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Mirrors > Home > MPE Home > Th. List > lediv12ad | Structured version Visualization version GIF version |
Description: Comparison of ratio of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
ltmul1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltmul1d.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
ltmul1d.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ+) |
lediv12ad.4 | ⊢ (𝜑 → 𝐷 ∈ ℝ) |
lediv12ad.5 | ⊢ (𝜑 → 0 ≤ 𝐴) |
lediv12ad.6 | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
lediv12ad.7 | ⊢ (𝜑 → 𝐶 ≤ 𝐷) |
Ref | Expression |
---|---|
lediv12ad | ⊢ (𝜑 → (𝐴 / 𝐷) ≤ (𝐵 / 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltmul1d.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | ltmul1d.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | 1, 2 | jca 509 | . 2 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) |
4 | lediv12ad.5 | . . 3 ⊢ (𝜑 → 0 ≤ 𝐴) | |
5 | lediv12ad.6 | . . 3 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
6 | 4, 5 | jca 509 | . 2 ⊢ (𝜑 → (0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) |
7 | ltmul1d.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ+) | |
8 | 7 | rpred 12155 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
9 | lediv12ad.4 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℝ) | |
10 | 8, 9 | jca 509 | . 2 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) |
11 | 7 | rpgt0d 12158 | . . 3 ⊢ (𝜑 → 0 < 𝐶) |
12 | lediv12ad.7 | . . 3 ⊢ (𝜑 → 𝐶 ≤ 𝐷) | |
13 | 11, 12 | jca 509 | . 2 ⊢ (𝜑 → (0 < 𝐶 ∧ 𝐶 ≤ 𝐷)) |
14 | lediv12a 11245 | . 2 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) ∧ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) ∧ (0 < 𝐶 ∧ 𝐶 ≤ 𝐷))) → (𝐴 / 𝐷) ≤ (𝐵 / 𝐶)) | |
15 | 3, 6, 10, 13, 14 | syl22anc 874 | 1 ⊢ (𝜑 → (𝐴 / 𝐷) ≤ (𝐵 / 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2166 class class class wbr 4872 (class class class)co 6904 ℝcr 10250 0cc0 10251 < clt 10390 ≤ cle 10391 / cdiv 11008 ℝ+crp 12111 |
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 df-rp 12112 |
This theorem is referenced by: lgamgulmlem5 25171 chpo1ubb 25582 selbergb 25650 selberg2b 25653 dvdivbd 40932 |
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