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Mirrors > Home > MPE Home > Th. List > Mathboxes > sn-ltmul2d | Structured version Visualization version GIF version |
Description: ltmul2d 12653 without ax-mulcom 10776. (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 40021 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 −ℝ 𝐴) ∈ ℝ) | |
5 | 2, 3, 4 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (𝐵 −ℝ 𝐴) ∈ ℝ) |
6 | sn-ltmul2d.1 | . . . 4 ⊢ (𝜑 → 0 < 𝐶) | |
7 | 1, 5, 6 | mulgt0b2d 40090 | . . 3 ⊢ (𝜑 → (0 < (𝐵 −ℝ 𝐴) ↔ 0 < (𝐶 · (𝐵 −ℝ 𝐴)))) |
8 | resubdi 40039 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐶 · (𝐵 −ℝ 𝐴)) = ((𝐶 · 𝐵) −ℝ (𝐶 · 𝐴))) | |
9 | 1, 2, 3, 8 | syl3anc 1373 | . . . 4 ⊢ (𝜑 → (𝐶 · (𝐵 −ℝ 𝐴)) = ((𝐶 · 𝐵) −ℝ (𝐶 · 𝐴))) |
10 | 9 | breq2d 5055 | . . 3 ⊢ (𝜑 → (0 < (𝐶 · (𝐵 −ℝ 𝐴)) ↔ 0 < ((𝐶 · 𝐵) −ℝ (𝐶 · 𝐴)))) |
11 | 7, 10 | bitr2d 283 | . 2 ⊢ (𝜑 → (0 < ((𝐶 · 𝐵) −ℝ (𝐶 · 𝐴)) ↔ 0 < (𝐵 −ℝ 𝐴))) |
12 | 1, 3 | remulcld 10846 | . . 3 ⊢ (𝜑 → (𝐶 · 𝐴) ∈ ℝ) |
13 | 1, 2 | remulcld 10846 | . . 3 ⊢ (𝜑 → (𝐶 · 𝐵) ∈ ℝ) |
14 | reposdif 40084 | . . 3 ⊢ (((𝐶 · 𝐴) ∈ ℝ ∧ (𝐶 · 𝐵) ∈ ℝ) → ((𝐶 · 𝐴) < (𝐶 · 𝐵) ↔ 0 < ((𝐶 · 𝐵) −ℝ (𝐶 · 𝐴)))) | |
15 | 12, 13, 14 | syl2anc 587 | . 2 ⊢ (𝜑 → ((𝐶 · 𝐴) < (𝐶 · 𝐵) ↔ 0 < ((𝐶 · 𝐵) −ℝ (𝐶 · 𝐴)))) |
16 | reposdif 40084 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 0 < (𝐵 −ℝ 𝐴))) | |
17 | 3, 2, 16 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 −ℝ 𝐴))) |
18 | 11, 15, 17 | 3bitr4d 314 | 1 ⊢ (𝜑 → ((𝐶 · 𝐴) < (𝐶 · 𝐵) ↔ 𝐴 < 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2110 class class class wbr 5043 (class class class)co 7202 ℝcr 10711 0cc0 10712 · cmul 10717 < clt 10850 −ℝ cresub 40008 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-po 5457 df-so 5458 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-pnf 10852 df-mnf 10853 df-ltxr 10855 df-2 11876 df-3 11877 df-resub 40009 |
This theorem is referenced by: (None) |
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