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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 11129 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ (𝐴 + 𝐵)) | |
6 | 1, 2, 3, 4, 5 | syl22anc 836 | 1 ⊢ (𝜑 → 0 ≤ (𝐴 + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 class class class wbr 5066 (class class class)co 7156 ℝcr 10536 0cc0 10537 + caddc 10540 ≤ cle 10676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 |
This theorem is referenced by: fldiv 13229 modaddmodlo 13304 cjmulge0 14505 absrele 14668 abstri 14690 nn0oddm1d2 15736 prdsxmetlem 22978 nmotri 23348 tcphcphlem1 23838 trirn 24003 minveclem4 24035 ibladdlem 24420 itgaddlem1 24423 itgaddlem2 24424 iblabs 24429 cxpaddle 25333 asinlem3a 25448 fsumharmonic 25589 lgamgulmlem3 25608 mulog2sumlem2 26111 selbergb 26125 selberg2b 26128 pntrlog2bndlem2 26154 pntrlog2bnd 26160 abvcxp 26191 smcnlem 28474 minvecolem4 28657 fsumrp0cl 30682 sqsscirc1 31151 omssubaddlem 31557 dnibndlem9 33825 itg2addnc 34961 ibladdnclem 34963 itgaddnclem1 34965 itgaddnclem2 34966 iblabsnc 34971 iblmulc2nc 34972 ftc1anclem4 34985 ftc1anclem7 34988 ftc1anc 34990 areacirc 35002 rmxypos 39564 wallispi2lem1 42376 fourierdlem15 42427 fourierdlem30 42442 fourierdlem47 42458 sge0xaddlem2 42736 hoidmvlelem2 42898 hoidmvlelem4 42900 ovolval5lem1 42954 flsqrt 43776 nn0eo 44608 2sphere 44756 itscnhlinecirc02plem3 44791 |
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