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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 11218 | . . 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 11691 | . . 3 ⊢ ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((0 + 𝐵) ≤ 𝐴 ↔ 0 ≤ (𝐴 − 𝐵))) | |
5 | 1, 2, 3, 4 | syl3anc 1368 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 + 𝐵) ≤ 𝐴 ↔ 0 ≤ (𝐴 − 𝐵))) |
6 | 2 | recnd 11243 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℂ) |
7 | 6 | addlidd 11416 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 + 𝐵) = 𝐵) |
8 | 7 | breq1d 5151 | . 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 2098 class class class wbr 5141 (class class class)co 7404 ℝcr 11108 0cc0 11109 + caddc 11112 ≤ cle 11250 − cmin 11445 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 |
This theorem is referenced by: subge0i 11768 subge0d 11805 mulsuble0b 12087 znn0sub 12610 uzsubsubfz 13526 difelfzle 13617 difelfznle 13618 fracge0 13772 modge0 13847 2submod 13900 expnbnd 14198 pfxccatin12lem2 14685 swrdccat 14689 repswswrd 14738 cshwidxmod 14757 abssubge0 15278 blcvx 24665 iirev 24801 iihalf2 24806 ovolfsf 25351 cosq14ge0 26397 sinord 26419 resinf1o 26421 ang180lem2 26693 acosbnd 26783 ftalem5 26960 mumullem2 27063 rpvmasumlem 27371 dchrisum0flblem1 27392 brbtwn2 28667 colinearalglem4 28671 ax5seglem3 28693 resconn 34765 fz0n 35234 sin2h 36989 cos2h 36990 tan2h 36991 ftc1anclem5 37076 dvasin 37083 jm2.23 42294 subsubelfzo0 46587 m1modmmod 47463 |
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