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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 767 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → 𝐴 ∈ ℝ) | |
2 | 1 | renegcld 11687 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → -𝐴 ∈ ℝ) |
3 | 1 | recnd 11286 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → 𝐴 ∈ ℂ) |
4 | abscl 15313 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → (abs‘𝐴) ∈ ℝ) |
6 | simplr 769 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → 𝐵 ∈ ℝ) | |
7 | leabs 15334 | . . . . . . . 8 ⊢ (-𝐴 ∈ ℝ → -𝐴 ≤ (abs‘-𝐴)) | |
8 | 2, 7 | syl 17 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → -𝐴 ≤ (abs‘-𝐴)) |
9 | absneg 15312 | . . . . . . . 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 484 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → (abs‘𝐴) < 𝐵) | |
13 | 2, 5, 6, 11, 12 | lelttrd 11416 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → -𝐴 < 𝐵) |
14 | leabs 15334 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ (abs‘𝐴)) | |
15 | 14 | ad2antrr 726 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → 𝐴 ≤ (abs‘𝐴)) |
16 | 1, 5, 6, 15, 12 | lelttrd 11416 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → 𝐴 < 𝐵) |
17 | 13, 16 | jca 511 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → (-𝐴 < 𝐵 ∧ 𝐴 < 𝐵)) |
18 | 17 | ex 412 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((abs‘𝐴) < 𝐵 → (-𝐴 < 𝐵 ∧ 𝐴 < 𝐵))) |
19 | absor 15335 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴)) | |
20 | 19 | adantr 480 | . . . 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 248 | . . . . . 6 ⊢ ((abs‘𝐴) = 𝐴 → (𝐴 < 𝐵 → (abs‘𝐴) < 𝐵)) |
23 | breq1 5150 | . . . . . . 7 ⊢ ((abs‘𝐴) = -𝐴 → ((abs‘𝐴) < 𝐵 ↔ -𝐴 < 𝐵)) | |
24 | 23 | biimprd 248 | . . . . . 6 ⊢ ((abs‘𝐴) = -𝐴 → (-𝐴 < 𝐵 → (abs‘𝐴) < 𝐵)) |
25 | 22, 24 | jaoa 957 | . . . . 5 ⊢ (((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴) → ((𝐴 < 𝐵 ∧ -𝐴 < 𝐵) → (abs‘𝐴) < 𝐵)) |
26 | 25 | ancomsd 465 | . . . 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 212 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((abs‘𝐴) < 𝐵 ↔ (-𝐴 < 𝐵 ∧ 𝐴 < 𝐵))) |
29 | ltnegcon1 11761 | . . 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 206 ∧ wa 395 ∨ wo 847 = wceq 1536 ∈ wcel 2105 class class class wbr 5147 ‘cfv 6562 ℂcc 11150 ℝcr 11151 < clt 11292 ≤ cle 11293 -cneg 11490 abscabs 15269 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-sup 9479 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-seq 14039 df-exp 14099 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 |
This theorem is referenced by: absdiflt 15352 abslti 15425 absltd 15464 tanregt0 26595 argregt0 26666 efopnlem2 26713 ftc1anclem1 37679 dvasin 37690 liminflimsupclim 45762 stoweidlem7 45962 |
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