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Mirrors > Home > MPE Home > Th. List > ledivmuld | Structured version Visualization version GIF version |
Description: 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
ltmul1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltmul1d.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
ltmul1d.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ+) |
Ref | Expression |
---|---|
ledivmuld | ⊢ (𝜑 → ((𝐴 / 𝐶) ≤ 𝐵 ↔ 𝐴 ≤ (𝐶 · 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltmul1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | ltmul1d.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | ltmul1d.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ+) | |
4 | 3 | rpregt0d 12425 | . 2 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 0 < 𝐶)) |
5 | ledivmul 11505 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐴 / 𝐶) ≤ 𝐵 ↔ 𝐴 ≤ (𝐶 · 𝐵))) | |
6 | 1, 2, 4, 5 | syl3anc 1368 | 1 ⊢ (𝜑 → ((𝐴 / 𝐶) ≤ 𝐵 ↔ 𝐴 ≤ (𝐶 · 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2111 class class class wbr 5030 (class class class)co 7135 ℝcr 10525 0cc0 10526 · cmul 10531 < clt 10664 ≤ 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: discr 13597 oddge22np1 15690 bitsfzo 15774 bitscmp 15777 c1liplem1 24599 aalioulem3 24930 aalioulem4 24931 aalioulem5 24932 aaliou3lem8 24941 logcnlem4 25236 lgamgulmlem2 25615 chtppilim 26059 rpvmasumlem 26071 dchrmusum2 26078 dchrisum0lem1 26100 mudivsum 26114 pntrlog2bndlem6 26167 ostth2lem3 26219 ostth2lem4 26220 ostth2 26221 ttgcontlem1 26679 poimirlem29 35086 poimirlem30 35087 poimirlem31 35088 poimirlem32 35089 ftc1anclem7 35136 areacirclem4 35148 hoidmvlelem3 43236 |
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