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Theorem abs3lem 4861

Description: Lemma involving absolute value of differences.

**Hypotheses**
Ref	Expression
releabs.1	⊢ A ∈ ℂ
abstri.2	⊢ B ∈ ℂ
abs3dif.3	⊢ C ∈ ℂ
abs3lem.4	⊢ D ∈ ℝ

**Assertion**
Ref	Expression
abs3lem	⊢ (((abs ‘(A − C)) < (D / 2) ∧ (abs ‘(C − B)) < (D / 2)) → (abs ‘(A − B)) < D)

**Proof of Theorem abs3lem**
Step	Hyp	Ref	Expression
1		releabs.1	. . . . . 6 ⊢ A ∈ ℂ
2		abs3dif.3	. . . . . 6 ⊢ C ∈ ℂ
3	1, 2	subcl 4139	. . . . 5 ⊢ (A − C) ∈ ℂ
4	3	abscl 4840	. . . 4 ⊢ (abs ‘(A − C)) ∈ ℝ
5		abstri.2	. . . . . 6 ⊢ B ∈ ℂ
6	2, 5	subcl 4139	. . . . 5 ⊢ (C − B) ∈ ℂ
7	6	abscl 4840	. . . 4 ⊢ (abs ‘(C − B)) ∈ ℝ
8		abs3lem.4	. . . . 5 ⊢ D ∈ ℝ
9		2re 4470	. . . . 5 ⊢ 2 ∈ ℝ
10		2pos 4479	. . . . . 6 ⊢ 0 < 2
11	9, 10	gt0ne0i 4345	. . . . 5 ⊢ 2 ≠ 0
12	8, 9, 11	redivcl 4274	. . . 4 ⊢ (D / 2) ∈ ℝ
13	4, 7, 12, 12	lt2add 4321	. . 3 ⊢ (((abs ‘(A − C)) < (D / 2) ∧ (abs ‘(C − B)) < (D / 2)) → ((abs ‘(A − C)) + (abs ‘(C − B))) < ((D / 2) + (D / 2)))
14	1, 5, 2	abs3dif 4860	. . . 4 ⊢ (abs ‘(A − B)) ≤ ((abs ‘(A − C)) + (abs ‘(C − B)))
15	1, 5	subcl 4139	. . . . . 6 ⊢ (A − B) ∈ ℂ
16	15	abscl 4840	. . . . 5 ⊢ (abs ‘(A − B)) ∈ ℝ
17	4, 7	readdcl 4118	. . . . 5 ⊢ ((abs ‘(A − C)) + (abs ‘(C − B))) ∈ ℝ
18	12, 12	readdcl 4118	. . . . 5 ⊢ ((D / 2) + (D / 2)) ∈ ℝ
19	16, 17, 18	lelttr 4308	. . . 4 ⊢ (((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)))
20	14, 19	mpan 518	. . 3 ⊢ (((abs ‘(A − C)) + (abs ‘(C − B))) < ((D / 2) + (D / 2)) → (abs ‘(A − B)) < ((D / 2) + (D / 2)))
21	13, 20	syl 12	. 2 ⊢ (((abs ‘(A − C)) < (D / 2) ∧ (abs ‘(C − B)) < (D / 2)) → (abs ‘(A − B)) < ((D / 2) + (D / 2)))
22	8	recn 4098	. . . . 5 ⊢ D ∈ ℂ
23		2cn 4471	. . . . 5 ⊢ 2 ∈ ℂ
24	22, 23, 11	divcl 4221	. . . 4 ⊢ (D / 2) ∈ ℂ
25	24	2times 4489	. . 3 ⊢ (2 · (D / 2)) = ((D / 2) + (D / 2))
26	23, 22, 11	divcan2 4224	. . 3 ⊢ (2 · (D / 2)) = D
27	25, 26	eqtr3 1121	. 2 ⊢ ((D / 2) + (D / 2)) = D
28	21, 27	syl6breq 2093	1 ⊢ (((abs ‘(A − C)) < (D / 2) ∧ (abs ‘(C − B)) < (D / 2)) → (abs ‘(A − B)) < D)

Colors of variables: wff set class

Syntax hints: → wi 2 ∧ wa 196 ∈ wcel 1092 class class class wbr 2054 ‘cfv 2422 (class class class)co 3001 ℂcc 4026 ℝcr 4027 + caddc 4031 · cmulc 4032 < clt 4033 − cmin 4089 / cdiv 4091 ≤ cle 4092 2c2 4454 abscabs 4789

This theorem is referenced by: abs3lemt 4865

This theorem was proved from axioms: ax-1 3 ax-2 4 ax-3 5 ax-mp 6 ax-4 673 ax-5 674 ax-6 675 ax-7 676 ax-gen 677 ax-8 798 ax-9 799 ax-10 800 ax-11 801 ax-12 802 ax-13 804 ax-14 805 ax-16 922 ax-17 925 ax-ext 1074 ax-rep 1075 ax-un 1076 ax-pow 1077 ax-reg 1078 ax-inf 1079

This theorem depends on definitions: df-bi 128 df-or 197 df-an 198 df-3or 582 df-3an 583 df-ex 679 df-sb 853 df-eu 1009 df-mo 1010 df-clab 1093 df-cleq 1097 df-clel 1099 df-ne 1192 df-ral 1205 df-rex 1206 df-reu 1207 df-rab 1208 df-v 1349 df-sbc 1441 df-dif 1489 df-un 1490 df-in 1491 df-ss 1492 df-pss 1494 df-nul 1708 df-if 1777 df-pw 1799 df-sn 1811 df-pr 1812 df-tp 1814 df-op 1815 df-uni 1920 df-int 1966 df-iun 1996 df-tr 2042 df-br 2063 df-opab 2098 df-eprel 2122 df-id 2125 df-po 2128 df-so 2138 df-sup 2154 df-fr 2169 df-we 2186 df-ord 2202 df-on 2203 df-lim 2204 df-suc 2205 df-om 2373 df-xp 2424 df-rel 2425 df-cnv 2426 df-co 2427 df-dm 2428 df-rn 2429 df-res 2430 df-ima 2431 df-fun 2432 df-fn 2433 df-f 2434 df-f1 2435 df-fo 2436 df-f1o 2437 df-fv 2438 df-rdg 2970 df-opr 3003 df-oprab 3004 df-1st 3087 df-2nd 3088 df-1o 3104 df-oadd 3106 df-omul 3107 df-er 3200 df-ec 3202 df-qs 3205 df-ni 3794 df-pli 3795 df-mi 3796 df-lti 3797 df-plpq 3829 df-mpq 3830 df-enq 3831 df-nq 3832 df-plq 3833 df-mq 3834 df-rq 3835 df-ltq 3836 df-1q 3837 df-np 3880 df-1p 3881 df-plp 3882 df-mp 3883 df-ltp 3884 df-plpr 3958 df-mpr 3959 df-enr 3960 df-nr 3961 df-plr 3962 df-mr 3963 df-ltr 3964 df-0r 3965 df-1r 3966 df-m1r 3967 df-c 4034 df-0 4035 df-1 4036 df-i 4037 df-r 4038 df-plus 4039 df-mul 4040 df-lt 4041 df-sub 4133 df-neg 4135 df-div 4216 df-le 4277 df-n 4423 df-2 4462 df-n0 4535 df-z 4564 df-seq 4661 df-exp 4676 df-sqr 4728 df-re 4790 df-im 4791 df-cj 4792 df-abs 4793

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