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Mirrors > Home > MPE Home > Th. List > gt0div | Structured version Visualization version GIF version |
Description: Division of a positive number by a positive number. (Contributed by NM, 28-Sep-2005.) |
Ref | Expression |
---|---|
gt0div | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐵) → (0 < 𝐴 ↔ 0 < (𝐴 / 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 10714 | . . . 4 ⊢ 0 ∈ ℝ | |
2 | ltdiv1 11575 | . . . 4 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (0 < 𝐴 ↔ (0 / 𝐵) < (𝐴 / 𝐵))) | |
3 | 1, 2 | mp3an1 1449 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (0 < 𝐴 ↔ (0 / 𝐵) < (𝐴 / 𝐵))) |
4 | 3 | 3impb 1116 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐵) → (0 < 𝐴 ↔ (0 / 𝐵) < (𝐴 / 𝐵))) |
5 | recn 10698 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
6 | gt0ne0 11176 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 0 < 𝐵) → 𝐵 ≠ 0) | |
7 | div0 11399 | . . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (0 / 𝐵) = 0) | |
8 | 5, 6, 7 | syl2an2r 685 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 0 < 𝐵) → (0 / 𝐵) = 0) |
9 | 8 | breq1d 5037 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ 0 < 𝐵) → ((0 / 𝐵) < (𝐴 / 𝐵) ↔ 0 < (𝐴 / 𝐵))) |
10 | 9 | 3adant1 1131 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐵) → ((0 / 𝐵) < (𝐴 / 𝐵) ↔ 0 < (𝐴 / 𝐵))) |
11 | 4, 10 | bitrd 282 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐵) → (0 < 𝐴 ↔ 0 < (𝐴 / 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2113 ≠ wne 2934 class class class wbr 5027 (class class class)co 7164 ℂcc 10606 ℝcr 10607 0cc0 10608 < clt 10746 / cdiv 11368 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-mpt 5108 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 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-div 11369 |
This theorem is referenced by: divgt0 11579 halfpos2 11938 elpq 12450 gt0divd 12544 dvferm1lem 24728 dvferm2lem 24730 dvgt0 24748 logbgt0b 25523 nnoALTV 44665 |
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