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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 11692 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐷) → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷))) | |
8 | 3, 4, 5, 6, 7 | syl22anc 836 | . 2 ⊢ (𝜑 → ((𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐷) → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷))) |
9 | 1, 2, 8 | mp2and 696 | 1 ⊢ (𝜑 → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2098 class class class wbr 5138 (class class class)co 7401 ℝcr 11104 + caddc 11108 ≤ cle 11245 |
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 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-po 5578 df-so 5579 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 |
This theorem is referenced by: supadd 12178 o1add 15554 o1sub 15556 o1fsum 15755 sadcaddlem 16394 4sqlem11 16884 4sqlem12 16885 4sqlem15 16888 4sqlem16 16889 prdsxmetlem 24184 nrmmetd 24393 nmotri 24566 pcoass 24861 minveclem2 25264 ovollb2lem 25327 ovolunlem1a 25335 ovoliunlem1 25341 nulmbl2 25375 ioombl1lem4 25400 uniioombllem5 25426 itg2splitlem 25588 itg2addlem 25598 ibladdlem 25659 ulmbdd 26239 cxpaddle 26591 ang180lem2 26646 fsumharmonic 26848 lgamgulmlem3 26867 lgamgulmlem5 26869 ppiub 27041 lgsdirprm 27168 lgsqrlem2 27184 lgseisenlem2 27213 2sqlem8 27263 vmadivsumb 27320 dchrisumlem2 27327 dchrisum0lem1b 27352 mulog2sumlem1 27371 mulog2sumlem2 27372 selbergb 27386 selberg2b 27389 chpdifbndlem1 27390 logdivbnd 27393 selberg3lem2 27395 pntrlog2bnd 27421 pntpbnd2 27424 pntibndlem2 27428 pntlemr 27439 ostth2lem2 27471 ostth3 27475 smcnlem 30374 minvecolem2 30552 stadd3i 31925 le2halvesd 32392 wrdt2ind 32541 dnibndlem9 35818 ismblfin 36985 itg2addnc 36998 ibladdnclem 37000 ftc1anclem7 37023 intlewftc 41385 aks4d1p1p2 41394 dvle2 41396 2np3bcnp1 41419 sticksstones7 41427 sticksstones12a 41432 sticksstones12 41433 metakunt29 41472 2xp3dxp2ge1d 41481 pell1qrgaplem 42066 pellqrex 42072 pellfundgt1 42076 areaquad 42420 imo72b2lem0 43372 int-ineq1stprincd 43399 dvdivbd 45090 fourierdlem30 45304 sge0xaddlem2 45601 carageniuncllem2 45689 |
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