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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 510 | . 2 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) |
4 | lediv12ad.5 | . . 3 ⊢ (𝜑 → 0 ≤ 𝐴) | |
5 | lediv12ad.6 | . . 3 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
6 | 4, 5 | jca 510 | . 2 ⊢ (𝜑 → (0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) |
7 | ltmul1d.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ+) | |
8 | 7 | rpred 13051 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
9 | lediv12ad.4 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℝ) | |
10 | 8, 9 | jca 510 | . 2 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) |
11 | 7 | rpgt0d 13054 | . . 3 ⊢ (𝜑 → 0 < 𝐶) |
12 | lediv12ad.7 | . . 3 ⊢ (𝜑 → 𝐶 ≤ 𝐷) | |
13 | 11, 12 | jca 510 | . 2 ⊢ (𝜑 → (0 < 𝐶 ∧ 𝐶 ≤ 𝐷)) |
14 | lediv12a 12140 | . 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 837 | 1 ⊢ (𝜑 → (𝐴 / 𝐷) ≤ (𝐵 / 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 class class class wbr 5149 (class class class)co 7419 ℝcr 11139 0cc0 11140 < clt 11280 ≤ cle 11281 / cdiv 11903 ℝ+crp 13009 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-rp 13010 |
This theorem is referenced by: lgamgulmlem5 27010 chpo1ubb 27459 selbergb 27527 selberg2b 27530 dvdivbd 45449 |
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