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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 10514 | . . . 4 ⊢ (𝜑 → 𝐵 ≠ 0) |
5 | 1, 2, 4 | redivcld 11206 | . . 3 ⊢ (𝜑 → (𝐴 / 𝐵) ∈ ℝ) |
6 | ltdiv23neg.4 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
7 | 5, 6, 2, 3 | ltmulneg 40535 | . 2 ⊢ (𝜑 → ((𝐴 / 𝐵) < 𝐶 ↔ (𝐶 · 𝐵) < ((𝐴 / 𝐵) · 𝐵))) |
8 | recn 10364 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
9 | 1, 8 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
10 | recn 10364 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
11 | 2, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
12 | 9, 11, 4 | divcan1d 11155 | . . 3 ⊢ (𝜑 → ((𝐴 / 𝐵) · 𝐵) = 𝐴) |
13 | 12 | breq2d 4900 | . 2 ⊢ (𝜑 → ((𝐶 · 𝐵) < ((𝐴 / 𝐵) · 𝐵) ↔ (𝐶 · 𝐵) < 𝐴)) |
14 | remulcl 10359 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 · 𝐵) ∈ ℝ) | |
15 | 6, 2, 14 | syl2anc 579 | . . . 4 ⊢ (𝜑 → (𝐶 · 𝐵) ∈ ℝ) |
16 | ltdiv23neg.5 | . . . . . 6 ⊢ (𝜑 → 𝐶 < 0) | |
17 | 6, 16 | ltned 10514 | . . . . 5 ⊢ (𝜑 → 𝐶 ≠ 0) |
18 | 6, 17 | rereccld 11205 | . . . 4 ⊢ (𝜑 → (1 / 𝐶) ∈ ℝ) |
19 | 6, 16 | reclt0d 40529 | . . . 4 ⊢ (𝜑 → (1 / 𝐶) < 0) |
20 | 15, 1, 18, 19 | ltmulneg 40535 | . . 3 ⊢ (𝜑 → ((𝐶 · 𝐵) < 𝐴 ↔ (𝐴 · (1 / 𝐶)) < ((𝐶 · 𝐵) · (1 / 𝐶)))) |
21 | recn 10364 | . . . . . . 7 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℂ) | |
22 | 6, 21 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
23 | 9, 22, 17 | divrecd 11157 | . . . . 5 ⊢ (𝜑 → (𝐴 / 𝐶) = (𝐴 · (1 / 𝐶))) |
24 | 23 | eqcomd 2784 | . . . 4 ⊢ (𝜑 → (𝐴 · (1 / 𝐶)) = (𝐴 / 𝐶)) |
25 | 22, 11 | mulcld 10399 | . . . . . 6 ⊢ (𝜑 → (𝐶 · 𝐵) ∈ ℂ) |
26 | 25, 22, 17 | divrecd 11157 | . . . . 5 ⊢ (𝜑 → ((𝐶 · 𝐵) / 𝐶) = ((𝐶 · 𝐵) · (1 / 𝐶))) |
27 | divcan3 11062 | . . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) → ((𝐶 · 𝐵) / 𝐶) = 𝐵) | |
28 | 27 | 3expb 1110 | . . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐶 · 𝐵) / 𝐶) = 𝐵) |
29 | 11, 22, 17, 28 | syl12anc 827 | . . . . 5 ⊢ (𝜑 → ((𝐶 · 𝐵) / 𝐶) = 𝐵) |
30 | 26, 29 | eqtr3d 2816 | . . . 4 ⊢ (𝜑 → ((𝐶 · 𝐵) · (1 / 𝐶)) = 𝐵) |
31 | 24, 30 | breq12d 4901 | . . 3 ⊢ (𝜑 → ((𝐴 · (1 / 𝐶)) < ((𝐶 · 𝐵) · (1 / 𝐶)) ↔ (𝐴 / 𝐶) < 𝐵)) |
32 | 20, 31 | bitrd 271 | . 2 ⊢ (𝜑 → ((𝐶 · 𝐵) < 𝐴 ↔ (𝐴 / 𝐶) < 𝐵)) |
33 | 7, 13, 32 | 3bitrd 297 | 1 ⊢ (𝜑 → ((𝐴 / 𝐵) < 𝐶 ↔ (𝐴 / 𝐶) < 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 class class class wbr 4888 (class class class)co 6924 ℂcc 10272 ℝcr 10273 0cc0 10274 1c1 10275 · cmul 10279 < clt 10413 / cdiv 11035 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-po 5276 df-so 5277 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11036 df-rp 12143 |
This theorem is referenced by: pimrecltneg 41874 |
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