β’ (πΉ β π΄ β π β Ring) |
3 | ringgrp 20178 | . . . 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 2728 | . . . . 5 β’ (-gβπ ) = (-gβπ ) | |
7 | 5, 6 | tngds 24577 | . . . 4 β’ (πΉ β π΄ β (πΉ β (-gβπ )) = (distβπ)) |
8 | eqid 2728 | . . . . 5 β’ (Baseβπ ) = (Baseβπ ) | |
9 | 8, 1, 6 | abvmet 24497 | . . . 4 β’ (πΉ β π΄ β (πΉ β (-gβπ )) β (Metβ(Baseβπ ))) |
10 | 7, 9 | eqeltrrd 2830 | . . 3 β’ (πΉ β π΄ β (distβπ) β (Metβ(Baseβπ ))) |
11 | 1, 8 | abvf 20703 | . . . 4 β’ (πΉ β π΄ β πΉ:(Baseβπ )βΆβ) |
12 | eqid 2728 | . . . . 5 β’ (distβπ) = (distβπ) | |
13 | 5, 8, 12 | tngngp2 24582 | . . . 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 11230 | . . . . . 6 β’ β β V | |
17 | 5, 8, 16 | tngnm 24581 | . . . . 5 β’ ((π β Grp β§ πΉ:(Baseβπ )βΆβ) β πΉ = (normβπ)) |
18 | 4, 11, 17 | syl2anc 583 | . . . 4 β’ (πΉ β π΄ β πΉ = (normβπ)) |
19 | eqidd 2729 | . . . . . 6 β’ (πΉ β π΄ β (Baseβπ ) = (Baseβπ )) | |
20 | 5, 8 | tngbas 24564 | . . . . . 6 β’ (πΉ β π΄ β (Baseβπ ) = (Baseβπ)) |
21 | eqid 2728 | . . . . . . . 8 β’ (+gβπ ) = (+gβπ ) | |
22 | 5, 21 | tngplusg 24566 | . . . . . . 7 β’ (πΉ β π΄ β (+gβπ ) = (+gβπ)) |
23 | 22 | oveqdr 7448 | . . . . . 6 β’ ((πΉ β π΄ β§ (π₯ β (Baseβπ ) β§ π¦ β (Baseβπ ))) β (π₯(+gβπ )π¦) = (π₯(+gβπ)π¦)) |
24 | eqid 2728 | . . . . . . . 8 β’ (.rβπ ) = (.rβπ ) | |
25 | 5, 24 | tngmulr 24569 | . . . . . . 7 β’ (πΉ β π΄ β (.rβπ ) = (.rβπ)) |
26 | 25 | oveqdr 7448 | . . . . . 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 20722 | . . . . 5 β’ (πΉ β π΄ β (AbsValβπ ) = (AbsValβπ)) |
28 | 1, 27 | eqtrid 2780 | . . . 4 β’ (πΉ β π΄ β π΄ = (AbsValβπ)) |
29 | 18, 28 | eleq12d 2823 | . . 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 2728 | . . 3 β’ (normβπ) = (normβπ) | |
32 | eqid 2728 | . . 3 β’ (AbsValβπ) = (AbsValβπ) | |
33 | 31, 32 | isnrg 24590 | . 2 β’ (π β NrmRing β (π β NrmGrp β§ (normβπ) β (AbsValβπ))) |
34 | 15, 30, 33 | sylanbrc 582 | 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 395 = wceq 1534 β wcel 2099 β ccom 5682 βΆwf 6544 βcfv 6548 (class class class)co 7420 βcr 11138 Basecbs 17180 +gcplusg 17233 .rcmulr 17234 distcds 17242 Grpcgrp 18890 -gcsg 18892 Ringcrg 20173 AbsValcabv 20696 Metcmet 21265 normcnm 24498 NrmGrpcngp 24499 toNrmGrp ctng 24500 NrmRingcnrg 24501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9466 df-inf 9467 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-ico 13363 df-seq 14000 df-exp 14060 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-plusg 17246 df-mulr 17247 df-tset 17252 df-ds 17255 df-rest 17404 df-topn 17405 df-0g 17423 df-topgen 17425 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-grp 18893 df-minusg 18894 df-sbg 18895 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-ring 20175 df-abv 20697 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22862 df-xms 24239 df-ms 24240 df-nm 24504 df-ngp 24505 df-tng 24506 df-nrg 24507 |
This theorem is referenced by: (None) |
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