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Mirrors > Home > ILE Home > Th. List > qavgle | GIF version |
Description: The average of two rational numbers is less than or equal to at least one of them. (Contributed by Jim Kingdon, 3-Nov-2021.) |
Ref | Expression |
---|---|
qavgle | ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (((𝐴 + 𝐵) / 2) ≤ 𝐴 ∨ ((𝐴 + 𝐵) / 2) ≤ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qletric 10273 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴)) | |
2 | 1 | orcomd 730 | . 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐵 ≤ 𝐴 ∨ 𝐴 ≤ 𝐵)) |
3 | qre 9654 | . . . . . 6 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℝ) | |
4 | 3 | adantl 277 | . . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → 𝐵 ∈ ℝ) |
5 | qre 9654 | . . . . . 6 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ) | |
6 | 5 | adantr 276 | . . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → 𝐴 ∈ ℝ) |
7 | avgle2 9189 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 ≤ 𝐴 ↔ ((𝐵 + 𝐴) / 2) ≤ 𝐴)) | |
8 | 4, 6, 7 | syl2anc 411 | . . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐵 ≤ 𝐴 ↔ ((𝐵 + 𝐴) / 2) ≤ 𝐴)) |
9 | qcn 9663 | . . . . . . . 8 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ) | |
10 | 9 | adantr 276 | . . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → 𝐴 ∈ ℂ) |
11 | qcn 9663 | . . . . . . . 8 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℂ) | |
12 | 11 | adantl 277 | . . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → 𝐵 ∈ ℂ) |
13 | 10, 12 | addcomd 8137 | . . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
14 | 13 | oveq1d 5910 | . . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((𝐴 + 𝐵) / 2) = ((𝐵 + 𝐴) / 2)) |
15 | 14 | breq1d 4028 | . . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (((𝐴 + 𝐵) / 2) ≤ 𝐴 ↔ ((𝐵 + 𝐴) / 2) ≤ 𝐴)) |
16 | 8, 15 | bitr4d 191 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐵 ≤ 𝐴 ↔ ((𝐴 + 𝐵) / 2) ≤ 𝐴)) |
17 | avgle2 9189 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ((𝐴 + 𝐵) / 2) ≤ 𝐵)) | |
18 | 6, 4, 17 | syl2anc 411 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 ≤ 𝐵 ↔ ((𝐴 + 𝐵) / 2) ≤ 𝐵)) |
19 | 16, 18 | orbi12d 794 | . 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((𝐵 ≤ 𝐴 ∨ 𝐴 ≤ 𝐵) ↔ (((𝐴 + 𝐵) / 2) ≤ 𝐴 ∨ ((𝐴 + 𝐵) / 2) ≤ 𝐵))) |
20 | 2, 19 | mpbid 147 | 1 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (((𝐴 + 𝐵) / 2) ≤ 𝐴 ∨ ((𝐴 + 𝐵) / 2) ≤ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 709 ∈ wcel 2160 class class class wbr 4018 (class class class)co 5895 ℂcc 7838 ℝcr 7839 + caddc 7843 ≤ cle 8022 / cdiv 8658 2c2 8999 ℚcq 9648 |
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 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7931 ax-resscn 7932 ax-1cn 7933 ax-1re 7934 ax-icn 7935 ax-addcl 7936 ax-addrcl 7937 ax-mulcl 7938 ax-mulrcl 7939 ax-addcom 7940 ax-mulcom 7941 ax-addass 7942 ax-mulass 7943 ax-distr 7944 ax-i2m1 7945 ax-0lt1 7946 ax-1rid 7947 ax-0id 7948 ax-rnegex 7949 ax-precex 7950 ax-cnre 7951 ax-pre-ltirr 7952 ax-pre-ltwlin 7953 ax-pre-lttrn 7954 ax-pre-apti 7955 ax-pre-ltadd 7956 ax-pre-mulgt0 7957 ax-pre-mulext 7958 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-po 4314 df-iso 4315 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fv 5243 df-riota 5851 df-ov 5898 df-oprab 5899 df-mpo 5900 df-1st 6164 df-2nd 6165 df-pnf 8023 df-mnf 8024 df-xr 8025 df-ltxr 8026 df-le 8027 df-sub 8159 df-neg 8160 df-reap 8561 df-ap 8568 df-div 8659 df-inn 8949 df-2 9007 df-n0 9206 df-z 9283 df-q 9649 df-rp 9683 |
This theorem is referenced by: facavg 10757 |
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