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| 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 11745 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐷) → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷))) | |
| 8 | 3, 4, 5, 6, 7 | syl22anc 839 | . 2 ⊢ (𝜑 → ((𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐷) → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷))) |
| 9 | 1, 2, 8 | mp2and 699 | 1 ⊢ (𝜑 → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 ℝcr 11154 + caddc 11158 ≤ cle 11296 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 |
| This theorem is referenced by: supadd 12236 o1add 15650 o1sub 15652 o1fsum 15849 sadcaddlem 16494 4sqlem11 16993 4sqlem12 16994 4sqlem15 16997 4sqlem16 16998 prdsxmetlem 24378 nrmmetd 24587 nmotri 24760 pcoass 25057 minveclem2 25460 ovollb2lem 25523 ovolunlem1a 25531 ovoliunlem1 25537 nulmbl2 25571 ioombl1lem4 25596 uniioombllem5 25622 itg2splitlem 25783 itg2addlem 25793 ibladdlem 25855 ulmbdd 26441 cxpaddle 26795 ang180lem2 26853 fsumharmonic 27055 lgamgulmlem3 27074 lgamgulmlem5 27076 ppiub 27248 lgsdirprm 27375 lgsqrlem2 27391 lgseisenlem2 27420 2sqlem8 27470 vmadivsumb 27527 dchrisumlem2 27534 dchrisum0lem1b 27559 mulog2sumlem1 27578 mulog2sumlem2 27579 selbergb 27593 selberg2b 27596 chpdifbndlem1 27597 logdivbnd 27600 selberg3lem2 27602 pntrlog2bnd 27628 pntpbnd2 27631 pntibndlem2 27635 pntlemr 27646 ostth2lem2 27678 ostth3 27682 smcnlem 30716 minvecolem2 30894 stadd3i 32267 le2halvesd 32759 wrdt2ind 32938 dnibndlem9 36487 ismblfin 37668 itg2addnc 37681 ibladdnclem 37683 ftc1anclem7 37706 intlewftc 42062 aks4d1p1p2 42071 dvle2 42073 posbezout 42101 2np3bcnp1 42145 sticksstones7 42153 sticksstones12a 42158 sticksstones12 42159 metakunt29 42234 2xp3dxp2ge1d 42242 pell1qrgaplem 42884 pellqrex 42890 pellfundgt1 42894 areaquad 43228 imo72b2lem0 44178 int-ineq1stprincd 44205 dvdivbd 45938 fourierdlem30 46152 sge0xaddlem2 46449 carageniuncllem2 46537 |
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