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Mirrors > Home > MPE Home > Th. List > subge0d | Structured version Visualization version GIF version |
Description: Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
leidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltnegd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
Ref | Expression |
---|---|
subge0d | ⊢ (𝜑 → (0 ≤ (𝐴 − 𝐵) ↔ 𝐵 ≤ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leidd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | ltnegd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | subge0 11141 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 ≤ (𝐴 − 𝐵) ↔ 𝐵 ≤ 𝐴)) | |
4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (0 ≤ (𝐴 − 𝐵) ↔ 𝐵 ≤ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∈ wcel 2105 class class class wbr 5057 (class class class)co 7145 ℝcr 10524 0cc0 10525 ≤ cle 10664 − cmin 10858 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 |
This theorem is referenced by: ofsubge0 11625 uzsubsubfz 12917 modsubdir 13296 modsumfzodifsn 13300 serle 13413 discr 13589 bcval5 13666 fzomaxdiflem 14690 sqreulem 14707 amgm2 14717 climle 14984 rlimle 14992 iseralt 15029 fsumle 15142 cvgcmp 15159 binomrisefac 15384 smuval2 15819 pcz 16205 4sqlem15 16283 mndodconglem 18598 ipcau2 23764 pjthlem1 23967 ovolicc2lem4 24048 vitalilem2 24137 itg1lea 24240 dvlip 24517 dvge0 24530 dvle 24531 dvivthlem1 24532 dvfsumlem2 24551 dvfsumlem4 24553 loglesqrt 25266 emcllem6 25505 harmoniclbnd 25513 basellem9 25593 gausslemma2dlem0h 25866 lgseisenlem1 25878 2sqmod 25939 vmadivsum 25985 rplogsumlem1 25987 dchrisumlem2 25993 rplogsum 26030 vmalogdivsum2 26041 selberg2lem 26053 logdivbnd 26059 pntpbnd2 26090 pntibndlem2 26094 pntlemg 26101 pntlemn 26103 ttgcontlem1 26598 brbtwn2 26618 axpaschlem 26653 axcontlem8 26684 crctcsh 27529 clwlkclwwlklem2a1 27697 clwlkclwwlklem2fv2 27701 pjhthlem1 29095 leop2 29828 pjssposi 29876 fdvposle 31771 rddif2 33713 dnibndlem4 33717 broucube 34807 areacirclem2 34864 areacirclem4 34866 areacirclem5 34867 areacirc 34868 acongrep 39455 lptre2pt 41797 dvnmul 42104 dvnprodlem1 42107 dvnprodlem2 42108 stoweidlem1 42163 stoweidlem26 42188 stoweidlem62 42224 wallispilem4 42230 fourierdlem26 42295 fourierdlem42 42311 fourierdlem65 42333 fourierdlem75 42343 elaa2lem 42395 etransclem3 42399 etransclem7 42403 etransclem10 42406 etransclem20 42416 etransclem21 42417 etransclem22 42418 etransclem24 42420 etransclem27 42423 hoidmvlelem1 42754 nnpw2pmod 44571 2itscp 44696 |
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