Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
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 9948 | . . . 4 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ) |
4 | lt4addmuld.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
5 | 3, 4 | readdcld 9948 | . . 3 ⊢ (𝜑 → ((𝐴 + 𝐵) + 𝐶) ∈ ℝ) |
6 | lt4addmuld.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℝ) | |
7 | 3re 10971 | . . . . 5 ⊢ 3 ∈ ℝ | |
8 | 7 | a1i 11 | . . . 4 ⊢ (𝜑 → 3 ∈ ℝ) |
9 | lt4addmuld.e | . . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ) | |
10 | 8, 9 | remulcld 9949 | . . 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 38456 | . . 3 ⊢ (𝜑 → ((𝐴 + 𝐵) + 𝐶) < (3 · 𝐸)) |
15 | lt4addmuld.dlte | . . 3 ⊢ (𝜑 → 𝐷 < 𝐸) | |
16 | 5, 6, 10, 9, 14, 15 | lt2addd 10529 | . 2 ⊢ (𝜑 → (((𝐴 + 𝐵) + 𝐶) + 𝐷) < ((3 · 𝐸) + 𝐸)) |
17 | df-4 10958 | . . . . 5 ⊢ 4 = (3 + 1) | |
18 | 17 | a1i 11 | . . . 4 ⊢ (𝜑 → 4 = (3 + 1)) |
19 | 18 | oveq1d 6564 | . . 3 ⊢ (𝜑 → (4 · 𝐸) = ((3 + 1) · 𝐸)) |
20 | 8 | recnd 9947 | . . . 4 ⊢ (𝜑 → 3 ∈ ℂ) |
21 | 9 | recnd 9947 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ ℂ) |
22 | 20, 21 | adddirp1d 9945 | . . 3 ⊢ (𝜑 → ((3 + 1) · 𝐸) = ((3 · 𝐸) + 𝐸)) |
23 | 19, 22 | eqtr2d 2645 | . 2 ⊢ (𝜑 → ((3 · 𝐸) + 𝐸) = (4 · 𝐸)) |
24 | 16, 23 | breqtrd 4609 | 1 ⊢ (𝜑 → (((𝐴 + 𝐵) + 𝐶) + 𝐷) < (4 · 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 (class class class)co 6549 ℝcr 9814 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 3c3 10948 4c4 10949 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-2 10956 df-3 10957 df-4 10958 |
This theorem is referenced by: limclner 38718 |
Copyright terms: Public domain | W3C validator |