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Mirrors > Home > MPE Home > Th. List > lemul12ad | Structured version Visualization version GIF version |
Description: Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
ltp1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
divgt0d.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
lemul1ad.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
ltmul12ad.3 | ⊢ (𝜑 → 𝐷 ∈ ℝ) |
lemul12ad.4 | ⊢ (𝜑 → 0 ≤ 𝐴) |
lemul12ad.5 | ⊢ (𝜑 → 0 ≤ 𝐶) |
lemul12ad.6 | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
lemul12ad.7 | ⊢ (𝜑 → 𝐶 ≤ 𝐷) |
Ref | Expression |
---|---|
lemul12ad | ⊢ (𝜑 → (𝐴 · 𝐶) ≤ (𝐵 · 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lemul12ad.6 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
2 | lemul12ad.7 | . 2 ⊢ (𝜑 → 𝐶 ≤ 𝐷) | |
3 | ltp1d.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | lemul12ad.4 | . . . 4 ⊢ (𝜑 → 0 ≤ 𝐴) | |
5 | 3, 4 | jca 510 | . . 3 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) |
6 | divgt0d.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
7 | lemul1ad.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
8 | lemul12ad.5 | . . . 4 ⊢ (𝜑 → 0 ≤ 𝐶) | |
9 | 7, 8 | jca 510 | . . 3 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) |
10 | ltmul12ad.3 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℝ) | |
11 | lemul12a 12117 | . . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ) ∧ ((𝐶 ∈ ℝ ∧ 0 ≤ 𝐶) ∧ 𝐷 ∈ ℝ)) → ((𝐴 ≤ 𝐵 ∧ 𝐶 ≤ 𝐷) → (𝐴 · 𝐶) ≤ (𝐵 · 𝐷))) | |
12 | 5, 6, 9, 10, 11 | syl22anc 837 | . 2 ⊢ (𝜑 → ((𝐴 ≤ 𝐵 ∧ 𝐶 ≤ 𝐷) → (𝐴 · 𝐶) ≤ (𝐵 · 𝐷))) |
13 | 1, 2, 12 | mp2and 697 | 1 ⊢ (𝜑 → (𝐴 · 𝐶) ≤ (𝐵 · 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2099 class class class wbr 5145 (class class class)co 7416 ℝcr 11148 0cc0 11149 · cmul 11154 ≤ cle 11290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-po 5586 df-so 5587 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 |
This theorem is referenced by: supmullem1 12230 faclbnd 14302 o1mul 15612 fprodge1 15992 lgamgulmlem2 27055 lgamgulmlem3 27056 dchrmusum2 27520 dchrvmasumlem3 27525 dchrisum0lem2a 27543 mudivsum 27556 mulogsumlem 27557 selberg2b 27578 selberg3lem2 27584 pntrlog2bndlem3 27605 pntrlog2bndlem4 27606 pntrlog2bnd 27610 smcnlem 30627 hgt750lemf 34512 aks6d1c2lem4 41839 dvdivbd 45580 dvbdfbdioolem1 45585 stoweidlem16 45673 fourierdlem39 45803 etransclem23 45914 |
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