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Mirrors > Home > MPE Home > Th. List > abs3lemi | Structured version Visualization version GIF version |
Description: Lemma involving absolute value of differences. (Contributed by NM, 2-Oct-1999.) |
Ref | Expression |
---|---|
absvalsqi.1 | ⊢ 𝐴 ∈ ℂ |
abssub.2 | ⊢ 𝐵 ∈ ℂ |
abs3dif.3 | ⊢ 𝐶 ∈ ℂ |
abs3lem.4 | ⊢ 𝐷 ∈ ℝ |
Ref | Expression |
---|---|
abs3lemi | ⊢ (((abs‘(𝐴 − 𝐶)) < (𝐷 / 2) ∧ (abs‘(𝐶 − 𝐵)) < (𝐷 / 2)) → (abs‘(𝐴 − 𝐵)) < 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | absvalsqi.1 | . . . 4 ⊢ 𝐴 ∈ ℂ | |
2 | abssub.2 | . . . 4 ⊢ 𝐵 ∈ ℂ | |
3 | abs3dif.3 | . . . 4 ⊢ 𝐶 ∈ ℂ | |
4 | 1, 2, 3 | abs3difi 14607 | . . 3 ⊢ (abs‘(𝐴 − 𝐵)) ≤ ((abs‘(𝐴 − 𝐶)) + (abs‘(𝐶 − 𝐵))) |
5 | 1, 3 | subcli 10816 | . . . . 5 ⊢ (𝐴 − 𝐶) ∈ ℂ |
6 | 5 | abscli 14593 | . . . 4 ⊢ (abs‘(𝐴 − 𝐶)) ∈ ℝ |
7 | 3, 2 | subcli 10816 | . . . . 5 ⊢ (𝐶 − 𝐵) ∈ ℂ |
8 | 7 | abscli 14593 | . . . 4 ⊢ (abs‘(𝐶 − 𝐵)) ∈ ℝ |
9 | abs3lem.4 | . . . . 5 ⊢ 𝐷 ∈ ℝ | |
10 | 9 | rehalfcli 11740 | . . . 4 ⊢ (𝐷 / 2) ∈ ℝ |
11 | 6, 8, 10, 10 | lt2addi 11056 | . . 3 ⊢ (((abs‘(𝐴 − 𝐶)) < (𝐷 / 2) ∧ (abs‘(𝐶 − 𝐵)) < (𝐷 / 2)) → ((abs‘(𝐴 − 𝐶)) + (abs‘(𝐶 − 𝐵))) < ((𝐷 / 2) + (𝐷 / 2))) |
12 | 1, 2 | subcli 10816 | . . . . 5 ⊢ (𝐴 − 𝐵) ∈ ℂ |
13 | 12 | abscli 14593 | . . . 4 ⊢ (abs‘(𝐴 − 𝐵)) ∈ ℝ |
14 | 6, 8 | readdcli 10509 | . . . 4 ⊢ ((abs‘(𝐴 − 𝐶)) + (abs‘(𝐶 − 𝐵))) ∈ ℝ |
15 | 10, 10 | readdcli 10509 | . . . 4 ⊢ ((𝐷 / 2) + (𝐷 / 2)) ∈ ℝ |
16 | 13, 14, 15 | lelttri 10620 | . . 3 ⊢ (((abs‘(𝐴 − 𝐵)) ≤ ((abs‘(𝐴 − 𝐶)) + (abs‘(𝐶 − 𝐵))) ∧ ((abs‘(𝐴 − 𝐶)) + (abs‘(𝐶 − 𝐵))) < ((𝐷 / 2) + (𝐷 / 2))) → (abs‘(𝐴 − 𝐵)) < ((𝐷 / 2) + (𝐷 / 2))) |
17 | 4, 11, 16 | sylancr 587 | . 2 ⊢ (((abs‘(𝐴 − 𝐶)) < (𝐷 / 2) ∧ (abs‘(𝐶 − 𝐵)) < (𝐷 / 2)) → (abs‘(𝐴 − 𝐵)) < ((𝐷 / 2) + (𝐷 / 2))) |
18 | 10 | recni 10508 | . . . 4 ⊢ (𝐷 / 2) ∈ ℂ |
19 | 18 | 2timesi 11629 | . . 3 ⊢ (2 · (𝐷 / 2)) = ((𝐷 / 2) + (𝐷 / 2)) |
20 | 9 | recni 10508 | . . . 4 ⊢ 𝐷 ∈ ℂ |
21 | 2cn 11566 | . . . 4 ⊢ 2 ∈ ℂ | |
22 | 2ne0 11595 | . . . 4 ⊢ 2 ≠ 0 | |
23 | 20, 21, 22 | divcan2i 11237 | . . 3 ⊢ (2 · (𝐷 / 2)) = 𝐷 |
24 | 19, 23 | eqtr3i 2823 | . 2 ⊢ ((𝐷 / 2) + (𝐷 / 2)) = 𝐷 |
25 | 17, 24 | syl6breq 5009 | 1 ⊢ (((abs‘(𝐴 − 𝐶)) < (𝐷 / 2) ∧ (abs‘(𝐶 − 𝐵)) < (𝐷 / 2)) → (abs‘(𝐴 − 𝐵)) < 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2083 class class class wbr 4968 ‘cfv 6232 (class class class)co 7023 ℂcc 10388 ℝcr 10389 + caddc 10393 · cmul 10395 < clt 10528 ≤ cle 10529 − cmin 10723 / cdiv 11151 2c2 11546 abscabs 14431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 ax-pre-sup 10468 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-2nd 7553 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-sup 8759 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-div 11152 df-nn 11493 df-2 11554 df-3 11555 df-n0 11752 df-z 11836 df-uz 12098 df-rp 12244 df-seq 13224 df-exp 13284 df-cj 14296 df-re 14297 df-im 14298 df-sqrt 14432 df-abs 14433 |
This theorem is referenced by: (None) |
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