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 10951 | . . . 4 ⊢ (𝜑 → 𝐵 ≠ 0) |
5 | 1, 2, 4 | redivcld 11643 | . . 3 ⊢ (𝜑 → (𝐴 / 𝐵) ∈ ℝ) |
6 | ltdiv23neg.4 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
7 | 5, 6, 2, 3 | ltmulneg 42557 | . 2 ⊢ (𝜑 → ((𝐴 / 𝐵) < 𝐶 ↔ (𝐶 · 𝐵) < ((𝐴 / 𝐵) · 𝐵))) |
8 | recn 10802 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
9 | 1, 8 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
10 | recn 10802 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
11 | 2, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
12 | 9, 11, 4 | divcan1d 11592 | . . 3 ⊢ (𝜑 → ((𝐴 / 𝐵) · 𝐵) = 𝐴) |
13 | 12 | breq2d 5055 | . 2 ⊢ (𝜑 → ((𝐶 · 𝐵) < ((𝐴 / 𝐵) · 𝐵) ↔ (𝐶 · 𝐵) < 𝐴)) |
14 | remulcl 10797 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 · 𝐵) ∈ ℝ) | |
15 | 6, 2, 14 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (𝐶 · 𝐵) ∈ ℝ) |
16 | ltdiv23neg.5 | . . . . . 6 ⊢ (𝜑 → 𝐶 < 0) | |
17 | 6, 16 | ltned 10951 | . . . . 5 ⊢ (𝜑 → 𝐶 ≠ 0) |
18 | 6, 17 | rereccld 11642 | . . . 4 ⊢ (𝜑 → (1 / 𝐶) ∈ ℝ) |
19 | 6, 16 | reclt0d 42551 | . . . 4 ⊢ (𝜑 → (1 / 𝐶) < 0) |
20 | 15, 1, 18, 19 | ltmulneg 42557 | . . 3 ⊢ (𝜑 → ((𝐶 · 𝐵) < 𝐴 ↔ (𝐴 · (1 / 𝐶)) < ((𝐶 · 𝐵) · (1 / 𝐶)))) |
21 | recn 10802 | . . . . . . 7 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℂ) | |
22 | 6, 21 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
23 | 9, 22, 17 | divrecd 11594 | . . . . 5 ⊢ (𝜑 → (𝐴 / 𝐶) = (𝐴 · (1 / 𝐶))) |
24 | 23 | eqcomd 2740 | . . . 4 ⊢ (𝜑 → (𝐴 · (1 / 𝐶)) = (𝐴 / 𝐶)) |
25 | 22, 11 | mulcld 10836 | . . . . . 6 ⊢ (𝜑 → (𝐶 · 𝐵) ∈ ℂ) |
26 | 25, 22, 17 | divrecd 11594 | . . . . 5 ⊢ (𝜑 → ((𝐶 · 𝐵) / 𝐶) = ((𝐶 · 𝐵) · (1 / 𝐶))) |
27 | divcan3 11499 | . . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) → ((𝐶 · 𝐵) / 𝐶) = 𝐵) | |
28 | 27 | 3expb 1122 | . . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐶 · 𝐵) / 𝐶) = 𝐵) |
29 | 11, 22, 17, 28 | syl12anc 837 | . . . . 5 ⊢ (𝜑 → ((𝐶 · 𝐵) / 𝐶) = 𝐵) |
30 | 26, 29 | eqtr3d 2776 | . . . 4 ⊢ (𝜑 → ((𝐶 · 𝐵) · (1 / 𝐶)) = 𝐵) |
31 | 24, 30 | breq12d 5056 | . . 3 ⊢ (𝜑 → ((𝐴 · (1 / 𝐶)) < ((𝐶 · 𝐵) · (1 / 𝐶)) ↔ (𝐴 / 𝐶) < 𝐵)) |
32 | 20, 31 | bitrd 282 | . 2 ⊢ (𝜑 → ((𝐶 · 𝐵) < 𝐴 ↔ (𝐴 / 𝐶) < 𝐵)) |
33 | 7, 13, 32 | 3bitrd 308 | 1 ⊢ (𝜑 → ((𝐴 / 𝐵) < 𝐶 ↔ (𝐴 / 𝐶) < 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2110 ≠ wne 2935 class class class wbr 5043 (class class class)co 7202 ℂcc 10710 ℝcr 10711 0cc0 10712 1c1 10713 · cmul 10717 < clt 10850 / cdiv 11472 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-po 5457 df-so 5458 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-rp 12570 |
This theorem is referenced by: pimrecltneg 43886 |
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