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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 14489 | . . 3 ⊢ (abs‘(𝐴 − 𝐵)) ≤ ((abs‘(𝐴 − 𝐶)) + (abs‘(𝐶 − 𝐵))) |
5 | 1, 3 | subcli 10649 | . . . . 5 ⊢ (𝐴 − 𝐶) ∈ ℂ |
6 | 5 | abscli 14475 | . . . 4 ⊢ (abs‘(𝐴 − 𝐶)) ∈ ℝ |
7 | 3, 2 | subcli 10649 | . . . . 5 ⊢ (𝐶 − 𝐵) ∈ ℂ |
8 | 7 | abscli 14475 | . . . 4 ⊢ (abs‘(𝐶 − 𝐵)) ∈ ℝ |
9 | abs3lem.4 | . . . . 5 ⊢ 𝐷 ∈ ℝ | |
10 | 9 | rehalfcli 11569 | . . . 4 ⊢ (𝐷 / 2) ∈ ℝ |
11 | 6, 8, 10, 10 | lt2addi 10882 | . . 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 10649 | . . . . 5 ⊢ (𝐴 − 𝐵) ∈ ℂ |
13 | 12 | abscli 14475 | . . . 4 ⊢ (abs‘(𝐴 − 𝐵)) ∈ ℝ |
14 | 6, 8 | readdcli 10344 | . . . 4 ⊢ ((abs‘(𝐴 − 𝐶)) + (abs‘(𝐶 − 𝐵))) ∈ ℝ |
15 | 10, 10 | readdcli 10344 | . . . 4 ⊢ ((𝐷 / 2) + (𝐷 / 2)) ∈ ℝ |
16 | 13, 14, 15 | lelttri 10454 | . . 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 582 | . 2 ⊢ (((abs‘(𝐴 − 𝐶)) < (𝐷 / 2) ∧ (abs‘(𝐶 − 𝐵)) < (𝐷 / 2)) → (abs‘(𝐴 − 𝐵)) < ((𝐷 / 2) + (𝐷 / 2))) |
18 | 10 | recni 10343 | . . . 4 ⊢ (𝐷 / 2) ∈ ℂ |
19 | 18 | 2timesi 11458 | . . 3 ⊢ (2 · (𝐷 / 2)) = ((𝐷 / 2) + (𝐷 / 2)) |
20 | 9 | recni 10343 | . . . 4 ⊢ 𝐷 ∈ ℂ |
21 | 2cn 11388 | . . . 4 ⊢ 2 ∈ ℂ | |
22 | 2ne0 11424 | . . . 4 ⊢ 2 ≠ 0 | |
23 | 20, 21, 22 | divcan2i 11060 | . . 3 ⊢ (2 · (𝐷 / 2)) = 𝐷 |
24 | 19, 23 | eqtr3i 2823 | . 2 ⊢ ((𝐷 / 2) + (𝐷 / 2)) = 𝐷 |
25 | 17, 24 | syl6breq 4884 | 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 385 ∈ wcel 2157 class class class wbr 4843 ‘cfv 6101 (class class class)co 6878 ℂcc 10222 ℝcr 10223 + caddc 10227 · cmul 10229 < clt 10363 ≤ cle 10364 − cmin 10556 / cdiv 10976 2c2 11368 abscabs 14315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-pre-sup 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-sup 8590 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-div 10977 df-nn 11313 df-2 11376 df-3 11377 df-n0 11581 df-z 11667 df-uz 11931 df-rp 12075 df-seq 13056 df-exp 13115 df-cj 14180 df-re 14181 df-im 14182 df-sqrt 14316 df-abs 14317 |
This theorem is referenced by: (None) |
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