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Mirrors > Home > MPE Home > Th. List > lt2halves | Structured version Visualization version GIF version |
Description: A sum is less than the whole if each term is less than half. (Contributed by NM, 13-Dec-2006.) |
Ref | Expression |
---|---|
lt2halves | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < (𝐶 / 2) ∧ 𝐵 < (𝐶 / 2)) → (𝐴 + 𝐵) < 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3simpa 1144 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) | |
2 | rehalfcl 11864 | . . . . 5 ⊢ (𝐶 ∈ ℝ → (𝐶 / 2) ∈ ℝ) | |
3 | 2, 2 | jca 514 | . . . 4 ⊢ (𝐶 ∈ ℝ → ((𝐶 / 2) ∈ ℝ ∧ (𝐶 / 2) ∈ ℝ)) |
4 | 3 | 3ad2ant3 1131 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐶 / 2) ∈ ℝ ∧ (𝐶 / 2) ∈ ℝ)) |
5 | lt2add 11125 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ ((𝐶 / 2) ∈ ℝ ∧ (𝐶 / 2) ∈ ℝ)) → ((𝐴 < (𝐶 / 2) ∧ 𝐵 < (𝐶 / 2)) → (𝐴 + 𝐵) < ((𝐶 / 2) + (𝐶 / 2)))) | |
6 | 1, 4, 5 | syl2anc 586 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < (𝐶 / 2) ∧ 𝐵 < (𝐶 / 2)) → (𝐴 + 𝐵) < ((𝐶 / 2) + (𝐶 / 2)))) |
7 | recn 10627 | . . . . 5 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℂ) | |
8 | 2halves 11866 | . . . . 5 ⊢ (𝐶 ∈ ℂ → ((𝐶 / 2) + (𝐶 / 2)) = 𝐶) | |
9 | 7, 8 | syl 17 | . . . 4 ⊢ (𝐶 ∈ ℝ → ((𝐶 / 2) + (𝐶 / 2)) = 𝐶) |
10 | 9 | breq2d 5078 | . . 3 ⊢ (𝐶 ∈ ℝ → ((𝐴 + 𝐵) < ((𝐶 / 2) + (𝐶 / 2)) ↔ (𝐴 + 𝐵) < 𝐶)) |
11 | 10 | 3ad2ant3 1131 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) < ((𝐶 / 2) + (𝐶 / 2)) ↔ (𝐴 + 𝐵) < 𝐶)) |
12 | 6, 11 | sylibd 241 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < (𝐶 / 2) ∧ 𝐵 < (𝐶 / 2)) → (𝐴 + 𝐵) < 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 (class class class)co 7156 ℂcc 10535 ℝcr 10536 + caddc 10540 < clt 10675 / cdiv 11297 2c2 11693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-2 11701 |
This theorem is referenced by: lt2halvesd 11886 ngptgp 23245 vacn 28471 bfplem2 35116 |
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