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Mirrors > Home > MPE Home > Th. List > divge1 | Structured version Visualization version GIF version |
Description: The ratio of a number over a smaller positive number is larger than 1. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
divge1 | ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 1 ≤ (𝐵 / 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpgecl 12405 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ+) | |
2 | rpcn 12387 | . . . . 5 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ∈ ℂ) | |
3 | rpne0 12393 | . . . . 5 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ≠ 0) | |
4 | 2, 3 | dividd 11403 | . . . 4 ⊢ (𝐵 ∈ ℝ+ → (𝐵 / 𝐵) = 1) |
5 | 4 | eqcomd 2804 | . . 3 ⊢ (𝐵 ∈ ℝ+ → 1 = (𝐵 / 𝐵)) |
6 | 1, 5 | syl 17 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 1 = (𝐵 / 𝐵)) |
7 | simp3 1135 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵) | |
8 | simp1 1133 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ+) | |
9 | 8, 1, 1 | lediv2d 12443 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (𝐴 ≤ 𝐵 ↔ (𝐵 / 𝐵) ≤ (𝐵 / 𝐴))) |
10 | 7, 9 | mpbid 235 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (𝐵 / 𝐵) ≤ (𝐵 / 𝐴)) |
11 | 6, 10 | eqbrtrd 5052 | 1 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 1 ≤ (𝐵 / 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 (class class class)co 7135 ℝcr 10525 1c1 10527 ≤ cle 10665 / cdiv 11286 ℝ+crp 12377 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-rp 12378 |
This theorem is referenced by: fldiv4lem1div2uz2 13201 fprodle 15342 gausslemma2dlem4 25953 2xp3dxp2ge1d 39387 |
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