![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > abslt | Structured version Visualization version GIF version |
Description: Absolute value and 'less than' relation. (Contributed by NM, 6-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.) |
Ref | Expression |
---|---|
abslt | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((abs‘𝐴) < 𝐵 ↔ (-𝐵 < 𝐴 ∧ 𝐴 < 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 766 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → 𝐴 ∈ ℝ) | |
2 | 1 | renegcld 11637 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → -𝐴 ∈ ℝ) |
3 | 1 | recnd 11238 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → 𝐴 ∈ ℂ) |
4 | abscl 15221 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → (abs‘𝐴) ∈ ℝ) |
6 | simplr 768 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → 𝐵 ∈ ℝ) | |
7 | leabs 15242 | . . . . . . . 8 ⊢ (-𝐴 ∈ ℝ → -𝐴 ≤ (abs‘-𝐴)) | |
8 | 2, 7 | syl 17 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → -𝐴 ≤ (abs‘-𝐴)) |
9 | absneg 15220 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (abs‘-𝐴) = (abs‘𝐴)) | |
10 | 3, 9 | syl 17 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → (abs‘-𝐴) = (abs‘𝐴)) |
11 | 8, 10 | breqtrd 5173 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → -𝐴 ≤ (abs‘𝐴)) |
12 | simpr 486 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → (abs‘𝐴) < 𝐵) | |
13 | 2, 5, 6, 11, 12 | lelttrd 11368 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → -𝐴 < 𝐵) |
14 | leabs 15242 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ (abs‘𝐴)) | |
15 | 14 | ad2antrr 725 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → 𝐴 ≤ (abs‘𝐴)) |
16 | 1, 5, 6, 15, 12 | lelttrd 11368 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → 𝐴 < 𝐵) |
17 | 13, 16 | jca 513 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → (-𝐴 < 𝐵 ∧ 𝐴 < 𝐵)) |
18 | 17 | ex 414 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((abs‘𝐴) < 𝐵 → (-𝐴 < 𝐵 ∧ 𝐴 < 𝐵))) |
19 | absor 15243 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴)) | |
20 | 19 | adantr 482 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴)) |
21 | breq1 5150 | . . . . . . 7 ⊢ ((abs‘𝐴) = 𝐴 → ((abs‘𝐴) < 𝐵 ↔ 𝐴 < 𝐵)) | |
22 | 21 | biimprd 247 | . . . . . 6 ⊢ ((abs‘𝐴) = 𝐴 → (𝐴 < 𝐵 → (abs‘𝐴) < 𝐵)) |
23 | breq1 5150 | . . . . . . 7 ⊢ ((abs‘𝐴) = -𝐴 → ((abs‘𝐴) < 𝐵 ↔ -𝐴 < 𝐵)) | |
24 | 23 | biimprd 247 | . . . . . 6 ⊢ ((abs‘𝐴) = -𝐴 → (-𝐴 < 𝐵 → (abs‘𝐴) < 𝐵)) |
25 | 22, 24 | jaoa 955 | . . . . 5 ⊢ (((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴) → ((𝐴 < 𝐵 ∧ -𝐴 < 𝐵) → (abs‘𝐴) < 𝐵)) |
26 | 25 | ancomsd 467 | . . . 4 ⊢ (((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴) → ((-𝐴 < 𝐵 ∧ 𝐴 < 𝐵) → (abs‘𝐴) < 𝐵)) |
27 | 20, 26 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((-𝐴 < 𝐵 ∧ 𝐴 < 𝐵) → (abs‘𝐴) < 𝐵)) |
28 | 18, 27 | impbid 211 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((abs‘𝐴) < 𝐵 ↔ (-𝐴 < 𝐵 ∧ 𝐴 < 𝐵))) |
29 | ltnegcon1 11711 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-𝐴 < 𝐵 ↔ -𝐵 < 𝐴)) | |
30 | 29 | anbi1d 631 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((-𝐴 < 𝐵 ∧ 𝐴 < 𝐵) ↔ (-𝐵 < 𝐴 ∧ 𝐴 < 𝐵))) |
31 | 28, 30 | bitrd 279 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((abs‘𝐴) < 𝐵 ↔ (-𝐵 < 𝐴 ∧ 𝐴 < 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 class class class wbr 5147 ‘cfv 6540 ℂcc 11104 ℝcr 11105 < clt 11244 ≤ cle 11245 -cneg 11441 abscabs 15177 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 |
This theorem is referenced by: absdiflt 15260 abslti 15333 absltd 15372 tanregt0 26030 argregt0 26100 efopnlem2 26147 ftc1anclem1 36499 dvasin 36510 liminflimsupclim 44458 stoweidlem7 44658 |
Copyright terms: Public domain | W3C validator |