![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > lt2addmuld | Structured version Visualization version GIF version |
Description: If two real numbers are less than a third real number, the sum of the two real numbers is less than twice the third real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
lt2addmuld.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
lt2addmuld.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
lt2addmuld.c | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
lt2addmuld.altc | ⊢ (𝜑 → 𝐴 < 𝐶) |
lt2addmuld.bltc | ⊢ (𝜑 → 𝐵 < 𝐶) |
Ref | Expression |
---|---|
lt2addmuld | ⊢ (𝜑 → (𝐴 + 𝐵) < (2 · 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lt2addmuld.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | lt2addmuld.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | lt2addmuld.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
4 | lt2addmuld.altc | . . 3 ⊢ (𝜑 → 𝐴 < 𝐶) | |
5 | lt2addmuld.bltc | . . 3 ⊢ (𝜑 → 𝐵 < 𝐶) | |
6 | 1, 2, 3, 3, 4, 5 | lt2addd 11874 | . 2 ⊢ (𝜑 → (𝐴 + 𝐵) < (𝐶 + 𝐶)) |
7 | 3 | recnd 11279 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
8 | 7 | 2timesd 12493 | . 2 ⊢ (𝜑 → (2 · 𝐶) = (𝐶 + 𝐶)) |
9 | 6, 8 | breqtrrd 5177 | 1 ⊢ (𝜑 → (𝐴 + 𝐵) < (2 · 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 class class class wbr 5149 (class class class)co 7419 ℝcr 11144 + caddc 11148 · cmul 11150 < clt 11285 2c2 12305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-2 12313 |
This theorem is referenced by: crctcshwlkn0lem5 29702 rmspecsqrtnq 42470 lt3addmuld 44823 |
Copyright terms: Public domain | W3C validator |