Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > le2addd | Structured version Visualization version GIF version |
Description: Adding both side of two inequalities. (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) |
le2addd.5 | ⊢ (𝜑 → 𝐴 ≤ 𝐶) |
le2addd.6 | ⊢ (𝜑 → 𝐵 ≤ 𝐷) |
Ref | Expression |
---|---|
le2addd | ⊢ (𝜑 → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | le2addd.5 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝐶) | |
2 | le2addd.6 | . 2 ⊢ (𝜑 → 𝐵 ≤ 𝐷) | |
3 | leidd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | ltnegd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
5 | ltadd1d.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
6 | lt2addd.4 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℝ) | |
7 | le2add 10389 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐷) → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷))) | |
8 | 3, 4, 5, 6, 7 | syl22anc 1319 | . 2 ⊢ (𝜑 → ((𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐷) → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷))) |
9 | 1, 2, 8 | mp2and 711 | 1 ⊢ (𝜑 → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 class class class wbr 4583 (class class class)co 6549 ℝcr 9814 + caddc 9818 ≤ cle 9954 |
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 |
This theorem is referenced by: supadd 10868 o1add 14192 o1sub 14194 o1fsum 14386 sadcaddlem 15017 4sqlem11 15497 4sqlem12 15498 4sqlem15 15501 4sqlem16 15502 prdsxmetlem 21983 nrmmetd 22189 nmotri 22353 pcoass 22632 minveclem2 23005 ovollb2lem 23063 ovolunlem1a 23071 ovoliunlem1 23077 nulmbl2 23111 ioombl1lem4 23136 uniioombllem5 23161 itg2splitlem 23321 itg2addlem 23331 ibladdlem 23392 ulmbdd 23956 cxpaddle 24293 ang180lem2 24340 fsumharmonic 24538 lgamgulmlem3 24557 lgamgulmlem5 24559 ppiub 24729 lgsdirprm 24856 lgsqrlem2 24872 lgseisenlem2 24901 2sqlem8 24951 vmadivsumb 24972 dchrisumlem2 24979 dchrisum0lem1b 25004 mulog2sumlem1 25023 mulog2sumlem2 25024 selbergb 25038 selberg2b 25041 chpdifbndlem1 25042 logdivbnd 25045 selberg3lem2 25047 pntrlog2bnd 25073 pntpbnd2 25076 pntibndlem2 25080 pntlemr 25091 ostth2lem2 25123 ostth3 25127 smcnlem 26936 minvecolem2 27115 stadd3i 28491 le2halvesd 28910 dnibndlem9 31646 ismblfin 32620 itg2addnc 32634 ibladdnclem 32636 ftc1anclem7 32661 pell1qrgaplem 36455 pellqrex 36461 pellfundgt1 36465 areaquad 36821 imo72b2lem0 37487 int-ineq1stprincd 37517 dvdivbd 38813 fourierdlem30 39030 sge0xaddlem2 39327 carageniuncllem2 39412 |
Copyright terms: Public domain | W3C validator |