![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 20660 | . . . 4 β’ (πΉ β π΄ β π β Ring) |
3 | ringgrp 20139 | . . . 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 2724 | . . . . 5 β’ (-gβπ ) = (-gβπ ) | |
7 | 5, 6 | tngds 24508 | . . . 4 β’ (πΉ β π΄ β (πΉ β (-gβπ )) = (distβπ)) |
8 | eqid 2724 | . . . . 5 β’ (Baseβπ ) = (Baseβπ ) | |
9 | 8, 1, 6 | abvmet 24428 | . . . 4 β’ (πΉ β π΄ β (πΉ β (-gβπ )) β (Metβ(Baseβπ ))) |
10 | 7, 9 | eqeltrrd 2826 | . . 3 β’ (πΉ β π΄ β (distβπ) β (Metβ(Baseβπ ))) |
11 | 1, 8 | abvf 20662 | . . . 4 β’ (πΉ β π΄ β πΉ:(Baseβπ )βΆβ) |
12 | eqid 2724 | . . . . 5 β’ (distβπ) = (distβπ) | |
13 | 5, 8, 12 | tngngp2 24513 | . . . 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 710 | . 2 β’ (πΉ β π΄ β π β NrmGrp) |
16 | reex 11198 | . . . . . 6 β’ β β V | |
17 | 5, 8, 16 | tngnm 24512 | . . . . 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 2725 | . . . . . 6 β’ (πΉ β π΄ β (Baseβπ ) = (Baseβπ )) | |
20 | 5, 8 | tngbas 24495 | . . . . . 6 β’ (πΉ β π΄ β (Baseβπ ) = (Baseβπ)) |
21 | eqid 2724 | . . . . . . . 8 β’ (+gβπ ) = (+gβπ ) | |
22 | 5, 21 | tngplusg 24497 | . . . . . . 7 β’ (πΉ β π΄ β (+gβπ ) = (+gβπ)) |
23 | 22 | oveqdr 7430 | . . . . . 6 β’ ((πΉ β π΄ β§ (π₯ β (Baseβπ ) β§ π¦ β (Baseβπ ))) β (π₯(+gβπ )π¦) = (π₯(+gβπ)π¦)) |
24 | eqid 2724 | . . . . . . . 8 β’ (.rβπ ) = (.rβπ ) | |
25 | 5, 24 | tngmulr 24500 | . . . . . . 7 β’ (πΉ β π΄ β (.rβπ ) = (.rβπ)) |
26 | 25 | oveqdr 7430 | . . . . . 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 20681 | . . . . 5 β’ (πΉ β π΄ β (AbsValβπ ) = (AbsValβπ)) |
28 | 1, 27 | eqtrid 2776 | . . . 4 β’ (πΉ β π΄ β π΄ = (AbsValβπ)) |
29 | 18, 28 | eleq12d 2819 | . . 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 2724 | . . 3 β’ (normβπ) = (normβπ) | |
32 | eqid 2724 | . . 3 β’ (AbsValβπ) = (AbsValβπ) | |
33 | 31, 32 | isnrg 24521 | . 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 1533 β wcel 2098 β ccom 5671 βΆwf 6530 βcfv 6534 (class class class)co 7402 βcr 11106 Basecbs 17149 +gcplusg 17202 .rcmulr 17203 distcds 17211 Grpcgrp 18859 -gcsg 18861 Ringcrg 20134 AbsValcabv 20655 Metcmet 21220 normcnm 24429 NrmGrpcngp 24430 toNrmGrp ctng 24431 NrmRingcnrg 24432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-sup 9434 df-inf 9435 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-q 12932 df-rp 12976 df-xneg 13093 df-xadd 13094 df-xmul 13095 df-ico 13331 df-seq 13968 df-exp 14029 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-plusg 17215 df-mulr 17216 df-tset 17221 df-ds 17224 df-rest 17373 df-topn 17374 df-0g 17392 df-topgen 17394 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-grp 18862 df-minusg 18863 df-sbg 18864 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-abv 20656 df-psmet 21226 df-xmet 21227 df-met 21228 df-bl 21229 df-mopn 21230 df-top 22740 df-topon 22757 df-topsp 22779 df-bases 22793 df-xms 24170 df-ms 24171 df-nm 24435 df-ngp 24436 df-tng 24437 df-nrg 24438 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |