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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 11247 | . . . 4 ⊢ 0 ∈ ℝ | |
2 | ltdiv1 12109 | . . . 4 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (0 < 𝐴 ↔ (0 / 𝐵) < (𝐴 / 𝐵))) | |
3 | 1, 2 | mp3an1 1445 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (0 < 𝐴 ↔ (0 / 𝐵) < (𝐴 / 𝐵))) |
4 | 3 | 3impb 1113 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐵) → (0 < 𝐴 ↔ (0 / 𝐵) < (𝐴 / 𝐵))) |
5 | recn 11229 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
6 | gt0ne0 11710 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 0 < 𝐵) → 𝐵 ≠ 0) | |
7 | div0 11933 | . . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (0 / 𝐵) = 0) | |
8 | 5, 6, 7 | syl2an2r 684 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 0 < 𝐵) → (0 / 𝐵) = 0) |
9 | 8 | breq1d 5158 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ 0 < 𝐵) → ((0 / 𝐵) < (𝐴 / 𝐵) ↔ 0 < (𝐴 / 𝐵))) |
10 | 9 | 3adant1 1128 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐵) → ((0 / 𝐵) < (𝐴 / 𝐵) ↔ 0 < (𝐴 / 𝐵))) |
11 | 4, 10 | bitrd 279 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐵) → (0 < 𝐴 ↔ 0 < (𝐴 / 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 class class class wbr 5148 (class class class)co 7420 ℂcc 11137 ℝcr 11138 0cc0 11139 < clt 11279 / cdiv 11902 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 |
This theorem is referenced by: divgt0 12113 halfpos2 12472 elpq 12990 gt0divd 13086 dvferm1lem 25929 dvferm2lem 25931 dvgt0 25950 logbgt0b 26738 nnoALTV 47035 |
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