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Mirrors > Home > MPE Home > Th. List > Mathboxes > sn-ltmul2d | Structured version Visualization version GIF version |
Description: ltmul2d 12953 without ax-mulcom 11073. (Contributed by SN, 26-Jun-2024.) |
Ref | Expression |
---|---|
sn-ltmul2d.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
sn-ltmul2d.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
sn-ltmul2d.c | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
sn-ltmul2d.1 | ⊢ (𝜑 → 0 < 𝐶) |
Ref | Expression |
---|---|
sn-ltmul2d | ⊢ (𝜑 → ((𝐶 · 𝐴) < (𝐶 · 𝐵) ↔ 𝐴 < 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sn-ltmul2d.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
2 | sn-ltmul2d.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | sn-ltmul2d.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | rersubcl 40750 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 −ℝ 𝐴) ∈ ℝ) | |
5 | 2, 3, 4 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝐵 −ℝ 𝐴) ∈ ℝ) |
6 | sn-ltmul2d.1 | . . . 4 ⊢ (𝜑 → 0 < 𝐶) | |
7 | 1, 5, 6 | mulgt0b2d 40832 | . . 3 ⊢ (𝜑 → (0 < (𝐵 −ℝ 𝐴) ↔ 0 < (𝐶 · (𝐵 −ℝ 𝐴)))) |
8 | resubdi 40768 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐶 · (𝐵 −ℝ 𝐴)) = ((𝐶 · 𝐵) −ℝ (𝐶 · 𝐴))) | |
9 | 1, 2, 3, 8 | syl3anc 1371 | . . . 4 ⊢ (𝜑 → (𝐶 · (𝐵 −ℝ 𝐴)) = ((𝐶 · 𝐵) −ℝ (𝐶 · 𝐴))) |
10 | 9 | breq2d 5115 | . . 3 ⊢ (𝜑 → (0 < (𝐶 · (𝐵 −ℝ 𝐴)) ↔ 0 < ((𝐶 · 𝐵) −ℝ (𝐶 · 𝐴)))) |
11 | 7, 10 | bitr2d 279 | . 2 ⊢ (𝜑 → (0 < ((𝐶 · 𝐵) −ℝ (𝐶 · 𝐴)) ↔ 0 < (𝐵 −ℝ 𝐴))) |
12 | 1, 3 | remulcld 11143 | . . 3 ⊢ (𝜑 → (𝐶 · 𝐴) ∈ ℝ) |
13 | 1, 2 | remulcld 11143 | . . 3 ⊢ (𝜑 → (𝐶 · 𝐵) ∈ ℝ) |
14 | reposdif 40815 | . . 3 ⊢ (((𝐶 · 𝐴) ∈ ℝ ∧ (𝐶 · 𝐵) ∈ ℝ) → ((𝐶 · 𝐴) < (𝐶 · 𝐵) ↔ 0 < ((𝐶 · 𝐵) −ℝ (𝐶 · 𝐴)))) | |
15 | 12, 13, 14 | syl2anc 584 | . 2 ⊢ (𝜑 → ((𝐶 · 𝐴) < (𝐶 · 𝐵) ↔ 0 < ((𝐶 · 𝐵) −ℝ (𝐶 · 𝐴)))) |
16 | reposdif 40815 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 0 < (𝐵 −ℝ 𝐴))) | |
17 | 3, 2, 16 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 −ℝ 𝐴))) |
18 | 11, 15, 17 | 3bitr4d 310 | 1 ⊢ (𝜑 → ((𝐶 · 𝐴) < (𝐶 · 𝐵) ↔ 𝐴 < 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 ∈ wcel 2106 class class class wbr 5103 (class class class)co 7351 ℝcr 11008 0cc0 11009 · cmul 11014 < clt 11147 −ℝ cresub 40737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-po 5543 df-so 5544 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-pnf 11149 df-mnf 11150 df-ltxr 11152 df-2 12174 df-3 12175 df-resub 40738 |
This theorem is referenced by: (None) |
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