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Mirrors > Home > MPE Home > Th. List > abs2dif | Structured version Visualization version GIF version |
Description: Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.) |
Ref | Expression |
---|---|
abs2dif | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘𝐴) − (abs‘𝐵)) ≤ (abs‘(𝐴 − 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subid1 11095 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 − 0) = 𝐴) | |
2 | 1 | fveq2d 6718 | . . 3 ⊢ (𝐴 ∈ ℂ → (abs‘(𝐴 − 0)) = (abs‘𝐴)) |
3 | subid1 11095 | . . . 4 ⊢ (𝐵 ∈ ℂ → (𝐵 − 0) = 𝐵) | |
4 | 3 | fveq2d 6718 | . . 3 ⊢ (𝐵 ∈ ℂ → (abs‘(𝐵 − 0)) = (abs‘𝐵)) |
5 | 2, 4 | oveqan12d 7229 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘(𝐴 − 0)) − (abs‘(𝐵 − 0))) = ((abs‘𝐴) − (abs‘𝐵))) |
6 | 0cn 10822 | . . . 4 ⊢ 0 ∈ ℂ | |
7 | abs3dif 14892 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (abs‘(𝐴 − 0)) ≤ ((abs‘(𝐴 − 𝐵)) + (abs‘(𝐵 − 0)))) | |
8 | 6, 7 | mp3an2 1451 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (abs‘(𝐴 − 0)) ≤ ((abs‘(𝐴 − 𝐵)) + (abs‘(𝐵 − 0)))) |
9 | subcl 11074 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → (𝐴 − 0) ∈ ℂ) | |
10 | 6, 9 | mpan2 691 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐴 − 0) ∈ ℂ) |
11 | abscl 14839 | . . . . . . 7 ⊢ ((𝐴 − 0) ∈ ℂ → (abs‘(𝐴 − 0)) ∈ ℝ) | |
12 | 10, 11 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (abs‘(𝐴 − 0)) ∈ ℝ) |
13 | subcl 11074 | . . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 0 ∈ ℂ) → (𝐵 − 0) ∈ ℂ) | |
14 | 6, 13 | mpan2 691 | . . . . . . 7 ⊢ (𝐵 ∈ ℂ → (𝐵 − 0) ∈ ℂ) |
15 | abscl 14839 | . . . . . . 7 ⊢ ((𝐵 − 0) ∈ ℂ → (abs‘(𝐵 − 0)) ∈ ℝ) | |
16 | 14, 15 | syl 17 | . . . . . 6 ⊢ (𝐵 ∈ ℂ → (abs‘(𝐵 − 0)) ∈ ℝ) |
17 | 12, 16 | anim12i 616 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘(𝐴 − 0)) ∈ ℝ ∧ (abs‘(𝐵 − 0)) ∈ ℝ)) |
18 | subcl 11074 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) ∈ ℂ) | |
19 | abscl 14839 | . . . . . 6 ⊢ ((𝐴 − 𝐵) ∈ ℂ → (abs‘(𝐴 − 𝐵)) ∈ ℝ) | |
20 | 18, 19 | syl 17 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (abs‘(𝐴 − 𝐵)) ∈ ℝ) |
21 | df-3an 1091 | . . . . 5 ⊢ (((abs‘(𝐴 − 0)) ∈ ℝ ∧ (abs‘(𝐵 − 0)) ∈ ℝ ∧ (abs‘(𝐴 − 𝐵)) ∈ ℝ) ↔ (((abs‘(𝐴 − 0)) ∈ ℝ ∧ (abs‘(𝐵 − 0)) ∈ ℝ) ∧ (abs‘(𝐴 − 𝐵)) ∈ ℝ)) | |
22 | 17, 20, 21 | sylanbrc 586 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘(𝐴 − 0)) ∈ ℝ ∧ (abs‘(𝐵 − 0)) ∈ ℝ ∧ (abs‘(𝐴 − 𝐵)) ∈ ℝ)) |
23 | lesubadd 11301 | . . . 4 ⊢ (((abs‘(𝐴 − 0)) ∈ ℝ ∧ (abs‘(𝐵 − 0)) ∈ ℝ ∧ (abs‘(𝐴 − 𝐵)) ∈ ℝ) → (((abs‘(𝐴 − 0)) − (abs‘(𝐵 − 0))) ≤ (abs‘(𝐴 − 𝐵)) ↔ (abs‘(𝐴 − 0)) ≤ ((abs‘(𝐴 − 𝐵)) + (abs‘(𝐵 − 0))))) | |
24 | 22, 23 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((abs‘(𝐴 − 0)) − (abs‘(𝐵 − 0))) ≤ (abs‘(𝐴 − 𝐵)) ↔ (abs‘(𝐴 − 0)) ≤ ((abs‘(𝐴 − 𝐵)) + (abs‘(𝐵 − 0))))) |
25 | 8, 24 | mpbird 260 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘(𝐴 − 0)) − (abs‘(𝐵 − 0))) ≤ (abs‘(𝐴 − 𝐵))) |
26 | 5, 25 | eqbrtrrd 5074 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘𝐴) − (abs‘𝐵)) ≤ (abs‘(𝐴 − 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 ∈ wcel 2110 class class class wbr 5050 ‘cfv 6377 (class class class)co 7210 ℂcc 10724 ℝcr 10725 0cc0 10726 + caddc 10729 ≤ cle 10865 − cmin 11059 abscabs 14794 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 ax-cnex 10782 ax-resscn 10783 ax-1cn 10784 ax-icn 10785 ax-addcl 10786 ax-addrcl 10787 ax-mulcl 10788 ax-mulrcl 10789 ax-mulcom 10790 ax-addass 10791 ax-mulass 10792 ax-distr 10793 ax-i2m1 10794 ax-1ne0 10795 ax-1rid 10796 ax-rnegex 10797 ax-rrecex 10798 ax-cnre 10799 ax-pre-lttri 10800 ax-pre-lttrn 10801 ax-pre-ltadd 10802 ax-pre-mulgt0 10803 ax-pre-sup 10804 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-nel 3044 df-ral 3063 df-rex 3064 df-reu 3065 df-rmo 3066 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-pss 3882 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-tp 4543 df-op 4545 df-uni 4817 df-iun 4903 df-br 5051 df-opab 5113 df-mpt 5133 df-tr 5159 df-id 5452 df-eprel 5457 df-po 5465 df-so 5466 df-fr 5506 df-we 5508 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6157 df-ord 6213 df-on 6214 df-lim 6215 df-suc 6216 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-riota 7167 df-ov 7213 df-oprab 7214 df-mpo 7215 df-om 7642 df-2nd 7759 df-wrecs 8044 df-recs 8105 df-rdg 8143 df-er 8388 df-en 8624 df-dom 8625 df-sdom 8626 df-sup 9055 df-pnf 10866 df-mnf 10867 df-xr 10868 df-ltxr 10869 df-le 10870 df-sub 11061 df-neg 11062 df-div 11487 df-nn 11828 df-2 11890 df-3 11891 df-n0 12088 df-z 12174 df-uz 12436 df-rp 12584 df-seq 13572 df-exp 13633 df-cj 14659 df-re 14660 df-im 14661 df-sqrt 14795 df-abs 14796 |
This theorem is referenced by: abs2difabs 14895 absrdbnd 14902 caubnd2 14918 abs2difd 15018 abelthlem2 25321 logfacbnd3 26101 log2sumbnd 26422 abs2difi 33350 |
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