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Mirrors > Home > MPE Home > Th. List > recgt0ii | Structured version Visualization version GIF version |
Description: The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.) |
Ref | Expression |
---|---|
ltplus1.1 | ⊢ 𝐴 ∈ ℝ |
recgt0i.2 | ⊢ 0 < 𝐴 |
Ref | Expression |
---|---|
recgt0ii | ⊢ 0 < (1 / 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1cn 9873 | . . . . 5 ⊢ 1 ∈ ℂ | |
2 | ltplus1.1 | . . . . . 6 ⊢ 𝐴 ∈ ℝ | |
3 | 2 | recni 9931 | . . . . 5 ⊢ 𝐴 ∈ ℂ |
4 | ax-1ne0 9884 | . . . . 5 ⊢ 1 ≠ 0 | |
5 | recgt0i.2 | . . . . . 6 ⊢ 0 < 𝐴 | |
6 | 2, 5 | gt0ne0ii 10443 | . . . . 5 ⊢ 𝐴 ≠ 0 |
7 | 1, 3, 4, 6 | divne0i 10652 | . . . 4 ⊢ (1 / 𝐴) ≠ 0 |
8 | 7 | nesymi 2839 | . . 3 ⊢ ¬ 0 = (1 / 𝐴) |
9 | 0lt1 10429 | . . . . 5 ⊢ 0 < 1 | |
10 | 0re 9919 | . . . . . 6 ⊢ 0 ∈ ℝ | |
11 | 1re 9918 | . . . . . 6 ⊢ 1 ∈ ℝ | |
12 | 10, 11 | ltnsymi 10035 | . . . . 5 ⊢ (0 < 1 → ¬ 1 < 0) |
13 | 9, 12 | ax-mp 5 | . . . 4 ⊢ ¬ 1 < 0 |
14 | 2, 6 | rereccli 10669 | . . . . . . . . 9 ⊢ (1 / 𝐴) ∈ ℝ |
15 | 14 | renegcli 10221 | . . . . . . . 8 ⊢ -(1 / 𝐴) ∈ ℝ |
16 | 15, 2 | mulgt0i 10048 | . . . . . . 7 ⊢ ((0 < -(1 / 𝐴) ∧ 0 < 𝐴) → 0 < (-(1 / 𝐴) · 𝐴)) |
17 | 5, 16 | mpan2 703 | . . . . . 6 ⊢ (0 < -(1 / 𝐴) → 0 < (-(1 / 𝐴) · 𝐴)) |
18 | 14 | recni 9931 | . . . . . . . 8 ⊢ (1 / 𝐴) ∈ ℂ |
19 | 18, 3 | mulneg1i 10355 | . . . . . . 7 ⊢ (-(1 / 𝐴) · 𝐴) = -((1 / 𝐴) · 𝐴) |
20 | 3, 6 | recidi 10635 | . . . . . . . . 9 ⊢ (𝐴 · (1 / 𝐴)) = 1 |
21 | 3, 18, 20 | mulcomli 9926 | . . . . . . . 8 ⊢ ((1 / 𝐴) · 𝐴) = 1 |
22 | 21 | negeqi 10153 | . . . . . . 7 ⊢ -((1 / 𝐴) · 𝐴) = -1 |
23 | 19, 22 | eqtri 2632 | . . . . . 6 ⊢ (-(1 / 𝐴) · 𝐴) = -1 |
24 | 17, 23 | syl6breq 4624 | . . . . 5 ⊢ (0 < -(1 / 𝐴) → 0 < -1) |
25 | lt0neg1 10413 | . . . . . 6 ⊢ ((1 / 𝐴) ∈ ℝ → ((1 / 𝐴) < 0 ↔ 0 < -(1 / 𝐴))) | |
26 | 14, 25 | ax-mp 5 | . . . . 5 ⊢ ((1 / 𝐴) < 0 ↔ 0 < -(1 / 𝐴)) |
27 | lt0neg1 10413 | . . . . . 6 ⊢ (1 ∈ ℝ → (1 < 0 ↔ 0 < -1)) | |
28 | 11, 27 | ax-mp 5 | . . . . 5 ⊢ (1 < 0 ↔ 0 < -1) |
29 | 24, 26, 28 | 3imtr4i 280 | . . . 4 ⊢ ((1 / 𝐴) < 0 → 1 < 0) |
30 | 13, 29 | mto 187 | . . 3 ⊢ ¬ (1 / 𝐴) < 0 |
31 | 8, 30 | pm3.2ni 895 | . 2 ⊢ ¬ (0 = (1 / 𝐴) ∨ (1 / 𝐴) < 0) |
32 | axlttri 9988 | . . 3 ⊢ ((0 ∈ ℝ ∧ (1 / 𝐴) ∈ ℝ) → (0 < (1 / 𝐴) ↔ ¬ (0 = (1 / 𝐴) ∨ (1 / 𝐴) < 0))) | |
33 | 10, 14, 32 | mp2an 704 | . 2 ⊢ (0 < (1 / 𝐴) ↔ ¬ (0 = (1 / 𝐴) ∨ (1 / 𝐴) < 0)) |
34 | 31, 33 | mpbir 220 | 1 ⊢ 0 < (1 / 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∨ wo 382 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 · cmul 9820 < clt 9953 -cneg 10146 / cdiv 10563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 |
This theorem is referenced by: halfgt0 11125 0.999... 14451 0.999...OLD 14452 sincos2sgn 14763 rpnnen2lem3 14784 rpnnen2lem4 14785 rpnnen2lem9 14790 pcoass 22632 log2tlbnd 24472 stoweidlem34 38927 stoweidlem59 38952 |
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