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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 765 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → 𝐴 ∈ ℝ) | |
2 | 1 | renegcld 11069 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → -𝐴 ∈ ℝ) |
3 | 1 | recnd 10671 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → 𝐴 ∈ ℂ) |
4 | abscl 14640 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → (abs‘𝐴) ∈ ℝ) |
6 | simplr 767 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → 𝐵 ∈ ℝ) | |
7 | leabs 14661 | . . . . . . . 8 ⊢ (-𝐴 ∈ ℝ → -𝐴 ≤ (abs‘-𝐴)) | |
8 | 2, 7 | syl 17 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → -𝐴 ≤ (abs‘-𝐴)) |
9 | absneg 14639 | . . . . . . . 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 5094 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → -𝐴 ≤ (abs‘𝐴)) |
12 | simpr 487 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → (abs‘𝐴) < 𝐵) | |
13 | 2, 5, 6, 11, 12 | lelttrd 10800 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → -𝐴 < 𝐵) |
14 | leabs 14661 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ (abs‘𝐴)) | |
15 | 14 | ad2antrr 724 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → 𝐴 ≤ (abs‘𝐴)) |
16 | 1, 5, 6, 15, 12 | lelttrd 10800 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → 𝐴 < 𝐵) |
17 | 13, 16 | jca 514 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (abs‘𝐴) < 𝐵) → (-𝐴 < 𝐵 ∧ 𝐴 < 𝐵)) |
18 | 17 | ex 415 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((abs‘𝐴) < 𝐵 → (-𝐴 < 𝐵 ∧ 𝐴 < 𝐵))) |
19 | absor 14662 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴)) | |
20 | 19 | adantr 483 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴)) |
21 | breq1 5071 | . . . . . . 7 ⊢ ((abs‘𝐴) = 𝐴 → ((abs‘𝐴) < 𝐵 ↔ 𝐴 < 𝐵)) | |
22 | 21 | biimprd 250 | . . . . . 6 ⊢ ((abs‘𝐴) = 𝐴 → (𝐴 < 𝐵 → (abs‘𝐴) < 𝐵)) |
23 | breq1 5071 | . . . . . . 7 ⊢ ((abs‘𝐴) = -𝐴 → ((abs‘𝐴) < 𝐵 ↔ -𝐴 < 𝐵)) | |
24 | 23 | biimprd 250 | . . . . . 6 ⊢ ((abs‘𝐴) = -𝐴 → (-𝐴 < 𝐵 → (abs‘𝐴) < 𝐵)) |
25 | 22, 24 | jaoa 952 | . . . . 5 ⊢ (((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴) → ((𝐴 < 𝐵 ∧ -𝐴 < 𝐵) → (abs‘𝐴) < 𝐵)) |
26 | 25 | ancomsd 468 | . . . 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 214 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((abs‘𝐴) < 𝐵 ↔ (-𝐴 < 𝐵 ∧ 𝐴 < 𝐵))) |
29 | ltnegcon1 11143 | . . 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 281 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((abs‘𝐴) < 𝐵 ↔ (-𝐵 < 𝐴 ∧ 𝐴 < 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ‘cfv 6357 ℂcc 10537 ℝcr 10538 < clt 10677 ≤ cle 10678 -cneg 10873 abscabs 14595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 |
This theorem is referenced by: absdiflt 14679 abslti 14752 absltd 14791 tanregt0 25125 argregt0 25195 efopnlem2 25242 ftc1anclem1 34969 dvasin 34980 liminflimsupclim 42095 stoweidlem7 42299 |
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