Mathbox for Stanislas Polu |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lemuldiv3d | Structured version Visualization version GIF version |
Description: 'Less than or equal to' relationship between division and multiplication. (Contributed by Stanislas Polu, 9-Mar-2020.) |
Ref | Expression |
---|---|
lemuldiv3d.1 | ⊢ (𝜑 → (𝐵 · 𝐴) ≤ 𝐶) |
lemuldiv3d.2 | ⊢ (𝜑 → 0 < 𝐴) |
lemuldiv3d.3 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
lemuldiv3d.4 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
lemuldiv3d.5 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
Ref | Expression |
---|---|
lemuldiv3d | ⊢ (𝜑 → 𝐵 ≤ (𝐶 / 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lemuldiv3d.1 | . 2 ⊢ (𝜑 → (𝐵 · 𝐴) ≤ 𝐶) | |
2 | lemuldiv3d.4 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | lemuldiv3d.5 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
4 | lemuldiv3d.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
5 | lemuldiv3d.2 | . . 3 ⊢ (𝜑 → 0 < 𝐴) | |
6 | lemuldiv 11522 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → ((𝐵 · 𝐴) ≤ 𝐶 ↔ 𝐵 ≤ (𝐶 / 𝐴))) | |
7 | 2, 3, 4, 5, 6 | syl112anc 1370 | . 2 ⊢ (𝜑 → ((𝐵 · 𝐴) ≤ 𝐶 ↔ 𝐵 ≤ (𝐶 / 𝐴))) |
8 | 1, 7 | mpbid 234 | 1 ⊢ (𝜑 → 𝐵 ≤ (𝐶 / 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2114 class class class wbr 5068 (class class class)co 7158 ℝcr 10538 0cc0 10539 · cmul 10544 < clt 10677 ≤ cle 10678 / cdiv 11299 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 |
This theorem is referenced by: imo72b2 40532 |
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