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 10776 | . . . 4 ⊢ (𝜑 → 𝐵 ≠ 0) |
5 | 1, 2, 4 | redivcld 11468 | . . 3 ⊢ (𝜑 → (𝐴 / 𝐵) ∈ ℝ) |
6 | ltdiv23neg.4 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
7 | 5, 6, 2, 3 | ltmulneg 41684 | . 2 ⊢ (𝜑 → ((𝐴 / 𝐵) < 𝐶 ↔ (𝐶 · 𝐵) < ((𝐴 / 𝐵) · 𝐵))) |
8 | recn 10627 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
9 | 1, 8 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
10 | recn 10627 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
11 | 2, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
12 | 9, 11, 4 | divcan1d 11417 | . . 3 ⊢ (𝜑 → ((𝐴 / 𝐵) · 𝐵) = 𝐴) |
13 | 12 | breq2d 5078 | . 2 ⊢ (𝜑 → ((𝐶 · 𝐵) < ((𝐴 / 𝐵) · 𝐵) ↔ (𝐶 · 𝐵) < 𝐴)) |
14 | remulcl 10622 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 · 𝐵) ∈ ℝ) | |
15 | 6, 2, 14 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝐶 · 𝐵) ∈ ℝ) |
16 | ltdiv23neg.5 | . . . . . 6 ⊢ (𝜑 → 𝐶 < 0) | |
17 | 6, 16 | ltned 10776 | . . . . 5 ⊢ (𝜑 → 𝐶 ≠ 0) |
18 | 6, 17 | rereccld 11467 | . . . 4 ⊢ (𝜑 → (1 / 𝐶) ∈ ℝ) |
19 | 6, 16 | reclt0d 41678 | . . . 4 ⊢ (𝜑 → (1 / 𝐶) < 0) |
20 | 15, 1, 18, 19 | ltmulneg 41684 | . . 3 ⊢ (𝜑 → ((𝐶 · 𝐵) < 𝐴 ↔ (𝐴 · (1 / 𝐶)) < ((𝐶 · 𝐵) · (1 / 𝐶)))) |
21 | recn 10627 | . . . . . . 7 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℂ) | |
22 | 6, 21 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
23 | 9, 22, 17 | divrecd 11419 | . . . . 5 ⊢ (𝜑 → (𝐴 / 𝐶) = (𝐴 · (1 / 𝐶))) |
24 | 23 | eqcomd 2827 | . . . 4 ⊢ (𝜑 → (𝐴 · (1 / 𝐶)) = (𝐴 / 𝐶)) |
25 | 22, 11 | mulcld 10661 | . . . . . 6 ⊢ (𝜑 → (𝐶 · 𝐵) ∈ ℂ) |
26 | 25, 22, 17 | divrecd 11419 | . . . . 5 ⊢ (𝜑 → ((𝐶 · 𝐵) / 𝐶) = ((𝐶 · 𝐵) · (1 / 𝐶))) |
27 | divcan3 11324 | . . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) → ((𝐶 · 𝐵) / 𝐶) = 𝐵) | |
28 | 27 | 3expb 1116 | . . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐶 · 𝐵) / 𝐶) = 𝐵) |
29 | 11, 22, 17, 28 | syl12anc 834 | . . . . 5 ⊢ (𝜑 → ((𝐶 · 𝐵) / 𝐶) = 𝐵) |
30 | 26, 29 | eqtr3d 2858 | . . . 4 ⊢ (𝜑 → ((𝐶 · 𝐵) · (1 / 𝐶)) = 𝐵) |
31 | 24, 30 | breq12d 5079 | . . 3 ⊢ (𝜑 → ((𝐴 · (1 / 𝐶)) < ((𝐶 · 𝐵) · (1 / 𝐶)) ↔ (𝐴 / 𝐶) < 𝐵)) |
32 | 20, 31 | bitrd 281 | . 2 ⊢ (𝜑 → ((𝐶 · 𝐵) < 𝐴 ↔ (𝐴 / 𝐶) < 𝐵)) |
33 | 7, 13, 32 | 3bitrd 307 | 1 ⊢ (𝜑 → ((𝐴 / 𝐵) < 𝐶 ↔ (𝐴 / 𝐶) < 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 class class class wbr 5066 (class class class)co 7156 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 · cmul 10542 < clt 10675 / cdiv 11297 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-rp 12391 |
This theorem is referenced by: pimrecltneg 43021 |
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