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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 10459 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 + 𝐴) < (𝐴 + 𝐵))) | |
2 | 1 | 3anidm13 1549 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 + 𝐴) < (𝐴 + 𝐵))) |
3 | simpl 476 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℝ) | |
4 | 3 | recnd 10384 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℂ) |
5 | times2 11494 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 · 2) = (𝐴 + 𝐴)) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 2) = (𝐴 + 𝐴)) |
7 | 6 | breq1d 4882 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 · 2) < (𝐴 + 𝐵) ↔ (𝐴 + 𝐴) < (𝐴 + 𝐵))) |
8 | readdcl 10334 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ) | |
9 | 2re 11424 | . . . . 5 ⊢ 2 ∈ ℝ | |
10 | 2pos 11460 | . . . . 5 ⊢ 0 < 2 | |
11 | 9, 10 | pm3.2i 464 | . . . 4 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
12 | 11 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (2 ∈ ℝ ∧ 0 < 2)) |
13 | ltmuldiv 11225 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝐴 + 𝐵) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((𝐴 · 2) < (𝐴 + 𝐵) ↔ 𝐴 < ((𝐴 + 𝐵) / 2))) | |
14 | 3, 8, 12, 13 | syl3anc 1496 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 · 2) < (𝐴 + 𝐵) ↔ 𝐴 < ((𝐴 + 𝐵) / 2))) |
15 | 2, 7, 14 | 3bitr2d 299 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 < ((𝐴 + 𝐵) / 2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 class class class wbr 4872 (class class class)co 6904 ℂcc 10249 ℝcr 10250 0cc0 10251 + caddc 10254 · cmul 10256 < clt 10390 / cdiv 11008 2c2 11405 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-op 4403 df-uni 4658 df-br 4873 df-opab 4935 df-mpt 4952 df-id 5249 df-po 5262 df-so 5263 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-er 8008 df-en 8222 df-dom 8223 df-sdom 8224 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-div 11009 df-2 11413 |
This theorem is referenced by: avgle2 11598 geomulcvg 14980 ruclem2 15334 ruclem3 15335 dvferm1lem 24145 dvferm2lem 24147 radcnvle 24572 psercnlem1 24577 psercn 24578 pserdvlem1 24579 logtayl 24804 iooelexlt 33754 ioomidp 40535 dvbdfbdioolem2 40938 dvbdfbdioo 40939 fourierdlem10 41127 |
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