![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 14192 | . . 3 ⊢ (abs‘(𝐴 − 𝐵)) ≤ ((abs‘(𝐴 − 𝐶)) + (abs‘(𝐶 − 𝐵))) |
5 | 1, 3 | subcli 10395 | . . . . 5 ⊢ (𝐴 − 𝐶) ∈ ℂ |
6 | 5 | abscli 14178 | . . . 4 ⊢ (abs‘(𝐴 − 𝐶)) ∈ ℝ |
7 | 3, 2 | subcli 10395 | . . . . 5 ⊢ (𝐶 − 𝐵) ∈ ℂ |
8 | 7 | abscli 14178 | . . . 4 ⊢ (abs‘(𝐶 − 𝐵)) ∈ ℝ |
9 | abs3lem.4 | . . . . 5 ⊢ 𝐷 ∈ ℝ | |
10 | 9 | rehalfcli 11319 | . . . 4 ⊢ (𝐷 / 2) ∈ ℝ |
11 | 6, 8, 10, 10 | lt2addi 10628 | . . 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 10395 | . . . . 5 ⊢ (𝐴 − 𝐵) ∈ ℂ |
13 | 12 | abscli 14178 | . . . 4 ⊢ (abs‘(𝐴 − 𝐵)) ∈ ℝ |
14 | 6, 8 | readdcli 10091 | . . . 4 ⊢ ((abs‘(𝐴 − 𝐶)) + (abs‘(𝐶 − 𝐵))) ∈ ℝ |
15 | 10, 10 | readdcli 10091 | . . . 4 ⊢ ((𝐷 / 2) + (𝐷 / 2)) ∈ ℝ |
16 | 13, 14, 15 | lelttri 10202 | . . 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 696 | . 2 ⊢ (((abs‘(𝐴 − 𝐶)) < (𝐷 / 2) ∧ (abs‘(𝐶 − 𝐵)) < (𝐷 / 2)) → (abs‘(𝐴 − 𝐵)) < ((𝐷 / 2) + (𝐷 / 2))) |
18 | 10 | recni 10090 | . . . 4 ⊢ (𝐷 / 2) ∈ ℂ |
19 | 18 | 2timesi 11185 | . . 3 ⊢ (2 · (𝐷 / 2)) = ((𝐷 / 2) + (𝐷 / 2)) |
20 | 9 | recni 10090 | . . . 4 ⊢ 𝐷 ∈ ℂ |
21 | 2cn 11129 | . . . 4 ⊢ 2 ∈ ℂ | |
22 | 2ne0 11151 | . . . 4 ⊢ 2 ≠ 0 | |
23 | 20, 21, 22 | divcan2i 10806 | . . 3 ⊢ (2 · (𝐷 / 2)) = 𝐷 |
24 | 19, 23 | eqtr3i 2675 | . 2 ⊢ ((𝐷 / 2) + (𝐷 / 2)) = 𝐷 |
25 | 17, 24 | syl6breq 4726 | 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 383 ∈ wcel 2030 class class class wbr 4685 ‘cfv 5926 (class class class)co 6690 ℂcc 9972 ℝcr 9973 + caddc 9977 · cmul 9979 < clt 10112 ≤ cle 10113 − cmin 10304 / cdiv 10722 2c2 11108 abscabs 14018 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-sup 8389 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-n0 11331 df-z 11416 df-uz 11726 df-rp 11871 df-seq 12842 df-exp 12901 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |