Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > lt3addmuld | Structured version Visualization version GIF version |
Description: If three real numbers are less than a fourth real number, the sum of the three real numbers is less than three times the third real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
lt3addmuld.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
lt3addmuld.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
lt3addmuld.c | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
lt3addmuld.d | ⊢ (𝜑 → 𝐷 ∈ ℝ) |
lt3addmuld.altd | ⊢ (𝜑 → 𝐴 < 𝐷) |
lt3addmuld.bltd | ⊢ (𝜑 → 𝐵 < 𝐷) |
lt3addmuld.cltd | ⊢ (𝜑 → 𝐶 < 𝐷) |
Ref | Expression |
---|---|
lt3addmuld | ⊢ (𝜑 → ((𝐴 + 𝐵) + 𝐶) < (3 · 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lt3addmuld.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | lt3addmuld.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | 1, 2 | readdcld 10658 | . . 3 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ) |
4 | lt3addmuld.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
5 | 2re 11699 | . . . . 5 ⊢ 2 ∈ ℝ | |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝜑 → 2 ∈ ℝ) |
7 | lt3addmuld.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ) | |
8 | 6, 7 | remulcld 10659 | . . 3 ⊢ (𝜑 → (2 · 𝐷) ∈ ℝ) |
9 | lt3addmuld.altd | . . . 4 ⊢ (𝜑 → 𝐴 < 𝐷) | |
10 | lt3addmuld.bltd | . . . 4 ⊢ (𝜑 → 𝐵 < 𝐷) | |
11 | 1, 2, 7, 9, 10 | lt2addmuld 11875 | . . 3 ⊢ (𝜑 → (𝐴 + 𝐵) < (2 · 𝐷)) |
12 | lt3addmuld.cltd | . . 3 ⊢ (𝜑 → 𝐶 < 𝐷) | |
13 | 3, 4, 8, 7, 11, 12 | lt2addd 11251 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐵) + 𝐶) < ((2 · 𝐷) + 𝐷)) |
14 | 6 | recnd 10657 | . . . 4 ⊢ (𝜑 → 2 ∈ ℂ) |
15 | 7 | recnd 10657 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
16 | 14, 15 | adddirp1d 10655 | . . 3 ⊢ (𝜑 → ((2 + 1) · 𝐷) = ((2 · 𝐷) + 𝐷)) |
17 | 2p1e3 11767 | . . . . 5 ⊢ (2 + 1) = 3 | |
18 | 17 | a1i 11 | . . . 4 ⊢ (𝜑 → (2 + 1) = 3) |
19 | 18 | oveq1d 7160 | . . 3 ⊢ (𝜑 → ((2 + 1) · 𝐷) = (3 · 𝐷)) |
20 | 16, 19 | eqtr3d 2855 | . 2 ⊢ (𝜑 → ((2 · 𝐷) + 𝐷) = (3 · 𝐷)) |
21 | 13, 20 | breqtrd 5083 | 1 ⊢ (𝜑 → ((𝐴 + 𝐵) + 𝐶) < (3 · 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 (class class class)co 7145 ℝcr 10524 1c1 10526 + caddc 10528 · cmul 10530 < clt 10663 2c2 11680 3c3 11681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-2 11688 df-3 11689 |
This theorem is referenced by: lt4addmuld 41449 |
Copyright terms: Public domain | W3C validator |