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Mirrors > Home > MPE Home > Th. List > mulsuble0b | Structured version Visualization version GIF version |
Description: A condition for multiplication of subtraction to be nonpositive. (Contributed by Scott Fenton, 25-Jun-2013.) |
Ref | Expression |
---|---|
mulsuble0b | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (((𝐴 − 𝐵) · (𝐶 − 𝐵)) ≤ 0 ↔ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) ∨ (𝐶 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resubcl 10665 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 − 𝐵) ∈ ℝ) | |
2 | 1 | 3adant3 1168 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 − 𝐵) ∈ ℝ) |
3 | resubcl 10665 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 − 𝐵) ∈ ℝ) | |
4 | 3 | ancoms 452 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 − 𝐵) ∈ ℝ) |
5 | 4 | 3adant1 1166 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 − 𝐵) ∈ ℝ) |
6 | mulle0b 11223 | . . 3 ⊢ (((𝐴 − 𝐵) ∈ ℝ ∧ (𝐶 − 𝐵) ∈ ℝ) → (((𝐴 − 𝐵) · (𝐶 − 𝐵)) ≤ 0 ↔ (((𝐴 − 𝐵) ≤ 0 ∧ 0 ≤ (𝐶 − 𝐵)) ∨ (0 ≤ (𝐴 − 𝐵) ∧ (𝐶 − 𝐵) ≤ 0)))) | |
7 | 2, 5, 6 | syl2anc 581 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (((𝐴 − 𝐵) · (𝐶 − 𝐵)) ≤ 0 ↔ (((𝐴 − 𝐵) ≤ 0 ∧ 0 ≤ (𝐶 − 𝐵)) ∨ (0 ≤ (𝐴 − 𝐵) ∧ (𝐶 − 𝐵) ≤ 0)))) |
8 | suble0 10865 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 − 𝐵) ≤ 0 ↔ 𝐴 ≤ 𝐵)) | |
9 | 8 | 3adant3 1168 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 − 𝐵) ≤ 0 ↔ 𝐴 ≤ 𝐵)) |
10 | subge0 10864 | . . . . . 6 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 ≤ (𝐶 − 𝐵) ↔ 𝐵 ≤ 𝐶)) | |
11 | 10 | ancoms 452 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (0 ≤ (𝐶 − 𝐵) ↔ 𝐵 ≤ 𝐶)) |
12 | 11 | 3adant1 1166 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (0 ≤ (𝐶 − 𝐵) ↔ 𝐵 ≤ 𝐶)) |
13 | 9, 12 | anbi12d 626 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (((𝐴 − 𝐵) ≤ 0 ∧ 0 ≤ (𝐶 − 𝐵)) ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶))) |
14 | subge0 10864 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 ≤ (𝐴 − 𝐵) ↔ 𝐵 ≤ 𝐴)) | |
15 | 14 | 3adant3 1168 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (0 ≤ (𝐴 − 𝐵) ↔ 𝐵 ≤ 𝐴)) |
16 | suble0 10865 | . . . . . . 7 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐶 − 𝐵) ≤ 0 ↔ 𝐶 ≤ 𝐵)) | |
17 | 16 | ancoms 452 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐶 − 𝐵) ≤ 0 ↔ 𝐶 ≤ 𝐵)) |
18 | 17 | 3adant1 1166 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐶 − 𝐵) ≤ 0 ↔ 𝐶 ≤ 𝐵)) |
19 | 15, 18 | anbi12d 626 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((0 ≤ (𝐴 − 𝐵) ∧ (𝐶 − 𝐵) ≤ 0) ↔ (𝐵 ≤ 𝐴 ∧ 𝐶 ≤ 𝐵))) |
20 | ancom 454 | . . . 4 ⊢ ((𝐵 ≤ 𝐴 ∧ 𝐶 ≤ 𝐵) ↔ (𝐶 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴)) | |
21 | 19, 20 | syl6bb 279 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((0 ≤ (𝐴 − 𝐵) ∧ (𝐶 − 𝐵) ≤ 0) ↔ (𝐶 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) |
22 | 13, 21 | orbi12d 949 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((((𝐴 − 𝐵) ≤ 0 ∧ 0 ≤ (𝐶 − 𝐵)) ∨ (0 ≤ (𝐴 − 𝐵) ∧ (𝐶 − 𝐵) ≤ 0)) ↔ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) ∨ (𝐶 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴)))) |
23 | 7, 22 | bitrd 271 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (((𝐴 − 𝐵) · (𝐶 − 𝐵)) ≤ 0 ↔ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) ∨ (𝐶 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∨ wo 880 ∧ w3a 1113 ∈ wcel 2166 class class class wbr 4872 (class class class)co 6904 ℝcr 10250 0cc0 10251 · cmul 10256 ≤ cle 10391 − cmin 10584 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-op 4403 df-uni 4658 df-br 4873 df-opab 4935 df-mpt 4952 df-id 5249 df-po 5262 df-so 5263 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-er 8008 df-en 8222 df-dom 8223 df-sdom 8224 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-div 11009 |
This theorem is referenced by: brbtwn2 26203 |
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