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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 23377. (In fact, the ring structure is not needed at all; the group properties abveq0 19590 and abvtri 19594, abvneg 19598 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 2798 | . 2 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
4 | abvmet.a | . . . 4 ⊢ 𝐴 = (AbsVal‘𝑅) | |
5 | 4 | abvrcl 19585 | . . 3 ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Ring) |
6 | ringgrp 19295 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Grp) |
8 | 4, 1 | abvf 19587 | . 2 ⊢ (𝐹 ∈ 𝐴 → 𝐹:𝑋⟶ℝ) |
9 | 4, 1, 3 | abveq0 19590 | . 2 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑅))) |
10 | 4, 1, 2 | abvsubtri 19599 | . . 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 23181 | 1 ⊢ (𝐹 ∈ 𝐴 → (𝐹 ∘ − ) ∈ (Met‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ∘ ccom 5523 ‘cfv 6324 (class class class)co 7135 + caddc 10529 ≤ cle 10665 Basecbs 16475 0gc0g 16705 Grpcgrp 18095 -gcsg 18097 Ringcrg 19290 AbsValcabv 19580 Metcmet 20077 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-ico 12732 df-seq 13365 df-exp 13426 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mgp 19233 df-ur 19245 df-ring 19292 df-abv 19581 df-met 20085 |
This theorem is referenced by: tngnrg 23280 |
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