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Mirrors > Home > MPE Home > Th. List > abvmet | Structured version Visualization version GIF version |
Description: An absolute value 𝐹 generates a metric defined by 𝑑(𝑥, 𝑦) = 𝐹(𝑥 − 𝑦), analogously to cnmet 23307. (In fact, the ring structure is not needed at all; the group properties abveq0 19526 and abvtri 19530, abvneg 19534 are sufficient.) (Contributed by Mario Carneiro, 9-Sep-2014.) (Revised by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
abvmet.x | ⊢ 𝑋 = (Base‘𝑅) |
abvmet.a | ⊢ 𝐴 = (AbsVal‘𝑅) |
abvmet.m | ⊢ − = (-g‘𝑅) |
Ref | Expression |
---|---|
abvmet | ⊢ (𝐹 ∈ 𝐴 → (𝐹 ∘ − ) ∈ (Met‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abvmet.x | . 2 ⊢ 𝑋 = (Base‘𝑅) | |
2 | abvmet.m | . 2 ⊢ − = (-g‘𝑅) | |
3 | eqid 2818 | . 2 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
4 | abvmet.a | . . . 4 ⊢ 𝐴 = (AbsVal‘𝑅) | |
5 | 4 | abvrcl 19521 | . . 3 ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Ring) |
6 | ringgrp 19231 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Grp) |
8 | 4, 1 | abvf 19523 | . 2 ⊢ (𝐹 ∈ 𝐴 → 𝐹:𝑋⟶ℝ) |
9 | 4, 1, 3 | abveq0 19526 | . 2 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑅))) |
10 | 4, 1, 2 | abvsubtri 19535 | . . 3 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 − 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))) |
11 | 10 | 3expb 1112 | . 2 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘(𝑥 − 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))) |
12 | 1, 2, 3, 7, 8, 9, 11 | nrmmetd 23111 | 1 ⊢ (𝐹 ∈ 𝐴 → (𝐹 ∘ − ) ∈ (Met‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ∘ ccom 5552 ‘cfv 6348 (class class class)co 7145 + caddc 10528 ≤ cle 10664 Basecbs 16471 0gc0g 16701 Grpcgrp 18041 -gcsg 18043 Ringcrg 19226 AbsValcabv 19516 Metcmet 20459 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-ico 12732 df-seq 13358 df-exp 13418 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-plusg 16566 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mgp 19169 df-ur 19181 df-ring 19228 df-abv 19517 df-met 20467 |
This theorem is referenced by: tngnrg 23210 |
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