![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 8205 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 + 𝐴) < (𝐴 + 𝐵))) | |
2 | 1 | 3anidm13 1275 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 + 𝐴) < (𝐴 + 𝐵))) |
3 | simpl 108 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℝ) | |
4 | 3 | recnd 7818 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℂ) |
5 | times2 8873 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 · 2) = (𝐴 + 𝐴)) | |
6 | 4, 5 | syl 14 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 2) = (𝐴 + 𝐴)) |
7 | 6 | breq1d 3947 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 · 2) < (𝐴 + 𝐵) ↔ (𝐴 + 𝐴) < (𝐴 + 𝐵))) |
8 | readdcl 7770 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ) | |
9 | 2re 8814 | . . . . 5 ⊢ 2 ∈ ℝ | |
10 | 2pos 8835 | . . . . 5 ⊢ 0 < 2 | |
11 | 9, 10 | pm3.2i 270 | . . . 4 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
12 | 11 | a1i 9 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (2 ∈ ℝ ∧ 0 < 2)) |
13 | ltmuldiv 8656 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝐴 + 𝐵) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((𝐴 · 2) < (𝐴 + 𝐵) ↔ 𝐴 < ((𝐴 + 𝐵) / 2))) | |
14 | 3, 8, 12, 13 | syl3anc 1217 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 · 2) < (𝐴 + 𝐵) ↔ 𝐴 < ((𝐴 + 𝐵) / 2))) |
15 | 2, 7, 14 | 3bitr2d 215 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 < ((𝐴 + 𝐵) / 2))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1332 ∈ wcel 1481 class class class wbr 3937 (class class class)co 5782 ℂcc 7642 ℝcr 7643 0cc0 7644 + caddc 7647 · cmul 7649 < clt 7824 / cdiv 8456 2c2 8795 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-po 4226 df-iso 4227 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-2 8803 |
This theorem is referenced by: avgle2 8985 apdifflemf 13414 |
Copyright terms: Public domain | W3C validator |