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Mirrors > Home > MPE Home > Th. List > ledivdivd | Structured version Visualization version GIF version |
Description: Invert ratios of positive numbers and swap their ordering. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
rpred.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
rpaddcld.1 | ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
ltdiv2d.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ+) |
ledivdivd.4 | ⊢ (𝜑 → 𝐷 ∈ ℝ+) |
ledivdivd.5 | ⊢ (𝜑 → (𝐴 / 𝐵) ≤ (𝐶 / 𝐷)) |
Ref | Expression |
---|---|
ledivdivd | ⊢ (𝜑 → (𝐷 / 𝐶) ≤ (𝐵 / 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ledivdivd.5 | . 2 ⊢ (𝜑 → (𝐴 / 𝐵) ≤ (𝐶 / 𝐷)) | |
2 | rpred.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
3 | 2 | rpregt0d 12776 | . . 3 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 0 < 𝐴)) |
4 | rpaddcld.1 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
5 | 4 | rpregt0d 12776 | . . 3 ⊢ (𝜑 → (𝐵 ∈ ℝ ∧ 0 < 𝐵)) |
6 | ltdiv2d.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ+) | |
7 | 6 | rpregt0d 12776 | . . 3 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 0 < 𝐶)) |
8 | ledivdivd.4 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ+) | |
9 | 8 | rpregt0d 12776 | . . 3 ⊢ (𝜑 → (𝐷 ∈ ℝ ∧ 0 < 𝐷)) |
10 | ledivdiv 11862 | . . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) ∧ ((𝐶 ∈ ℝ ∧ 0 < 𝐶) ∧ (𝐷 ∈ ℝ ∧ 0 < 𝐷))) → ((𝐴 / 𝐵) ≤ (𝐶 / 𝐷) ↔ (𝐷 / 𝐶) ≤ (𝐵 / 𝐴))) | |
11 | 3, 5, 7, 9, 10 | syl22anc 836 | . 2 ⊢ (𝜑 → ((𝐴 / 𝐵) ≤ (𝐶 / 𝐷) ↔ (𝐷 / 𝐶) ≤ (𝐵 / 𝐴))) |
12 | 1, 11 | mpbid 231 | 1 ⊢ (𝜑 → (𝐷 / 𝐶) ≤ (𝐵 / 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 class class class wbr 5076 (class class class)co 7277 ℝcr 10868 0cc0 10869 < clt 11007 ≤ cle 11008 / cdiv 11630 ℝ+crp 12728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-mulcom 10933 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 ax-pre-mulgt0 10946 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-br 5077 df-opab 5139 df-mpt 5160 df-id 5491 df-po 5505 df-so 5506 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-er 8496 df-en 8732 df-dom 8733 df-sdom 8734 df-pnf 11009 df-mnf 11010 df-xr 11011 df-ltxr 11012 df-le 11013 df-sub 11205 df-neg 11206 df-div 11631 df-rp 12729 |
This theorem is referenced by: (None) |
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