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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 23923. (In fact, the ring structure is not needed at all; the group properties abveq0 20074 and abvtri 20078, abvneg 20082 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 2738 | . 2 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
4 | abvmet.a | . . . 4 ⊢ 𝐴 = (AbsVal‘𝑅) | |
5 | 4 | abvrcl 20069 | . . 3 ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Ring) |
6 | ringgrp 19776 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Grp) |
8 | 4, 1 | abvf 20071 | . 2 ⊢ (𝐹 ∈ 𝐴 → 𝐹:𝑋⟶ℝ) |
9 | 4, 1, 3 | abveq0 20074 | . 2 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑅))) |
10 | 4, 1, 2 | abvsubtri 20083 | . . 3 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 − 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))) |
11 | 10 | 3expb 1119 | . 2 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘(𝑥 − 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))) |
12 | 1, 2, 3, 7, 8, 9, 11 | nrmmetd 23718 | 1 ⊢ (𝐹 ∈ 𝐴 → (𝐹 ∘ − ) ∈ (Met‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 class class class wbr 5074 ∘ ccom 5589 ‘cfv 6427 (class class class)co 7268 + caddc 10862 ≤ cle 10998 Basecbs 16900 0gc0g 17138 Grpcgrp 18565 -gcsg 18567 Ringcrg 19771 AbsValcabv 20064 Metcmet 20571 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-cnex 10915 ax-resscn 10916 ax-1cn 10917 ax-icn 10918 ax-addcl 10919 ax-addrcl 10920 ax-mulcl 10921 ax-mulrcl 10922 ax-mulcom 10923 ax-addass 10924 ax-mulass 10925 ax-distr 10926 ax-i2m1 10927 ax-1ne0 10928 ax-1rid 10929 ax-rnegex 10930 ax-rrecex 10931 ax-cnre 10932 ax-pre-lttri 10933 ax-pre-lttrn 10934 ax-pre-ltadd 10935 ax-pre-mulgt0 10936 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-iun 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6263 df-on 6264 df-lim 6265 df-suc 6266 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7704 df-1st 7821 df-2nd 7822 df-frecs 8085 df-wrecs 8116 df-recs 8190 df-rdg 8229 df-er 8486 df-map 8605 df-en 8722 df-dom 8723 df-sdom 8724 df-pnf 10999 df-mnf 11000 df-xr 11001 df-ltxr 11002 df-le 11003 df-sub 11195 df-neg 11196 df-div 11621 df-nn 11962 df-2 12024 df-n0 12222 df-z 12308 df-uz 12571 df-ico 13073 df-seq 13710 df-exp 13771 df-sets 16853 df-slot 16871 df-ndx 16883 df-base 16901 df-plusg 16963 df-0g 17140 df-mgm 18314 df-sgrp 18363 df-mnd 18374 df-grp 18568 df-minusg 18569 df-sbg 18570 df-mgp 19709 df-ur 19726 df-ring 19773 df-abv 20065 df-met 20579 |
This theorem is referenced by: tngnrg 23826 |
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