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Mirrors > Home > MPE Home > Th. List > lt2halvesd | Structured version Visualization version GIF version |
Description: A sum is less than the whole if each term is less than half. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
rehalfcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
lt2halvesd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
lt2halvesd.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
lt2halvesd.4 | ⊢ (𝜑 → 𝐴 < (𝐶 / 2)) |
lt2halvesd.5 | ⊢ (𝜑 → 𝐵 < (𝐶 / 2)) |
Ref | Expression |
---|---|
lt2halvesd | ⊢ (𝜑 → (𝐴 + 𝐵) < 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lt2halvesd.4 | . 2 ⊢ (𝜑 → 𝐴 < (𝐶 / 2)) | |
2 | lt2halvesd.5 | . 2 ⊢ (𝜑 → 𝐵 < (𝐶 / 2)) | |
3 | rehalfcld.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | lt2halvesd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
5 | lt2halvesd.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
6 | lt2halves 12469 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < (𝐶 / 2) ∧ 𝐵 < (𝐶 / 2)) → (𝐴 + 𝐵) < 𝐶)) | |
7 | 3, 4, 5, 6 | syl3anc 1369 | . 2 ⊢ (𝜑 → ((𝐴 < (𝐶 / 2) ∧ 𝐵 < (𝐶 / 2)) → (𝐴 + 𝐵) < 𝐶)) |
8 | 1, 2, 7 | mp2and 698 | 1 ⊢ (𝜑 → (𝐴 + 𝐵) < 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2099 class class class wbr 5142 (class class class)co 7414 ℝcr 11129 + caddc 11133 < clt 11270 / cdiv 11893 2c2 12289 |
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 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-2 12297 |
This theorem is referenced by: abs3lem 15309 metustexhalf 24452 nlmvscnlem2 24589 metdcnlem 24739 cntotbnd 37204 addlimc 44959 fourierdlem103 45520 fourierdlem104 45521 |
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