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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 11750 | . 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 2106 class class class wbr 5148 (class class class)co 7431 ℝcr 11152 0cc0 11153 + caddc 11156 ≤ cle 11294 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 |
This theorem is referenced by: fldiv 13897 modaddmodlo 13973 cjmulge0 15182 absrele 15344 abstri 15366 nn0oddm1d2 16419 prdsxmetlem 24394 nmotri 24776 tcphcphlem1 25283 trirn 25448 minveclem4 25480 ibladdlem 25870 itgaddlem1 25873 itgaddlem2 25874 iblabs 25879 cxpaddle 26810 asinlem3a 26928 fsumharmonic 27070 lgamgulmlem3 27089 mulog2sumlem2 27594 selbergb 27608 selberg2b 27611 pntrlog2bndlem2 27637 pntrlog2bnd 27643 abvcxp 27674 smcnlem 30726 minvecolem4 30909 fsumrp0cl 33009 sqsscirc1 33869 omssubaddlem 34281 dnibndlem9 36469 itg2addnc 37661 ibladdnclem 37663 itgaddnclem1 37665 itgaddnclem2 37666 iblabsnc 37671 iblmulc2nc 37672 ftc1anclem4 37683 ftc1anclem7 37686 ftc1anc 37688 areacirc 37700 lcmineqlem18 42028 posbezout 42082 aks6d1c1 42098 2np3bcnp1 42126 rmxypos 42936 wallispi2lem1 46027 fourierdlem15 46078 fourierdlem30 46093 fourierdlem47 46109 sge0xaddlem2 46390 hoidmvlelem2 46552 hoidmvlelem4 46554 ovolval5lem1 46608 flsqrt 47518 nn0eo 48378 2sphere 48599 itscnhlinecirc02plem3 48634 |
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