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| 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 11265 | . . 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 11740 | . . 3 ⊢ ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((0 + 𝐵) ≤ 𝐴 ↔ 0 ≤ (𝐴 − 𝐵))) | |
| 5 | 1, 2, 3, 4 | syl3anc 1372 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 + 𝐵) ≤ 𝐴 ↔ 0 ≤ (𝐴 − 𝐵))) | 
| 6 | 2 | recnd 11290 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℂ) | 
| 7 | 6 | addlidd 11463 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 + 𝐵) = 𝐵) | 
| 8 | 7 | breq1d 5152 | . 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 206 ∧ wa 395 ∈ wcel 2107 class class class wbr 5142 (class class class)co 7432 ℝcr 11155 0cc0 11156 + caddc 11159 ≤ cle 11297 − cmin 11493 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 | 
| This theorem is referenced by: subge0i 11817 subge0d 11854 mulsuble0b 12141 znn0sub 12666 uzsubsubfz 13587 difelfzle 13682 difelfznle 13683 fracge0 13845 modge0 13920 2submod 13974 expnbnd 14272 pfxccatin12lem2 14770 swrdccat 14774 repswswrd 14823 cshwidxmod 14842 abssubge0 15367 blcvx 24820 iirev 24957 iihalf2 24962 ovolfsf 25507 cosq14ge0 26554 sinord 26577 resinf1o 26579 ang180lem2 26854 acosbnd 26944 ftalem5 27121 mumullem2 27224 rpvmasumlem 27532 dchrisum0flblem1 27553 brbtwn2 28921 colinearalglem4 28925 ax5seglem3 28947 resconn 35252 fz0n 35732 sin2h 37618 cos2h 37619 tan2h 37620 ftc1anclem5 37705 dvasin 37712 jm2.23 43013 subsubelfzo0 47343 gpgedgvtx1 48025 m1modmmod 48447 | 
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