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| Mirrors > Home > MPE Home > Th. List > addge0d | Structured version Visualization version GIF version | ||
| Description: Addition of 2 nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.) | 
| Ref | Expression | 
|---|---|
| leidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) | 
| ltnegd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) | 
| addge0d.3 | ⊢ (𝜑 → 0 ≤ 𝐴) | 
| addge0d.4 | ⊢ (𝜑 → 0 ≤ 𝐵) | 
| Ref | Expression | 
|---|---|
| addge0d | ⊢ (𝜑 → 0 ≤ (𝐴 + 𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | leidd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
| 2 | ltnegd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
| 3 | addge0d.3 | . 2 ⊢ (𝜑 → 0 ≤ 𝐴) | |
| 4 | addge0d.4 | . 2 ⊢ (𝜑 → 0 ≤ 𝐵) | |
| 5 | addge0 11752 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ (𝐴 + 𝐵)) | |
| 6 | 1, 2, 3, 4, 5 | syl22anc 839 | 1 ⊢ (𝜑 → 0 ≤ (𝐴 + 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 ℝcr 11154 0cc0 11155 + 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: fldiv 13900 modaddmodlo 13976 cjmulge0 15185 absrele 15347 abstri 15369 nn0oddm1d2 16422 prdsxmetlem 24378 nmotri 24760 tcphcphlem1 25269 trirn 25434 minveclem4 25466 ibladdlem 25855 itgaddlem1 25858 itgaddlem2 25859 iblabs 25864 cxpaddle 26795 asinlem3a 26913 fsumharmonic 27055 lgamgulmlem3 27074 mulog2sumlem2 27579 selbergb 27593 selberg2b 27596 pntrlog2bndlem2 27622 pntrlog2bnd 27628 abvcxp 27659 smcnlem 30716 minvecolem4 30899 fsumrp0cl 33026 sqsscirc1 33907 omssubaddlem 34301 dnibndlem9 36487 itg2addnc 37681 ibladdnclem 37683 itgaddnclem1 37685 itgaddnclem2 37686 iblabsnc 37691 iblmulc2nc 37692 ftc1anclem4 37703 ftc1anclem7 37706 ftc1anc 37708 areacirc 37720 lcmineqlem18 42047 posbezout 42101 aks6d1c1 42117 2np3bcnp1 42145 rmxypos 42959 wallispi2lem1 46086 fourierdlem15 46137 fourierdlem30 46152 fourierdlem47 46168 sge0xaddlem2 46449 hoidmvlelem2 46611 hoidmvlelem4 46613 ovolval5lem1 46667 ormkglobd 46890 flsqrt 47580 nn0eo 48449 2sphere 48670 itscnhlinecirc02plem3 48705 | 
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