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Mirrors > Home > MPE Home > Th. List > elicc4abs | Structured version Visualization version GIF version |
Description: Membership in a symmetric closed real interval. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
Ref | Expression |
---|---|
elicc4abs | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 ∈ ((𝐴 − 𝐵)[,](𝐴 + 𝐵)) ↔ (abs‘(𝐶 − 𝐴)) ≤ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resubcl 10942 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 − 𝐵) ∈ ℝ) | |
2 | 1 | 3adant3 1127 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 − 𝐵) ∈ ℝ) |
3 | 2 | rexrd 10683 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 − 𝐵) ∈ ℝ*) |
4 | readdcl 10612 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ) | |
5 | 4 | 3adant3 1127 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ) |
6 | 5 | rexrd 10683 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ*) |
7 | rexr 10679 | . . . 4 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℝ*) | |
8 | 7 | 3ad2ant3 1130 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐶 ∈ ℝ*) |
9 | elicc4 12795 | . . 3 ⊢ (((𝐴 − 𝐵) ∈ ℝ* ∧ (𝐴 + 𝐵) ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐶 ∈ ((𝐴 − 𝐵)[,](𝐴 + 𝐵)) ↔ ((𝐴 − 𝐵) ≤ 𝐶 ∧ 𝐶 ≤ (𝐴 + 𝐵)))) | |
10 | 3, 6, 8, 9 | syl3anc 1366 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 ∈ ((𝐴 − 𝐵)[,](𝐴 + 𝐵)) ↔ ((𝐴 − 𝐵) ≤ 𝐶 ∧ 𝐶 ≤ (𝐴 + 𝐵)))) |
11 | absdifle 14670 | . . 3 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((abs‘(𝐶 − 𝐴)) ≤ 𝐵 ↔ ((𝐴 − 𝐵) ≤ 𝐶 ∧ 𝐶 ≤ (𝐴 + 𝐵)))) | |
12 | 11 | 3coml 1122 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((abs‘(𝐶 − 𝐴)) ≤ 𝐵 ↔ ((𝐴 − 𝐵) ≤ 𝐶 ∧ 𝐶 ≤ (𝐴 + 𝐵)))) |
13 | 10, 12 | bitr4d 284 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 ∈ ((𝐴 − 𝐵)[,](𝐴 + 𝐵)) ↔ (abs‘(𝐶 − 𝐴)) ≤ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1082 ∈ wcel 2108 class class class wbr 5057 ‘cfv 6348 (class class class)co 7148 ℝcr 10528 + caddc 10532 ℝ*cxr 10666 ≤ cle 10668 − cmin 10862 [,]cicc 12733 abscabs 14585 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 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 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-sup 8898 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-n0 11890 df-z 11974 df-uz 12236 df-rp 12382 df-icc 12737 df-seq 13362 df-exp 13422 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 |
This theorem is referenced by: aalioulem3 24915 |
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