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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 13006 | . . 3 ⊢ (𝜑 → -𝐶 ∈ ℝ+) |
6 | 1, 2, 5 | ltmul1d 13055 | . 2 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐴 · -𝐶) < (𝐵 · -𝐶))) |
7 | 3 | renegcld 11639 | . . . 4 ⊢ (𝜑 → -𝐶 ∈ ℝ) |
8 | 1, 7 | remulcld 11242 | . . 3 ⊢ (𝜑 → (𝐴 · -𝐶) ∈ ℝ) |
9 | 2, 7 | remulcld 11242 | . . 3 ⊢ (𝜑 → (𝐵 · -𝐶) ∈ ℝ) |
10 | 8, 9 | ltnegd 11790 | . 2 ⊢ (𝜑 → ((𝐴 · -𝐶) < (𝐵 · -𝐶) ↔ -(𝐵 · -𝐶) < -(𝐴 · -𝐶))) |
11 | 2 | recnd 11240 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
12 | 7 | recnd 11240 | . . . . 5 ⊢ (𝜑 → -𝐶 ∈ ℂ) |
13 | 11, 12 | mulneg2d 11666 | . . . 4 ⊢ (𝜑 → (𝐵 · --𝐶) = -(𝐵 · -𝐶)) |
14 | 3 | recnd 11240 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
15 | 14 | negnegd 11560 | . . . . 5 ⊢ (𝜑 → --𝐶 = 𝐶) |
16 | 15 | oveq2d 7418 | . . . 4 ⊢ (𝜑 → (𝐵 · --𝐶) = (𝐵 · 𝐶)) |
17 | 13, 16 | eqtr3d 2766 | . . 3 ⊢ (𝜑 → -(𝐵 · -𝐶) = (𝐵 · 𝐶)) |
18 | 1 | recnd 11240 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
19 | 18, 12 | mulneg2d 11666 | . . . 4 ⊢ (𝜑 → (𝐴 · --𝐶) = -(𝐴 · -𝐶)) |
20 | 15 | oveq2d 7418 | . . . 4 ⊢ (𝜑 → (𝐴 · --𝐶) = (𝐴 · 𝐶)) |
21 | 19, 20 | eqtr3d 2766 | . . 3 ⊢ (𝜑 → -(𝐴 · -𝐶) = (𝐴 · 𝐶)) |
22 | 17, 21 | breq12d 5152 | . 2 ⊢ (𝜑 → (-(𝐵 · -𝐶) < -(𝐴 · -𝐶) ↔ (𝐵 · 𝐶) < (𝐴 · 𝐶))) |
23 | 6, 10, 22 | 3bitrd 305 | 1 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐵 · 𝐶) < (𝐴 · 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2098 class class class wbr 5139 (class class class)co 7402 ℝcr 11106 0cc0 11107 · cmul 11112 < clt 11246 -cneg 11443 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-po 5579 df-so 5580 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-rp 12973 |
This theorem is referenced by: ltdiv23neg 44614 |
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