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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 15335 | . . 3 ⊢ (abs‘(𝐴 − 𝐵)) ≤ ((abs‘(𝐴 − 𝐶)) + (abs‘(𝐶 − 𝐵))) |
5 | 1, 3 | subcli 11515 | . . . . 5 ⊢ (𝐴 − 𝐶) ∈ ℂ |
6 | 5 | abscli 15321 | . . . 4 ⊢ (abs‘(𝐴 − 𝐶)) ∈ ℝ |
7 | 3, 2 | subcli 11515 | . . . . 5 ⊢ (𝐶 − 𝐵) ∈ ℂ |
8 | 7 | abscli 15321 | . . . 4 ⊢ (abs‘(𝐶 − 𝐵)) ∈ ℝ |
9 | abs3lem.4 | . . . . 5 ⊢ 𝐷 ∈ ℝ | |
10 | 9 | rehalfcli 12440 | . . . 4 ⊢ (𝐷 / 2) ∈ ℝ |
11 | 6, 8, 10, 10 | lt2addi 11755 | . . 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 11515 | . . . . 5 ⊢ (𝐴 − 𝐵) ∈ ℂ |
13 | 12 | abscli 15321 | . . . 4 ⊢ (abs‘(𝐴 − 𝐵)) ∈ ℝ |
14 | 6, 8 | readdcli 11208 | . . . 4 ⊢ ((abs‘(𝐴 − 𝐶)) + (abs‘(𝐶 − 𝐵))) ∈ ℝ |
15 | 10, 10 | readdcli 11208 | . . . 4 ⊢ ((𝐷 / 2) + (𝐷 / 2)) ∈ ℝ |
16 | 13, 14, 15 | lelttri 11320 | . . 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 11207 | . . . 4 ⊢ (𝐷 / 2) ∈ ℂ |
19 | 18 | 2timesi 12329 | . . 3 ⊢ (2 · (𝐷 / 2)) = ((𝐷 / 2) + (𝐷 / 2)) |
20 | 9 | recni 11207 | . . . 4 ⊢ 𝐷 ∈ ℂ |
21 | 2cn 12266 | . . . 4 ⊢ 2 ∈ ℂ | |
22 | 2ne0 12295 | . . . 4 ⊢ 2 ≠ 0 | |
23 | 20, 21, 22 | divcan2i 11936 | . . 3 ⊢ (2 · (𝐷 / 2)) = 𝐷 |
24 | 19, 23 | eqtr3i 2761 | . 2 ⊢ ((𝐷 / 2) + (𝐷 / 2)) = 𝐷 |
25 | 17, 24 | breqtrdi 5179 | 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 2106 class class class wbr 5138 ‘cfv 6529 (class class class)co 7390 ℂcc 11087 ℝcr 11088 + caddc 11092 · cmul 11094 < clt 11227 ≤ cle 11228 − cmin 11423 / cdiv 11850 2c2 12246 abscabs 15160 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2702 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7705 ax-cnex 11145 ax-resscn 11146 ax-1cn 11147 ax-icn 11148 ax-addcl 11149 ax-addrcl 11150 ax-mulcl 11151 ax-mulrcl 11152 ax-mulcom 11153 ax-addass 11154 ax-mulass 11155 ax-distr 11156 ax-i2m1 11157 ax-1ne0 11158 ax-1rid 11159 ax-rnegex 11160 ax-rrecex 11161 ax-cnre 11162 ax-pre-lttri 11163 ax-pre-lttrn 11164 ax-pre-ltadd 11165 ax-pre-mulgt0 11166 ax-pre-sup 11167 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3430 df-v 3472 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4520 df-pw 4595 df-sn 4620 df-pr 4622 df-op 4626 df-uni 4899 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6286 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6531 df-fn 6532 df-f 6533 df-f1 6534 df-fo 6535 df-f1o 6536 df-fv 6537 df-riota 7346 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7836 df-2nd 7955 df-frecs 8245 df-wrecs 8276 df-recs 8350 df-rdg 8389 df-er 8683 df-en 8920 df-dom 8921 df-sdom 8922 df-sup 9416 df-pnf 11229 df-mnf 11230 df-xr 11231 df-ltxr 11232 df-le 11233 df-sub 11425 df-neg 11426 df-div 11851 df-nn 12192 df-2 12254 df-3 12255 df-n0 12452 df-z 12538 df-uz 12802 df-rp 12954 df-seq 13946 df-exp 14007 df-cj 15025 df-re 15026 df-im 15027 df-sqrt 15161 df-abs 15162 |
This theorem is referenced by: (None) |
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