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Mirrors > Home > MPE Home > Th. List > lt2addd | Structured version Visualization version GIF version |
Description: Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
leidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltnegd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
ltadd1d.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
lt2addd.4 | ⊢ (𝜑 → 𝐷 ∈ ℝ) |
lt2addd.5 | ⊢ (𝜑 → 𝐴 < 𝐶) |
lt2addd.6 | ⊢ (𝜑 → 𝐵 < 𝐷) |
Ref | Expression |
---|---|
lt2addd | ⊢ (𝜑 → (𝐴 + 𝐵) < (𝐶 + 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leidd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | ltnegd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | ltadd1d.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
4 | lt2addd.4 | . 2 ⊢ (𝜑 → 𝐷 ∈ ℝ) | |
5 | lt2addd.5 | . 2 ⊢ (𝜑 → 𝐴 < 𝐶) | |
6 | lt2addd.6 | . . 3 ⊢ (𝜑 → 𝐵 < 𝐷) | |
7 | 2, 4, 6 | ltled 10511 | . 2 ⊢ (𝜑 → 𝐵 ≤ 𝐷) |
8 | 1, 2, 3, 4, 5, 7 | ltleaddd 10980 | 1 ⊢ (𝜑 → (𝐴 + 𝐵) < (𝐶 + 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2164 class class class wbr 4875 (class class class)co 6910 ℝcr 10258 + caddc 10262 < clt 10398 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-po 5265 df-so 5266 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-ov 6913 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 |
This theorem is referenced by: lt2addmuld 11615 mertenslem1 14996 sadcaddlem 15559 uniioombllem3 23758 itg2cnlem2 23935 ang180lem2 24957 sqsscirc1 30495 hgt750lemd 31271 unbdqndv2lem1 33027 heicant 33987 mblfinlem3 33991 ftc1anclem7 34033 isbnd3 34124 dffltz 38096 lt3addmuld 40311 lt4addmuld 40316 stoweidlem13 41022 |
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