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Mirrors > Home > MPE Home > Th. List > abvmet | Structured version Visualization version GIF version |
Description: An absolute value πΉ generates a metric defined by π(π₯, π¦) = πΉ(π₯ β π¦), analogously to cnmet 24632. (In fact, the ring structure is not needed at all; the group properties abveq0 20665 and abvtri 20669, abvneg 20673 are sufficient.) (Contributed by Mario Carneiro, 9-Sep-2014.) (Revised by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
abvmet.x | β’ π = (Baseβπ ) |
abvmet.a | β’ π΄ = (AbsValβπ ) |
abvmet.m | β’ β = (-gβπ ) |
Ref | Expression |
---|---|
abvmet | β’ (πΉ β π΄ β (πΉ β β ) β (Metβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abvmet.x | . 2 β’ π = (Baseβπ ) | |
2 | abvmet.m | . 2 β’ β = (-gβπ ) | |
3 | eqid 2724 | . 2 β’ (0gβπ ) = (0gβπ ) | |
4 | abvmet.a | . . . 4 β’ π΄ = (AbsValβπ ) | |
5 | 4 | abvrcl 20660 | . . 3 β’ (πΉ β π΄ β π β Ring) |
6 | ringgrp 20139 | . . 3 β’ (π β Ring β π β Grp) | |
7 | 5, 6 | syl 17 | . 2 β’ (πΉ β π΄ β π β Grp) |
8 | 4, 1 | abvf 20662 | . 2 β’ (πΉ β π΄ β πΉ:πβΆβ) |
9 | 4, 1, 3 | abveq0 20665 | . 2 β’ ((πΉ β π΄ β§ π₯ β π) β ((πΉβπ₯) = 0 β π₯ = (0gβπ ))) |
10 | 4, 1, 2 | abvsubtri 20674 | . . 3 β’ ((πΉ β π΄ β§ π₯ β π β§ π¦ β π) β (πΉβ(π₯ β π¦)) β€ ((πΉβπ₯) + (πΉβπ¦))) |
11 | 10 | 3expb 1117 | . 2 β’ ((πΉ β π΄ β§ (π₯ β π β§ π¦ β π)) β (πΉβ(π₯ β π¦)) β€ ((πΉβπ₯) + (πΉβπ¦))) |
12 | 1, 2, 3, 7, 8, 9, 11 | nrmmetd 24427 | 1 β’ (πΉ β π΄ β (πΉ β β ) β (Metβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 class class class wbr 5139 β ccom 5671 βcfv 6534 (class class class)co 7402 + caddc 11110 β€ cle 11248 Basecbs 17149 0gc0g 17390 Grpcgrp 18859 -gcsg 18861 Ringcrg 20134 AbsValcabv 20655 Metcmet 21220 |
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-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 |
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-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-n0 12472 df-z 12558 df-uz 12822 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-0g 17392 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-met 21228 |
This theorem is referenced by: tngnrg 24535 |
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