![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > avglt2 | GIF version |
Description: Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.) |
Ref | Expression |
---|---|
avglt2 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ((𝐴 + 𝐵) / 2) < 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 110 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ) | |
2 | 1 | recnd 8048 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℂ) |
3 | 2times 9110 | . . . 4 ⊢ (𝐵 ∈ ℂ → (2 · 𝐵) = (𝐵 + 𝐵)) | |
4 | 2, 3 | syl 14 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (2 · 𝐵) = (𝐵 + 𝐵)) |
5 | 4 | breq2d 4041 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 + 𝐵) < (2 · 𝐵) ↔ (𝐴 + 𝐵) < (𝐵 + 𝐵))) |
6 | readdcl 7998 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ) | |
7 | 2re 9052 | . . . . 5 ⊢ 2 ∈ ℝ | |
8 | 2pos 9073 | . . . . 5 ⊢ 0 < 2 | |
9 | 7, 8 | pm3.2i 272 | . . . 4 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
10 | 9 | a1i 9 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (2 ∈ ℝ ∧ 0 < 2)) |
11 | ltdivmul 8895 | . . 3 ⊢ (((𝐴 + 𝐵) ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (((𝐴 + 𝐵) / 2) < 𝐵 ↔ (𝐴 + 𝐵) < (2 · 𝐵))) | |
12 | 6, 1, 10, 11 | syl3anc 1249 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (((𝐴 + 𝐵) / 2) < 𝐵 ↔ (𝐴 + 𝐵) < (2 · 𝐵))) |
13 | ltadd1 8448 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 + 𝐵) < (𝐵 + 𝐵))) | |
14 | 13 | 3anidm23 1308 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 + 𝐵) < (𝐵 + 𝐵))) |
15 | 5, 12, 14 | 3bitr4rd 221 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ((𝐴 + 𝐵) / 2) < 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∈ wcel 2164 class class class wbr 4029 (class class class)co 5918 ℂcc 7870 ℝcr 7871 0cc0 7872 + caddc 7875 · cmul 7877 < clt 8054 / cdiv 8691 2c2 9033 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-br 4030 df-opab 4091 df-id 4324 df-po 4327 df-iso 4328 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-iota 5215 df-fun 5256 df-fv 5262 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-2 9041 |
This theorem is referenced by: avgle1 9223 apdifflemf 15536 |
Copyright terms: Public domain | W3C validator |