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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ltmulneg | Structured version Visualization version GIF version |
Description: Multiplying by a negative number, swaps the order. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
ltmulneg.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltmulneg.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
ltmulneg.c | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
ltmulneg.n | ⊢ (𝜑 → 𝐶 < 0) |
Ref | Expression |
---|---|
ltmulneg | ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐵 · 𝐶) < (𝐴 · 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltmulneg.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | ltmulneg.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | ltmulneg.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
4 | ltmulneg.n | . . . 4 ⊢ (𝜑 → 𝐶 < 0) | |
5 | 3, 4 | negelrpd 12148 | . . 3 ⊢ (𝜑 → -𝐶 ∈ ℝ+) |
6 | 1, 2, 5 | ltmul1d 12197 | . 2 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐴 · -𝐶) < (𝐵 · -𝐶))) |
7 | 3 | renegcld 10781 | . . . 4 ⊢ (𝜑 → -𝐶 ∈ ℝ) |
8 | 1, 7 | remulcld 10387 | . . 3 ⊢ (𝜑 → (𝐴 · -𝐶) ∈ ℝ) |
9 | 2, 7 | remulcld 10387 | . . 3 ⊢ (𝜑 → (𝐵 · -𝐶) ∈ ℝ) |
10 | 8, 9 | ltnegd 10930 | . 2 ⊢ (𝜑 → ((𝐴 · -𝐶) < (𝐵 · -𝐶) ↔ -(𝐵 · -𝐶) < -(𝐴 · -𝐶))) |
11 | 2 | recnd 10385 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
12 | 7 | recnd 10385 | . . . . 5 ⊢ (𝜑 → -𝐶 ∈ ℂ) |
13 | 11, 12 | mulneg2d 10808 | . . . 4 ⊢ (𝜑 → (𝐵 · --𝐶) = -(𝐵 · -𝐶)) |
14 | 3 | recnd 10385 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
15 | 14 | negnegd 10704 | . . . . 5 ⊢ (𝜑 → --𝐶 = 𝐶) |
16 | 15 | oveq2d 6921 | . . . 4 ⊢ (𝜑 → (𝐵 · --𝐶) = (𝐵 · 𝐶)) |
17 | 13, 16 | eqtr3d 2863 | . . 3 ⊢ (𝜑 → -(𝐵 · -𝐶) = (𝐵 · 𝐶)) |
18 | 1 | recnd 10385 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
19 | 18, 12 | mulneg2d 10808 | . . . 4 ⊢ (𝜑 → (𝐴 · --𝐶) = -(𝐴 · -𝐶)) |
20 | 15 | oveq2d 6921 | . . . 4 ⊢ (𝜑 → (𝐴 · --𝐶) = (𝐴 · 𝐶)) |
21 | 19, 20 | eqtr3d 2863 | . . 3 ⊢ (𝜑 → -(𝐴 · -𝐶) = (𝐴 · 𝐶)) |
22 | 17, 21 | breq12d 4886 | . 2 ⊢ (𝜑 → (-(𝐵 · -𝐶) < -(𝐴 · -𝐶) ↔ (𝐵 · 𝐶) < (𝐴 · 𝐶))) |
23 | 6, 10, 22 | 3bitrd 297 | 1 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐵 · 𝐶) < (𝐴 · 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∈ wcel 2164 class class class wbr 4873 (class class class)co 6905 ℝcr 10251 0cc0 10252 · cmul 10257 < clt 10391 -cneg 10586 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-po 5263 df-so 5264 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-rp 12113 |
This theorem is referenced by: ltdiv23neg 40405 |
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