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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 11664 | . 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 205 ∈ wcel 2106 class class class wbr 5103 (class class class)co 7353 ℝcr 11046 0cc0 11047 ≤ cle 11186 − cmin 11381 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-po 5543 df-so 5544 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 |
This theorem is referenced by: ofsubge0 12148 uzsubsubfz 13455 modsubdir 13837 modsumfzodifsn 13841 serle 13955 discr 14135 bcval5 14210 fzomaxdiflem 15219 sqreulem 15236 amgm2 15246 climle 15514 rlimle 15524 iseralt 15561 fsumle 15676 cvgcmp 15693 binomrisefac 15917 smuval2 16354 pcz 16745 4sqlem15 16823 mndodconglem 19314 ipcau2 24582 pjthlem1 24785 ovolicc2lem4 24868 vitalilem2 24957 itg1lea 25061 dvlip 25341 dvge0 25354 dvle 25355 dvivthlem1 25356 dvfsumlem2 25375 dvfsumlem4 25377 loglesqrt 26095 emcllem6 26334 harmoniclbnd 26342 basellem9 26422 gausslemma2dlem0h 26695 lgseisenlem1 26707 2sqmod 26768 vmadivsum 26814 rplogsumlem1 26816 dchrisumlem2 26822 rplogsum 26859 vmalogdivsum2 26870 selberg2lem 26882 logdivbnd 26888 pntpbnd2 26919 pntibndlem2 26923 pntlemg 26930 pntlemn 26932 ttgcontlem1 27719 brbtwn2 27740 axpaschlem 27775 axcontlem8 27806 crctcsh 28655 clwlkclwwlklem2a1 28822 clwlkclwwlklem2fv2 28826 pjhthlem1 30219 leop2 30952 pjssposi 31000 fdvposle 33083 rddif2 34907 dnibndlem4 34911 broucube 36079 areacirclem2 36134 areacirclem4 36136 areacirclem5 36137 areacirc 36138 metakunt29 40572 acongrep 41242 sqrtcvallem2 41851 sqrtcvallem4 41853 lptre2pt 43813 dvnmul 44116 dvnprodlem1 44119 dvnprodlem2 44120 stoweidlem1 44174 stoweidlem26 44199 stoweidlem62 44235 wallispilem4 44241 fourierdlem26 44306 fourierdlem42 44322 fourierdlem65 44344 fourierdlem75 44354 elaa2lem 44406 etransclem3 44410 etransclem7 44414 etransclem10 44417 etransclem20 44427 etransclem21 44428 etransclem22 44429 etransclem24 44431 etransclem27 44434 hoidmvlelem1 44768 nnpw2pmod 46601 2itscp 46799 |
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