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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 10406 | . . 3 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ) |
4 | lt3addmuld.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
5 | 2re 11449 | . . . . 5 ⊢ 2 ∈ ℝ | |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝜑 → 2 ∈ ℝ) |
7 | lt3addmuld.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ) | |
8 | 6, 7 | remulcld 10407 | . . 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 11632 | . . 3 ⊢ (𝜑 → (𝐴 + 𝐵) < (2 · 𝐷)) |
12 | lt3addmuld.cltd | . . 3 ⊢ (𝜑 → 𝐶 < 𝐷) | |
13 | 3, 4, 8, 7, 11, 12 | lt2addd 10998 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐵) + 𝐶) < ((2 · 𝐷) + 𝐷)) |
14 | 6 | recnd 10405 | . . . 4 ⊢ (𝜑 → 2 ∈ ℂ) |
15 | 7 | recnd 10405 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
16 | 14, 15 | adddirp1d 10403 | . . 3 ⊢ (𝜑 → ((2 + 1) · 𝐷) = ((2 · 𝐷) + 𝐷)) |
17 | 2p1e3 11524 | . . . . 5 ⊢ (2 + 1) = 3 | |
18 | 17 | a1i 11 | . . . 4 ⊢ (𝜑 → (2 + 1) = 3) |
19 | 18 | oveq1d 6937 | . . 3 ⊢ (𝜑 → ((2 + 1) · 𝐷) = (3 · 𝐷)) |
20 | 16, 19 | eqtr3d 2816 | . 2 ⊢ (𝜑 → ((2 · 𝐷) + 𝐷) = (3 · 𝐷)) |
21 | 13, 20 | breqtrd 4912 | 1 ⊢ (𝜑 → ((𝐴 + 𝐵) + 𝐶) < (3 · 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 class class class wbr 4886 (class class class)co 6922 ℝcr 10271 1c1 10273 + caddc 10275 · cmul 10277 < clt 10411 2c2 11430 3c3 11431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-po 5274 df-so 5275 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-ov 6925 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-2 11438 df-3 11439 |
This theorem is referenced by: lt4addmuld 40429 |
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