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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 11719 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ (𝐴 + 𝐵)) | |
6 | 1, 2, 3, 4, 5 | syl22anc 838 | 1 ⊢ (𝜑 → 0 ≤ (𝐴 + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 class class class wbr 5142 (class class class)co 7414 ℝcr 11123 0cc0 11124 + caddc 11127 ≤ cle 11265 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 |
This theorem is referenced by: fldiv 13843 modaddmodlo 13918 cjmulge0 15111 absrele 15273 abstri 15295 nn0oddm1d2 16347 prdsxmetlem 24248 nmotri 24630 tcphcphlem1 25137 trirn 25302 minveclem4 25334 ibladdlem 25723 itgaddlem1 25726 itgaddlem2 25727 iblabs 25732 cxpaddle 26661 asinlem3a 26776 fsumharmonic 26918 lgamgulmlem3 26937 mulog2sumlem2 27442 selbergb 27456 selberg2b 27459 pntrlog2bndlem2 27485 pntrlog2bnd 27491 abvcxp 27522 smcnlem 30481 minvecolem4 30664 fsumrp0cl 32720 sqsscirc1 33432 omssubaddlem 33842 dnibndlem9 35884 itg2addnc 37069 ibladdnclem 37071 itgaddnclem1 37073 itgaddnclem2 37074 iblabsnc 37079 iblmulc2nc 37080 ftc1anclem4 37091 ftc1anclem7 37094 ftc1anc 37096 areacirc 37108 lcmineqlem18 41441 posbezout 41494 aks6d1c1 41507 2np3bcnp1 41535 rmxypos 42280 wallispi2lem1 45372 fourierdlem15 45423 fourierdlem30 45438 fourierdlem47 45454 sge0xaddlem2 45735 hoidmvlelem2 45897 hoidmvlelem4 45899 ovolval5lem1 45953 flsqrt 46846 nn0eo 47514 2sphere 47735 itscnhlinecirc02plem3 47770 |
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