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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 11118 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ (𝐴 + 𝐵)) | |
6 | 1, 2, 3, 4, 5 | syl22anc 837 | 1 ⊢ (𝜑 → 0 ≤ (𝐴 + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 class class class wbr 5030 (class class class)co 7135 ℝcr 10525 0cc0 10526 + caddc 10529 ≤ cle 10665 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 |
This theorem is referenced by: fldiv 13223 modaddmodlo 13298 cjmulge0 14497 absrele 14660 abstri 14682 nn0oddm1d2 15726 prdsxmetlem 22975 nmotri 23345 tcphcphlem1 23839 trirn 24004 minveclem4 24036 ibladdlem 24423 itgaddlem1 24426 itgaddlem2 24427 iblabs 24432 cxpaddle 25341 asinlem3a 25456 fsumharmonic 25597 lgamgulmlem3 25616 mulog2sumlem2 26119 selbergb 26133 selberg2b 26136 pntrlog2bndlem2 26162 pntrlog2bnd 26168 abvcxp 26199 smcnlem 28480 minvecolem4 28663 fsumrp0cl 30729 sqsscirc1 31261 omssubaddlem 31667 dnibndlem9 33938 itg2addnc 35111 ibladdnclem 35113 itgaddnclem1 35115 itgaddnclem2 35116 iblabsnc 35121 iblmulc2nc 35122 ftc1anclem4 35133 ftc1anclem7 35136 ftc1anc 35138 areacirc 35150 lcmineqlem18 39334 2np3bcnp1 39348 rmxypos 39888 wallispi2lem1 42713 fourierdlem15 42764 fourierdlem30 42779 fourierdlem47 42795 sge0xaddlem2 43073 hoidmvlelem2 43235 hoidmvlelem4 43237 ovolval5lem1 43291 flsqrt 44110 nn0eo 44942 2sphere 45163 itscnhlinecirc02plem3 45198 |
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