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Mirrors > Home > MPE Home > Th. List > avglt1 | Structured version Visualization version GIF version |
Description: Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.) |
Ref | Expression |
---|---|
avglt1 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 < ((𝐴 + 𝐵) / 2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltadd2 10815 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 + 𝐴) < (𝐴 + 𝐵))) | |
2 | 1 | 3anidm13 1421 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 + 𝐴) < (𝐴 + 𝐵))) |
3 | simpl 486 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℝ) | |
4 | 3 | recnd 10740 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℂ) |
5 | times2 11846 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 · 2) = (𝐴 + 𝐴)) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 2) = (𝐴 + 𝐴)) |
7 | 6 | breq1d 5037 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 · 2) < (𝐴 + 𝐵) ↔ (𝐴 + 𝐴) < (𝐴 + 𝐵))) |
8 | readdcl 10691 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ) | |
9 | 2re 11783 | . . . . 5 ⊢ 2 ∈ ℝ | |
10 | 2pos 11812 | . . . . 5 ⊢ 0 < 2 | |
11 | 9, 10 | pm3.2i 474 | . . . 4 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
12 | 11 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (2 ∈ ℝ ∧ 0 < 2)) |
13 | ltmuldiv 11584 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝐴 + 𝐵) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((𝐴 · 2) < (𝐴 + 𝐵) ↔ 𝐴 < ((𝐴 + 𝐵) / 2))) | |
14 | 3, 8, 12, 13 | syl3anc 1372 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 · 2) < (𝐴 + 𝐵) ↔ 𝐴 < ((𝐴 + 𝐵) / 2))) |
15 | 2, 7, 14 | 3bitr2d 310 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 < ((𝐴 + 𝐵) / 2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2113 class class class wbr 5027 (class class class)co 7164 ℂcc 10606 ℝcr 10607 0cc0 10608 + caddc 10611 · cmul 10613 < clt 10746 / cdiv 11368 2c2 11764 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-div 11369 df-2 11772 |
This theorem is referenced by: avgle2 11950 geomulcvg 15317 ruclem2 15670 ruclem3 15671 dvferm1lem 24728 dvferm2lem 24730 radcnvle 25159 psercnlem1 25164 psercn 25165 pserdvlem1 25166 logtayl 25395 iooelexlt 35145 ioomidp 42576 dvbdfbdioolem2 42996 dvbdfbdioo 42997 fourierdlem10 43184 |
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