![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > xmetrtri2 | Structured version Visualization version GIF version |
Description: The reverse triangle inequality for the distance function of an extended metric. In order to express the "extended absolute value function", we use the distance function xrsdsval 19984 defined on the extended real structure. (Contributed by Mario Carneiro, 4-Sep-2015.) |
Ref | Expression |
---|---|
xmetrtri2.1 | ⊢ 𝐾 = (dist‘ℝ*𝑠) |
Ref | Expression |
---|---|
xmetrtri2 | ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷𝐶)𝐾(𝐵𝐷𝐶)) ≤ (𝐴𝐷𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xmetcl 22329 | . . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐷𝐶) ∈ ℝ*) | |
2 | 1 | 3adant3r2 1196 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐶) ∈ ℝ*) |
3 | xmetcl 22329 | . . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵𝐷𝐶) ∈ ℝ*) | |
4 | 3 | 3adant3r1 1195 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵𝐷𝐶) ∈ ℝ*) |
5 | xmetrtri2.1 | . . . 4 ⊢ 𝐾 = (dist‘ℝ*𝑠) | |
6 | 5 | xrsdsval 19984 | . . 3 ⊢ (((𝐴𝐷𝐶) ∈ ℝ* ∧ (𝐵𝐷𝐶) ∈ ℝ*) → ((𝐴𝐷𝐶)𝐾(𝐵𝐷𝐶)) = if((𝐴𝐷𝐶) ≤ (𝐵𝐷𝐶), ((𝐵𝐷𝐶) +𝑒 -𝑒(𝐴𝐷𝐶)), ((𝐴𝐷𝐶) +𝑒 -𝑒(𝐵𝐷𝐶)))) |
7 | 2, 4, 6 | syl2anc 696 | . 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷𝐶)𝐾(𝐵𝐷𝐶)) = if((𝐴𝐷𝐶) ≤ (𝐵𝐷𝐶), ((𝐵𝐷𝐶) +𝑒 -𝑒(𝐴𝐷𝐶)), ((𝐴𝐷𝐶) +𝑒 -𝑒(𝐵𝐷𝐶)))) |
8 | 3ancoma 1084 | . . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ↔ (𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) | |
9 | xmetrtri 22353 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐵𝐷𝐶) +𝑒 -𝑒(𝐴𝐷𝐶)) ≤ (𝐵𝐷𝐴)) | |
10 | 8, 9 | sylan2b 493 | . . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐵𝐷𝐶) +𝑒 -𝑒(𝐴𝐷𝐶)) ≤ (𝐵𝐷𝐴)) |
11 | xmetsym 22345 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴)) | |
12 | 11 | 3adant3r3 1197 | . . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴)) |
13 | 10, 12 | breqtrrd 4824 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐵𝐷𝐶) +𝑒 -𝑒(𝐴𝐷𝐶)) ≤ (𝐴𝐷𝐵)) |
14 | xmetrtri 22353 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷𝐶) +𝑒 -𝑒(𝐵𝐷𝐶)) ≤ (𝐴𝐷𝐵)) | |
15 | breq1 4799 | . . . 4 ⊢ (((𝐵𝐷𝐶) +𝑒 -𝑒(𝐴𝐷𝐶)) = if((𝐴𝐷𝐶) ≤ (𝐵𝐷𝐶), ((𝐵𝐷𝐶) +𝑒 -𝑒(𝐴𝐷𝐶)), ((𝐴𝐷𝐶) +𝑒 -𝑒(𝐵𝐷𝐶))) → (((𝐵𝐷𝐶) +𝑒 -𝑒(𝐴𝐷𝐶)) ≤ (𝐴𝐷𝐵) ↔ if((𝐴𝐷𝐶) ≤ (𝐵𝐷𝐶), ((𝐵𝐷𝐶) +𝑒 -𝑒(𝐴𝐷𝐶)), ((𝐴𝐷𝐶) +𝑒 -𝑒(𝐵𝐷𝐶))) ≤ (𝐴𝐷𝐵))) | |
16 | breq1 4799 | . . . 4 ⊢ (((𝐴𝐷𝐶) +𝑒 -𝑒(𝐵𝐷𝐶)) = if((𝐴𝐷𝐶) ≤ (𝐵𝐷𝐶), ((𝐵𝐷𝐶) +𝑒 -𝑒(𝐴𝐷𝐶)), ((𝐴𝐷𝐶) +𝑒 -𝑒(𝐵𝐷𝐶))) → (((𝐴𝐷𝐶) +𝑒 -𝑒(𝐵𝐷𝐶)) ≤ (𝐴𝐷𝐵) ↔ if((𝐴𝐷𝐶) ≤ (𝐵𝐷𝐶), ((𝐵𝐷𝐶) +𝑒 -𝑒(𝐴𝐷𝐶)), ((𝐴𝐷𝐶) +𝑒 -𝑒(𝐵𝐷𝐶))) ≤ (𝐴𝐷𝐵))) | |
17 | 15, 16 | ifboth 4260 | . . 3 ⊢ ((((𝐵𝐷𝐶) +𝑒 -𝑒(𝐴𝐷𝐶)) ≤ (𝐴𝐷𝐵) ∧ ((𝐴𝐷𝐶) +𝑒 -𝑒(𝐵𝐷𝐶)) ≤ (𝐴𝐷𝐵)) → if((𝐴𝐷𝐶) ≤ (𝐵𝐷𝐶), ((𝐵𝐷𝐶) +𝑒 -𝑒(𝐴𝐷𝐶)), ((𝐴𝐷𝐶) +𝑒 -𝑒(𝐵𝐷𝐶))) ≤ (𝐴𝐷𝐵)) |
18 | 13, 14, 17 | syl2anc 696 | . 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → if((𝐴𝐷𝐶) ≤ (𝐵𝐷𝐶), ((𝐵𝐷𝐶) +𝑒 -𝑒(𝐴𝐷𝐶)), ((𝐴𝐷𝐶) +𝑒 -𝑒(𝐵𝐷𝐶))) ≤ (𝐴𝐷𝐵)) |
19 | 7, 18 | eqbrtrd 4818 | 1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷𝐶)𝐾(𝐵𝐷𝐶)) ≤ (𝐴𝐷𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1072 = wceq 1624 ∈ wcel 2131 ifcif 4222 class class class wbr 4796 ‘cfv 6041 (class class class)co 6805 ℝ*cxr 10257 ≤ cle 10259 -𝑒cxne 12128 +𝑒 cxad 12129 distcds 16144 ℝ*𝑠cxrs 16354 ∞Metcxmt 19925 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 ax-un 7106 ax-cnex 10176 ax-resscn 10177 ax-1cn 10178 ax-icn 10179 ax-addcl 10180 ax-addrcl 10181 ax-mulcl 10182 ax-mulrcl 10183 ax-mulcom 10184 ax-addass 10185 ax-mulass 10186 ax-distr 10187 ax-i2m1 10188 ax-1ne0 10189 ax-1rid 10190 ax-rnegex 10191 ax-rrecex 10192 ax-cnre 10193 ax-pre-lttri 10194 ax-pre-lttrn 10195 ax-pre-ltadd 10196 ax-pre-mulgt0 10197 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-nel 3028 df-ral 3047 df-rex 3048 df-reu 3049 df-rmo 3050 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-pss 3723 df-nul 4051 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-tp 4318 df-op 4320 df-uni 4581 df-int 4620 df-iun 4666 df-br 4797 df-opab 4857 df-mpt 4874 df-tr 4897 df-id 5166 df-eprel 5171 df-po 5179 df-so 5180 df-fr 5217 df-we 5219 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-pred 5833 df-ord 5879 df-on 5880 df-lim 5881 df-suc 5882 df-iota 6004 df-fun 6043 df-fn 6044 df-f 6045 df-f1 6046 df-fo 6047 df-f1o 6048 df-fv 6049 df-riota 6766 df-ov 6808 df-oprab 6809 df-mpt2 6810 df-om 7223 df-1st 7325 df-2nd 7326 df-wrecs 7568 df-recs 7629 df-rdg 7667 df-1o 7721 df-oadd 7725 df-er 7903 df-map 8017 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-pnf 10260 df-mnf 10261 df-xr 10262 df-ltxr 10263 df-le 10264 df-sub 10452 df-neg 10453 df-div 10869 df-nn 11205 df-2 11263 df-3 11264 df-4 11265 df-5 11266 df-6 11267 df-7 11268 df-8 11269 df-9 11270 df-n0 11477 df-z 11562 df-dec 11678 df-uz 11872 df-rp 12018 df-xneg 12131 df-xadd 12132 df-xmul 12133 df-fz 12512 df-struct 16053 df-ndx 16054 df-slot 16055 df-base 16057 df-plusg 16148 df-mulr 16149 df-tset 16154 df-ple 16155 df-ds 16158 df-xrs 16356 df-xmet 19933 |
This theorem is referenced by: metrtri 22355 metdcnlem 22832 |
Copyright terms: Public domain | W3C validator |