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Mirrors > Home > MPE Home > Th. List > ledivmul2d | 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 |
---|---|
ledivmul2d | ⊢ (𝜑 → ((𝐴 / 𝐶) ≤ 𝐵 ↔ 𝐴 ≤ (𝐵 · 𝐶))) |
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 12431 | . 2 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 0 < 𝐶)) |
5 | ledivmul2 11512 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐴 / 𝐶) ≤ 𝐵 ↔ 𝐴 ≤ (𝐵 · 𝐶))) | |
6 | 1, 2, 4, 5 | syl3anc 1366 | 1 ⊢ (𝜑 → ((𝐴 / 𝐶) ≤ 𝐵 ↔ 𝐴 ≤ (𝐵 · 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2113 class class class wbr 5059 (class class class)co 7149 ℝcr 10529 0cc0 10530 · cmul 10535 < clt 10668 ≤ cle 10669 / cdiv 11290 ℝ+crp 12383 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-rp 12384 |
This theorem is referenced by: nmoi 23332 nmoleub 23335 nmoleub2lem3 23714 nmoleub3 23718 ulmdvlem1 24986 fsumharmonic 25587 bposlem1 25858 chpo1ubb 26055 chpdifbndlem1 26127 selberg3lem1 26131 pntrlog2bndlem2 26152 pntrlog2bndlem4 26154 pntrlog2bndlem6 26157 pntibndlem2 26165 pntlemo 26181 knoppndvlem18 33889 geomcau 35067 |
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