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Mathbox for Glauco Siliprandi |
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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 10659 | . . 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 10660 | . . 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 11252 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐵) + 𝐶) < ((2 · 𝐷) + 𝐷)) |
14 | 6 | recnd 10658 | . . . 4 ⊢ (𝜑 → 2 ∈ ℂ) |
15 | 7 | recnd 10658 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
16 | 14, 15 | adddirp1d 10656 | . . 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 7150 | . . 3 ⊢ (𝜑 → ((2 + 1) · 𝐷) = (3 · 𝐷)) |
20 | 16, 19 | eqtr3d 2835 | . 2 ⊢ (𝜑 → ((2 · 𝐷) + 𝐷) = (3 · 𝐷)) |
21 | 13, 20 | breqtrd 5056 | 1 ⊢ (𝜑 → ((𝐴 + 𝐵) + 𝐶) < (3 · 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 (class class class)co 7135 ℝcr 10525 1c1 10527 + caddc 10529 · cmul 10531 < clt 10664 2c2 11680 3c3 11681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-2 11688 df-3 11689 |
This theorem is referenced by: lt4addmuld 41938 |
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