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Mirrors > Home > MPE Home > Th. List > avgle2 | Structured version Visualization version GIF version |
Description: Ordering property for average. (Contributed by Jeff Hankins, 15-Sep-2013.) (Revised by Mario Carneiro, 28-May-2014.) |
Ref | Expression |
---|---|
avgle2 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ((𝐴 + 𝐵) / 2) ≤ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | avglt1 11473 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 < 𝐴 ↔ 𝐵 < ((𝐵 + 𝐴) / 2))) | |
2 | 1 | ancoms 446 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴 ↔ 𝐵 < ((𝐵 + 𝐴) / 2))) |
3 | recn 10228 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
4 | recn 10228 | . . . . . . 7 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
5 | addcom 10424 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) | |
6 | 3, 4, 5 | syl2an 583 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
7 | 6 | oveq1d 6807 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 + 𝐵) / 2) = ((𝐵 + 𝐴) / 2)) |
8 | 7 | breq2d 4798 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 < ((𝐴 + 𝐵) / 2) ↔ 𝐵 < ((𝐵 + 𝐴) / 2))) |
9 | 2, 8 | bitr4d 271 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴 ↔ 𝐵 < ((𝐴 + 𝐵) / 2))) |
10 | 9 | notbid 307 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ 𝐵 < 𝐴 ↔ ¬ 𝐵 < ((𝐴 + 𝐵) / 2))) |
11 | lenlt 10318 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | |
12 | readdcl 10221 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ) | |
13 | rehalfcl 11461 | . . . 4 ⊢ ((𝐴 + 𝐵) ∈ ℝ → ((𝐴 + 𝐵) / 2) ∈ ℝ) | |
14 | 12, 13 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 + 𝐵) / 2) ∈ ℝ) |
15 | lenlt 10318 | . . 3 ⊢ ((((𝐴 + 𝐵) / 2) ∈ ℝ ∧ 𝐵 ∈ ℝ) → (((𝐴 + 𝐵) / 2) ≤ 𝐵 ↔ ¬ 𝐵 < ((𝐴 + 𝐵) / 2))) | |
16 | 14, 15 | sylancom 576 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (((𝐴 + 𝐵) / 2) ≤ 𝐵 ↔ ¬ 𝐵 < ((𝐴 + 𝐵) / 2))) |
17 | 10, 11, 16 | 3bitr4d 300 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ((𝐴 + 𝐵) / 2) ≤ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 class class class wbr 4786 (class class class)co 6792 ℂcc 10136 ℝcr 10137 + caddc 10141 < clt 10276 ≤ cle 10277 / cdiv 10886 2c2 11272 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-po 5170 df-so 5171 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-2 11281 |
This theorem is referenced by: avgle 11477 pilem3 24423 pilem3OLD 24424 |
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