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Mirrors > Home > MPE Home > Th. List > addge01d | Structured version Visualization version GIF version |
Description: A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
leidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltnegd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
Ref | Expression |
---|---|
addge01d | ⊢ (𝜑 → (0 ≤ 𝐵 ↔ 𝐴 ≤ (𝐴 + 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leidd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | ltnegd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | addge01 11307 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 ≤ 𝐵 ↔ 𝐴 ≤ (𝐴 + 𝐵))) | |
4 | 1, 2, 3 | syl2anc 587 | 1 ⊢ (𝜑 → (0 ≤ 𝐵 ↔ 𝐴 ≤ (𝐴 + 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∈ wcel 2112 class class class wbr 5039 (class class class)co 7191 ℝcr 10693 0cc0 10694 + caddc 10697 ≤ cle 10833 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-po 5453 df-so 5454 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 |
This theorem is referenced by: xralrple 12760 2tnp1ge0ge0 13369 sermono 13573 bernneq 13761 sqrlem7 14777 absrele 14837 climserle 15191 iseraltlem3 15212 fsumless 15323 sinbnd 15704 sadcaddlem 15979 mndodconglem 18887 isabvd 19810 ovolicc2lem4 24371 ioombl1lem4 24412 ioorcl2 24423 mbfi1fseqlem6 24572 coemulhi 25102 cxpaddle 25592 jensenlem2 25824 padicabv 26465 axpaschlem 26985 chscllem2 29673 hstle1 30261 esumpcvgval 31712 itg2addnclem 35514 itg2addnc 35517 areacirclem5 35555 lcmineqlem18 39737 sticksstones6 39776 sticksstones7 39777 pell1qrge1 40336 ltrmxnn0 40415 xralrple4 42526 xralrple3 42527 mccllem 42756 wallispilem4 43227 fourierdlem42 43308 fourierdlem65 43330 etransclem35 43428 smfmullem1 43940 smfmullem2 43941 smfmullem3 43942 |
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