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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 11262 | . . 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 11737 | . . 3 ⊢ ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((0 + 𝐵) ≤ 𝐴 ↔ 0 ≤ (𝐴 − 𝐵))) | |
5 | 1, 2, 3, 4 | syl3anc 1370 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 + 𝐵) ≤ 𝐴 ↔ 0 ≤ (𝐴 − 𝐵))) |
6 | 2 | recnd 11287 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℂ) |
7 | 6 | addlidd 11460 | . . 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 206 ∧ wa 395 ∈ wcel 2106 class class class wbr 5148 (class class class)co 7431 ℝcr 11152 0cc0 11153 + caddc 11156 ≤ cle 11294 − cmin 11490 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 |
This theorem is referenced by: subge0i 11814 subge0d 11851 mulsuble0b 12138 znn0sub 12662 uzsubsubfz 13583 difelfzle 13678 difelfznle 13679 fracge0 13841 modge0 13916 2submod 13970 expnbnd 14268 pfxccatin12lem2 14766 swrdccat 14770 repswswrd 14819 cshwidxmod 14838 abssubge0 15363 blcvx 24834 iirev 24970 iihalf2 24975 ovolfsf 25520 cosq14ge0 26568 sinord 26591 resinf1o 26593 ang180lem2 26868 acosbnd 26958 ftalem5 27135 mumullem2 27238 rpvmasumlem 27546 dchrisum0flblem1 27567 brbtwn2 28935 colinearalglem4 28939 ax5seglem3 28961 resconn 35231 fz0n 35711 sin2h 37597 cos2h 37598 tan2h 37599 ftc1anclem5 37684 dvasin 37691 jm2.23 42985 subsubelfzo0 47276 gpgedgvtx1 47955 m1modmmod 48371 |
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