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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > le2halvesd | Structured version Visualization version GIF version |
Description: A sum is less than the whole if each term is less than half. (Contributed by Thierry Arnoux, 29-Nov-2017.) |
Ref | Expression |
---|---|
le2halvesd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
le2halvesd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
le2halvesd.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
le2halvesd.4 | ⊢ (𝜑 → 𝐴 ≤ (𝐶 / 2)) |
le2halvesd.5 | ⊢ (𝜑 → 𝐵 ≤ (𝐶 / 2)) |
Ref | Expression |
---|---|
le2halvesd | ⊢ (𝜑 → (𝐴 + 𝐵) ≤ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | le2halvesd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | le2halvesd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | le2halvesd.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
4 | 3 | rehalfcld 12513 | . . 3 ⊢ (𝜑 → (𝐶 / 2) ∈ ℝ) |
5 | le2halvesd.4 | . . 3 ⊢ (𝜑 → 𝐴 ≤ (𝐶 / 2)) | |
6 | le2halvesd.5 | . . 3 ⊢ (𝜑 → 𝐵 ≤ (𝐶 / 2)) | |
7 | 1, 2, 4, 4, 5, 6 | le2addd 11885 | . 2 ⊢ (𝜑 → (𝐴 + 𝐵) ≤ ((𝐶 / 2) + (𝐶 / 2))) |
8 | 3 | recnd 11294 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
9 | 8 | 2halvesd 12512 | . 2 ⊢ (𝜑 → ((𝐶 / 2) + (𝐶 / 2)) = 𝐶) |
10 | 7, 9 | breqtrd 5181 | 1 ⊢ (𝜑 → (𝐴 + 𝐵) ≤ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 class class class wbr 5155 (class class class)co 7426 ℝcr 11159 + caddc 11163 ≤ cle 11301 / cdiv 11923 2c2 12321 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-br 5156 df-opab 5218 df-mpt 5239 df-id 5582 df-po 5596 df-so 5597 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-er 8736 df-en 8977 df-dom 8978 df-sdom 8979 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11498 df-neg 11499 df-div 11924 df-2 12329 |
This theorem is referenced by: (None) |
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