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Mirrors > Home > ILE Home > Th. List > avglt1 | 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 7523 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 + 𝐴) < (𝐴 + 𝐵))) | |
2 | 1 | 3anidm13 1227 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 + 𝐴) < (𝐴 + 𝐵))) |
3 | simpl 107 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℝ) | |
4 | 3 | recnd 7147 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℂ) |
5 | times2 8161 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 · 2) = (𝐴 + 𝐴)) | |
6 | 4, 5 | syl 14 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 2) = (𝐴 + 𝐴)) |
7 | 6 | breq1d 3795 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 · 2) < (𝐴 + 𝐵) ↔ (𝐴 + 𝐴) < (𝐴 + 𝐵))) |
8 | readdcl 7099 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ) | |
9 | 2re 8109 | . . . . 5 ⊢ 2 ∈ ℝ | |
10 | 2pos 8130 | . . . . 5 ⊢ 0 < 2 | |
11 | 9, 10 | pm3.2i 266 | . . . 4 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
12 | 11 | a1i 9 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (2 ∈ ℝ ∧ 0 < 2)) |
13 | ltmuldiv 7952 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝐴 + 𝐵) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((𝐴 · 2) < (𝐴 + 𝐵) ↔ 𝐴 < ((𝐴 + 𝐵) / 2))) | |
14 | 3, 8, 12, 13 | syl3anc 1169 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 · 2) < (𝐴 + 𝐵) ↔ 𝐴 < ((𝐴 + 𝐵) / 2))) |
15 | 2, 7, 14 | 3bitr2d 214 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 < ((𝐴 + 𝐵) / 2))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1284 ∈ wcel 1433 class class class wbr 3785 (class class class)co 5532 ℂcc 6979 ℝcr 6980 0cc0 6981 + caddc 6984 · cmul 6986 < clt 7153 / cdiv 7760 2c2 8089 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-po 4051 df-iso 4052 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-2 8098 |
This theorem is referenced by: avgle2 8272 |
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