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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 24782. (In fact, the ring structure is not needed at all; the group properties abveq0 20799 and abvtri 20803, abvneg 20807 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 2726 | . 2 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
4 | abvmet.a | . . . 4 ⊢ 𝐴 = (AbsVal‘𝑅) | |
5 | 4 | abvrcl 20794 | . . 3 ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Ring) |
6 | ringgrp 20223 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Grp) |
8 | 4, 1 | abvf 20796 | . 2 ⊢ (𝐹 ∈ 𝐴 → 𝐹:𝑋⟶ℝ) |
9 | 4, 1, 3 | abveq0 20799 | . 2 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑅))) |
10 | 4, 1, 2 | abvsubtri 20808 | . . 3 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 − 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))) |
11 | 10 | 3expb 1117 | . 2 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘(𝑥 − 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))) |
12 | 1, 2, 3, 7, 8, 9, 11 | nrmmetd 24577 | 1 ⊢ (𝐹 ∈ 𝐴 → (𝐹 ∘ − ) ∈ (Met‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 class class class wbr 5155 ∘ ccom 5688 ‘cfv 6556 (class class class)co 7426 + caddc 11163 ≤ cle 11301 Basecbs 17215 0gc0g 17456 Grpcgrp 18930 -gcsg 18932 Ringcrg 20218 AbsValcabv 20789 Metcmet 21331 |
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 2697 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-cnex 11216 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-iun 5005 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8005 df-2nd 8006 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-er 8736 df-map 8859 df-en 8977 df-dom 8978 df-sdom 8979 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11498 df-neg 11499 df-div 11924 df-nn 12267 df-2 12329 df-n0 12527 df-z 12613 df-uz 12877 df-ico 13386 df-seq 14024 df-exp 14084 df-sets 17168 df-slot 17186 df-ndx 17198 df-base 17216 df-plusg 17281 df-0g 17458 df-mgm 18635 df-sgrp 18714 df-mnd 18730 df-grp 18933 df-minusg 18934 df-sbg 18935 df-cmn 19782 df-abl 19783 df-mgp 20120 df-rng 20138 df-ur 20167 df-ring 20220 df-abv 20790 df-met 21339 |
This theorem is referenced by: tngnrg 24685 |
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