Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ltdiv23neg | Structured version Visualization version GIF version |
Description: Swap denominator with other side of 'less than', when both are negative. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
ltdiv23neg.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltdiv23neg.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
ltdiv23neg.3 | ⊢ (𝜑 → 𝐵 < 0) |
ltdiv23neg.4 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
ltdiv23neg.5 | ⊢ (𝜑 → 𝐶 < 0) |
Ref | Expression |
---|---|
ltdiv23neg | ⊢ (𝜑 → ((𝐴 / 𝐵) < 𝐶 ↔ (𝐴 / 𝐶) < 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltdiv23neg.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | ltdiv23neg.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | ltdiv23neg.3 | . . . . 5 ⊢ (𝜑 → 𝐵 < 0) | |
4 | 2, 3 | ltned 10052 | . . . 4 ⊢ (𝜑 → 𝐵 ≠ 0) |
5 | 1, 2, 4 | redivcld 10732 | . . 3 ⊢ (𝜑 → (𝐴 / 𝐵) ∈ ℝ) |
6 | ltdiv23neg.4 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
7 | 5, 6, 2, 3 | ltmulneg 38556 | . 2 ⊢ (𝜑 → ((𝐴 / 𝐵) < 𝐶 ↔ (𝐶 · 𝐵) < ((𝐴 / 𝐵) · 𝐵))) |
8 | recn 9905 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
9 | 1, 8 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
10 | recn 9905 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
11 | 2, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
12 | 9, 11, 4 | divcan1d 10681 | . . 3 ⊢ (𝜑 → ((𝐴 / 𝐵) · 𝐵) = 𝐴) |
13 | 12 | breq2d 4595 | . 2 ⊢ (𝜑 → ((𝐶 · 𝐵) < ((𝐴 / 𝐵) · 𝐵) ↔ (𝐶 · 𝐵) < 𝐴)) |
14 | remulcl 9900 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 · 𝐵) ∈ ℝ) | |
15 | 6, 2, 14 | syl2anc 691 | . . . 4 ⊢ (𝜑 → (𝐶 · 𝐵) ∈ ℝ) |
16 | ltdiv23neg.5 | . . . . . 6 ⊢ (𝜑 → 𝐶 < 0) | |
17 | 6, 16 | ltned 10052 | . . . . 5 ⊢ (𝜑 → 𝐶 ≠ 0) |
18 | 6, 17 | rereccld 10731 | . . . 4 ⊢ (𝜑 → (1 / 𝐶) ∈ ℝ) |
19 | 6, 16 | reclt0d 38548 | . . . 4 ⊢ (𝜑 → (1 / 𝐶) < 0) |
20 | 15, 1, 18, 19 | ltmulneg 38556 | . . 3 ⊢ (𝜑 → ((𝐶 · 𝐵) < 𝐴 ↔ (𝐴 · (1 / 𝐶)) < ((𝐶 · 𝐵) · (1 / 𝐶)))) |
21 | recn 9905 | . . . . . . 7 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℂ) | |
22 | 6, 21 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
23 | 9, 22, 17 | divrecd 10683 | . . . . 5 ⊢ (𝜑 → (𝐴 / 𝐶) = (𝐴 · (1 / 𝐶))) |
24 | 23 | eqcomd 2616 | . . . 4 ⊢ (𝜑 → (𝐴 · (1 / 𝐶)) = (𝐴 / 𝐶)) |
25 | 22, 11 | mulcld 9939 | . . . . . 6 ⊢ (𝜑 → (𝐶 · 𝐵) ∈ ℂ) |
26 | 25, 22, 17 | divrecd 10683 | . . . . 5 ⊢ (𝜑 → ((𝐶 · 𝐵) / 𝐶) = ((𝐶 · 𝐵) · (1 / 𝐶))) |
27 | divcan3 10590 | . . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) → ((𝐶 · 𝐵) / 𝐶) = 𝐵) | |
28 | 27 | 3expb 1258 | . . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐶 · 𝐵) / 𝐶) = 𝐵) |
29 | 11, 22, 17, 28 | syl12anc 1316 | . . . . 5 ⊢ (𝜑 → ((𝐶 · 𝐵) / 𝐶) = 𝐵) |
30 | 26, 29 | eqtr3d 2646 | . . . 4 ⊢ (𝜑 → ((𝐶 · 𝐵) · (1 / 𝐶)) = 𝐵) |
31 | 24, 30 | breq12d 4596 | . . 3 ⊢ (𝜑 → ((𝐴 · (1 / 𝐶)) < ((𝐶 · 𝐵) · (1 / 𝐶)) ↔ (𝐴 / 𝐶) < 𝐵)) |
32 | 20, 31 | bitrd 267 | . 2 ⊢ (𝜑 → ((𝐶 · 𝐵) < 𝐴 ↔ (𝐴 / 𝐶) < 𝐵)) |
33 | 7, 13, 32 | 3bitrd 293 | 1 ⊢ (𝜑 → ((𝐴 / 𝐵) < 𝐶 ↔ (𝐴 / 𝐶) < 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 class class class wbr 4583 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 · cmul 9820 < clt 9953 / cdiv 10563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-rp 11709 |
This theorem is referenced by: pimrecltneg 39610 |
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