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 10670 | . . . 4 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ) |
4 | lt4addmuld.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
5 | 3, 4 | readdcld 10670 | . . 3 ⊢ (𝜑 → ((𝐴 + 𝐵) + 𝐶) ∈ ℝ) |
6 | lt4addmuld.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℝ) | |
7 | 3re 11718 | . . . . 5 ⊢ 3 ∈ ℝ | |
8 | 7 | a1i 11 | . . . 4 ⊢ (𝜑 → 3 ∈ ℝ) |
9 | lt4addmuld.e | . . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ) | |
10 | 8, 9 | remulcld 10671 | . . 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 41588 | . . 3 ⊢ (𝜑 → ((𝐴 + 𝐵) + 𝐶) < (3 · 𝐸)) |
15 | lt4addmuld.dlte | . . 3 ⊢ (𝜑 → 𝐷 < 𝐸) | |
16 | 5, 6, 10, 9, 14, 15 | lt2addd 11263 | . 2 ⊢ (𝜑 → (((𝐴 + 𝐵) + 𝐶) + 𝐷) < ((3 · 𝐸) + 𝐸)) |
17 | df-4 11703 | . . . . 5 ⊢ 4 = (3 + 1) | |
18 | 17 | a1i 11 | . . . 4 ⊢ (𝜑 → 4 = (3 + 1)) |
19 | 18 | oveq1d 7171 | . . 3 ⊢ (𝜑 → (4 · 𝐸) = ((3 + 1) · 𝐸)) |
20 | 8 | recnd 10669 | . . . 4 ⊢ (𝜑 → 3 ∈ ℂ) |
21 | 9 | recnd 10669 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ ℂ) |
22 | 20, 21 | adddirp1d 10667 | . . 3 ⊢ (𝜑 → ((3 + 1) · 𝐸) = ((3 · 𝐸) + 𝐸)) |
23 | 19, 22 | eqtr2d 2857 | . 2 ⊢ (𝜑 → ((3 · 𝐸) + 𝐸) = (4 · 𝐸)) |
24 | 16, 23 | breqtrd 5092 | 1 ⊢ (𝜑 → (((𝐴 + 𝐵) + 𝐶) + 𝐷) < (4 · 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 (class class class)co 7156 ℝcr 10536 1c1 10538 + caddc 10540 · cmul 10542 < clt 10675 3c3 11694 4c4 11695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-2 11701 df-3 11702 df-4 11703 |
This theorem is referenced by: limclner 41952 |
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