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Mirrors > Home > MPE Home > Th. List > lemul12bd | 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) |
lemul12bd.4 | ⊢ (𝜑 → 0 ≤ 𝐴) |
lemul12bd.5 | ⊢ (𝜑 → 0 ≤ 𝐷) |
lemul12bd.6 | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
lemul12bd.7 | ⊢ (𝜑 → 𝐶 ≤ 𝐷) |
Ref | Expression |
---|---|
lemul12bd | ⊢ (𝜑 → (𝐴 · 𝐶) ≤ (𝐵 · 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lemul12bd.6 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
2 | lemul12bd.7 | . 2 ⊢ (𝜑 → 𝐶 ≤ 𝐷) | |
3 | ltp1d.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | lemul12bd.4 | . . . 4 ⊢ (𝜑 → 0 ≤ 𝐴) | |
5 | 3, 4 | jca 515 | . . 3 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) |
6 | divgt0d.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
7 | lemul1ad.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
8 | ltmul12ad.3 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ) | |
9 | lemul12bd.5 | . . . 4 ⊢ (𝜑 → 0 ≤ 𝐷) | |
10 | 8, 9 | jca 515 | . . 3 ⊢ (𝜑 → (𝐷 ∈ ℝ ∧ 0 ≤ 𝐷)) |
11 | lemul12b 11540 | . . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ (𝐷 ∈ ℝ ∧ 0 ≤ 𝐷))) → ((𝐴 ≤ 𝐵 ∧ 𝐶 ≤ 𝐷) → (𝐴 · 𝐶) ≤ (𝐵 · 𝐷))) | |
12 | 5, 6, 7, 10, 11 | syl22anc 837 | . 2 ⊢ (𝜑 → ((𝐴 ≤ 𝐵 ∧ 𝐶 ≤ 𝐷) → (𝐴 · 𝐶) ≤ (𝐵 · 𝐷))) |
13 | 1, 2, 12 | mp2and 698 | 1 ⊢ (𝜑 → (𝐴 · 𝐶) ≤ (𝐵 · 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 class class class wbr 5035 (class class class)co 7155 ℝcr 10579 0cc0 10580 · cmul 10585 ≤ cle 10719 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-sep 5172 ax-nul 5179 ax-pow 5237 ax-pr 5301 ax-un 7464 ax-resscn 10637 ax-1cn 10638 ax-icn 10639 ax-addcl 10640 ax-addrcl 10641 ax-mulcl 10642 ax-mulrcl 10643 ax-mulcom 10644 ax-addass 10645 ax-mulass 10646 ax-distr 10647 ax-i2m1 10648 ax-1ne0 10649 ax-1rid 10650 ax-rnegex 10651 ax-rrecex 10652 ax-cnre 10653 ax-pre-lttri 10654 ax-pre-lttrn 10655 ax-pre-ltadd 10656 ax-pre-mulgt0 10657 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4802 df-br 5036 df-opab 5098 df-mpt 5116 df-id 5433 df-po 5446 df-so 5447 df-xp 5533 df-rel 5534 df-cnv 5535 df-co 5536 df-dm 5537 df-rn 5538 df-res 5539 df-ima 5540 df-iota 6298 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-er 8304 df-en 8533 df-dom 8534 df-sdom 8535 df-pnf 10720 df-mnf 10721 df-xr 10722 df-ltxr 10723 df-le 10724 df-sub 10915 df-neg 10916 |
This theorem is referenced by: expmulnbnd 13651 fourierdlem77 43219 |
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