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Mirrors > Home > ILE Home > Th. List > lesub0 | GIF version |
Description: Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
lesub0 | ⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → ((0 ≤ A ∧ B ≤ (B − A)) ↔ A = 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0red 6826 | . . 3 ⊢ (B ∈ ℝ → 0 ∈ ℝ) | |
2 | letri3 6896 | . . 3 ⊢ ((A ∈ ℝ ∧ 0 ∈ ℝ) → (A = 0 ↔ (A ≤ 0 ∧ 0 ≤ A))) | |
3 | 1, 2 | sylan2 270 | . 2 ⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → (A = 0 ↔ (A ≤ 0 ∧ 0 ≤ A))) |
4 | ancom 253 | . . 3 ⊢ ((A ≤ 0 ∧ 0 ≤ A) ↔ (0 ≤ A ∧ A ≤ 0)) | |
5 | simpr 103 | . . . . . . 7 ⊢ ((B ∈ ℝ ∧ A ∈ ℝ) → A ∈ ℝ) | |
6 | 0red 6826 | . . . . . . 7 ⊢ ((B ∈ ℝ ∧ A ∈ ℝ) → 0 ∈ ℝ) | |
7 | simpl 102 | . . . . . . 7 ⊢ ((B ∈ ℝ ∧ A ∈ ℝ) → B ∈ ℝ) | |
8 | lesub2 7247 | . . . . . . 7 ⊢ ((A ∈ ℝ ∧ 0 ∈ ℝ ∧ B ∈ ℝ) → (A ≤ 0 ↔ (B − 0) ≤ (B − A))) | |
9 | 5, 6, 7, 8 | syl3anc 1134 | . . . . . 6 ⊢ ((B ∈ ℝ ∧ A ∈ ℝ) → (A ≤ 0 ↔ (B − 0) ≤ (B − A))) |
10 | 7 | recnd 6851 | . . . . . . . 8 ⊢ ((B ∈ ℝ ∧ A ∈ ℝ) → B ∈ ℂ) |
11 | 10 | subid1d 7107 | . . . . . . 7 ⊢ ((B ∈ ℝ ∧ A ∈ ℝ) → (B − 0) = B) |
12 | 11 | breq1d 3765 | . . . . . 6 ⊢ ((B ∈ ℝ ∧ A ∈ ℝ) → ((B − 0) ≤ (B − A) ↔ B ≤ (B − A))) |
13 | 9, 12 | bitrd 177 | . . . . 5 ⊢ ((B ∈ ℝ ∧ A ∈ ℝ) → (A ≤ 0 ↔ B ≤ (B − A))) |
14 | 13 | ancoms 255 | . . . 4 ⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → (A ≤ 0 ↔ B ≤ (B − A))) |
15 | 14 | anbi2d 437 | . . 3 ⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → ((0 ≤ A ∧ A ≤ 0) ↔ (0 ≤ A ∧ B ≤ (B − A)))) |
16 | 4, 15 | syl5bb 181 | . 2 ⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → ((A ≤ 0 ∧ 0 ≤ A) ↔ (0 ≤ A ∧ B ≤ (B − A)))) |
17 | 3, 16 | bitr2d 178 | 1 ⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → ((0 ≤ A ∧ B ≤ (B − A)) ↔ A = 0)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ∈ wcel 1390 class class class wbr 3755 (class class class)co 5455 ℝcr 6710 0cc0 6711 ≤ cle 6858 − cmin 6979 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-cnex 6774 ax-resscn 6775 ax-1cn 6776 ax-1re 6777 ax-icn 6778 ax-addcl 6779 ax-addrcl 6780 ax-mulcl 6781 ax-addcom 6783 ax-addass 6785 ax-distr 6787 ax-i2m1 6788 ax-0id 6791 ax-rnegex 6792 ax-cnre 6794 ax-pre-ltirr 6795 ax-pre-apti 6798 ax-pre-ltadd 6799 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-nel 2204 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-iota 4810 df-fun 4847 df-fv 4853 df-riota 5411 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-pnf 6859 df-mnf 6860 df-xr 6861 df-ltxr 6862 df-le 6863 df-sub 6981 df-neg 6982 |
This theorem is referenced by: lesub0i 7283 |
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