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Mirrors > Home > MPE Home > Th. List > subge0 | Structured version Visualization version GIF version |
Description: Nonnegative subtraction. (Contributed by NM, 14-Mar-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
subge0 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 ≤ (𝐴 − 𝐵) ↔ 𝐵 ≤ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0red 11248 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 0 ∈ ℝ) | |
2 | simpr 484 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ) | |
3 | simpl 482 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℝ) | |
4 | leaddsub 11721 | . . 3 ⊢ ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((0 + 𝐵) ≤ 𝐴 ↔ 0 ≤ (𝐴 − 𝐵))) | |
5 | 1, 2, 3, 4 | syl3anc 1369 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 + 𝐵) ≤ 𝐴 ↔ 0 ≤ (𝐴 − 𝐵))) |
6 | 2 | recnd 11273 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℂ) |
7 | 6 | addlidd 11446 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 + 𝐵) = 𝐵) |
8 | 7 | breq1d 5158 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 + 𝐵) ≤ 𝐴 ↔ 𝐵 ≤ 𝐴)) |
9 | 5, 8 | bitr3d 281 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 ≤ (𝐴 − 𝐵) ↔ 𝐵 ≤ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2099 class class class wbr 5148 (class class class)co 7420 ℝcr 11138 0cc0 11139 + caddc 11142 ≤ cle 11280 − cmin 11475 |
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 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 |
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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 |
This theorem is referenced by: subge0i 11798 subge0d 11835 mulsuble0b 12117 znn0sub 12640 uzsubsubfz 13556 difelfzle 13647 difelfznle 13648 fracge0 13802 modge0 13877 2submod 13930 expnbnd 14227 pfxccatin12lem2 14714 swrdccat 14718 repswswrd 14767 cshwidxmod 14786 abssubge0 15307 blcvx 24727 iirev 24863 iihalf2 24868 ovolfsf 25413 cosq14ge0 26459 sinord 26481 resinf1o 26483 ang180lem2 26755 acosbnd 26845 ftalem5 27022 mumullem2 27125 rpvmasumlem 27433 dchrisum0flblem1 27454 brbtwn2 28729 colinearalglem4 28733 ax5seglem3 28755 resconn 34856 fz0n 35325 sin2h 37083 cos2h 37084 tan2h 37085 ftc1anclem5 37170 dvasin 37177 jm2.23 42417 subsubelfzo0 46706 m1modmmod 47594 |
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