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 10664 | . . . 4 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ) |
4 | lt4addmuld.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
5 | 3, 4 | readdcld 10664 | . . 3 ⊢ (𝜑 → ((𝐴 + 𝐵) + 𝐶) ∈ ℝ) |
6 | lt4addmuld.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℝ) | |
7 | 3re 11711 | . . . . 5 ⊢ 3 ∈ ℝ | |
8 | 7 | a1i 11 | . . . 4 ⊢ (𝜑 → 3 ∈ ℝ) |
9 | lt4addmuld.e | . . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ) | |
10 | 8, 9 | remulcld 10665 | . . 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 41561 | . . 3 ⊢ (𝜑 → ((𝐴 + 𝐵) + 𝐶) < (3 · 𝐸)) |
15 | lt4addmuld.dlte | . . 3 ⊢ (𝜑 → 𝐷 < 𝐸) | |
16 | 5, 6, 10, 9, 14, 15 | lt2addd 11257 | . 2 ⊢ (𝜑 → (((𝐴 + 𝐵) + 𝐶) + 𝐷) < ((3 · 𝐸) + 𝐸)) |
17 | df-4 11696 | . . . . 5 ⊢ 4 = (3 + 1) | |
18 | 17 | a1i 11 | . . . 4 ⊢ (𝜑 → 4 = (3 + 1)) |
19 | 18 | oveq1d 7165 | . . 3 ⊢ (𝜑 → (4 · 𝐸) = ((3 + 1) · 𝐸)) |
20 | 8 | recnd 10663 | . . . 4 ⊢ (𝜑 → 3 ∈ ℂ) |
21 | 9 | recnd 10663 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ ℂ) |
22 | 20, 21 | adddirp1d 10661 | . . 3 ⊢ (𝜑 → ((3 + 1) · 𝐸) = ((3 · 𝐸) + 𝐸)) |
23 | 19, 22 | eqtr2d 2857 | . 2 ⊢ (𝜑 → ((3 · 𝐸) + 𝐸) = (4 · 𝐸)) |
24 | 16, 23 | breqtrd 5084 | 1 ⊢ (𝜑 → (((𝐴 + 𝐵) + 𝐶) + 𝐷) < (4 · 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 class class class wbr 5058 (class class class)co 7150 ℝcr 10530 1c1 10532 + caddc 10534 · cmul 10536 < clt 10669 3c3 11687 4c4 11688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-2 11694 df-3 11695 df-4 11696 |
This theorem is referenced by: limclner 41925 |
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