Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lt4addmuld | Structured version Visualization version GIF version |
Description: If four real numbers are less than a fifth real number, the sum of the four real numbers is less than four times the fifth real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
lt4addmuld.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
lt4addmuld.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
lt4addmuld.c | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
lt4addmuld.d | ⊢ (𝜑 → 𝐷 ∈ ℝ) |
lt4addmuld.e | ⊢ (𝜑 → 𝐸 ∈ ℝ) |
lt4addmuld.alte | ⊢ (𝜑 → 𝐴 < 𝐸) |
lt4addmuld.blte | ⊢ (𝜑 → 𝐵 < 𝐸) |
lt4addmuld.clte | ⊢ (𝜑 → 𝐶 < 𝐸) |
lt4addmuld.dlte | ⊢ (𝜑 → 𝐷 < 𝐸) |
Ref | Expression |
---|---|
lt4addmuld | ⊢ (𝜑 → (((𝐴 + 𝐵) + 𝐶) + 𝐷) < (4 · 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lt4addmuld.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | lt4addmuld.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | 1, 2 | readdcld 10760 | . . . 4 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ) |
4 | lt4addmuld.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
5 | 3, 4 | readdcld 10760 | . . 3 ⊢ (𝜑 → ((𝐴 + 𝐵) + 𝐶) ∈ ℝ) |
6 | lt4addmuld.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℝ) | |
7 | 3re 11808 | . . . . 5 ⊢ 3 ∈ ℝ | |
8 | 7 | a1i 11 | . . . 4 ⊢ (𝜑 → 3 ∈ ℝ) |
9 | lt4addmuld.e | . . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ) | |
10 | 8, 9 | remulcld 10761 | . . 3 ⊢ (𝜑 → (3 · 𝐸) ∈ ℝ) |
11 | lt4addmuld.alte | . . . 4 ⊢ (𝜑 → 𝐴 < 𝐸) | |
12 | lt4addmuld.blte | . . . 4 ⊢ (𝜑 → 𝐵 < 𝐸) | |
13 | lt4addmuld.clte | . . . 4 ⊢ (𝜑 → 𝐶 < 𝐸) | |
14 | 1, 2, 4, 9, 11, 12, 13 | lt3addmuld 42418 | . . 3 ⊢ (𝜑 → ((𝐴 + 𝐵) + 𝐶) < (3 · 𝐸)) |
15 | lt4addmuld.dlte | . . 3 ⊢ (𝜑 → 𝐷 < 𝐸) | |
16 | 5, 6, 10, 9, 14, 15 | lt2addd 11353 | . 2 ⊢ (𝜑 → (((𝐴 + 𝐵) + 𝐶) + 𝐷) < ((3 · 𝐸) + 𝐸)) |
17 | df-4 11793 | . . . . 5 ⊢ 4 = (3 + 1) | |
18 | 17 | a1i 11 | . . . 4 ⊢ (𝜑 → 4 = (3 + 1)) |
19 | 18 | oveq1d 7197 | . . 3 ⊢ (𝜑 → (4 · 𝐸) = ((3 + 1) · 𝐸)) |
20 | 8 | recnd 10759 | . . . 4 ⊢ (𝜑 → 3 ∈ ℂ) |
21 | 9 | recnd 10759 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ ℂ) |
22 | 20, 21 | adddirp1d 10757 | . . 3 ⊢ (𝜑 → ((3 + 1) · 𝐸) = ((3 · 𝐸) + 𝐸)) |
23 | 19, 22 | eqtr2d 2775 | . 2 ⊢ (𝜑 → ((3 · 𝐸) + 𝐸) = (4 · 𝐸)) |
24 | 16, 23 | breqtrd 5066 | 1 ⊢ (𝜑 → (((𝐴 + 𝐵) + 𝐶) + 𝐷) < (4 · 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 class class class wbr 5040 (class class class)co 7182 ℝcr 10626 1c1 10628 + caddc 10630 · cmul 10632 < clt 10765 3c3 11784 4c4 11785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-po 5452 df-so 5453 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7185 df-er 8332 df-en 8568 df-dom 8569 df-sdom 8570 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-2 11791 df-3 11792 df-4 11793 |
This theorem is referenced by: limclner 42774 |
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