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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 10584 | . . . . 5 ⊢ 1 ∈ ℂ | |
2 | ltplus1.1 | . . . . . 6 ⊢ 𝐴 ∈ ℝ | |
3 | 2 | recni 10644 | . . . . 5 ⊢ 𝐴 ∈ ℂ |
4 | ax-1ne0 10595 | . . . . 5 ⊢ 1 ≠ 0 | |
5 | recgt0i.2 | . . . . . 6 ⊢ 0 < 𝐴 | |
6 | 2, 5 | gt0ne0ii 11165 | . . . . 5 ⊢ 𝐴 ≠ 0 |
7 | 1, 3, 4, 6 | divne0i 11377 | . . . 4 ⊢ (1 / 𝐴) ≠ 0 |
8 | 7 | nesymi 3044 | . . 3 ⊢ ¬ 0 = (1 / 𝐴) |
9 | 0lt1 11151 | . . . . 5 ⊢ 0 < 1 | |
10 | 0re 10632 | . . . . . 6 ⊢ 0 ∈ ℝ | |
11 | 1re 10630 | . . . . . 6 ⊢ 1 ∈ ℝ | |
12 | 10, 11 | ltnsymi 10748 | . . . . 5 ⊢ (0 < 1 → ¬ 1 < 0) |
13 | 9, 12 | ax-mp 5 | . . . 4 ⊢ ¬ 1 < 0 |
14 | 2, 6 | rereccli 11394 | . . . . . . . . 9 ⊢ (1 / 𝐴) ∈ ℝ |
15 | 14 | renegcli 10936 | . . . . . . . 8 ⊢ -(1 / 𝐴) ∈ ℝ |
16 | 15, 2 | mulgt0i 10761 | . . . . . . 7 ⊢ ((0 < -(1 / 𝐴) ∧ 0 < 𝐴) → 0 < (-(1 / 𝐴) · 𝐴)) |
17 | 5, 16 | mpan2 690 | . . . . . 6 ⊢ (0 < -(1 / 𝐴) → 0 < (-(1 / 𝐴) · 𝐴)) |
18 | 14 | recni 10644 | . . . . . . . 8 ⊢ (1 / 𝐴) ∈ ℂ |
19 | 18, 3 | mulneg1i 11075 | . . . . . . 7 ⊢ (-(1 / 𝐴) · 𝐴) = -((1 / 𝐴) · 𝐴) |
20 | 3, 6 | recidi 11360 | . . . . . . . . 9 ⊢ (𝐴 · (1 / 𝐴)) = 1 |
21 | 3, 18, 20 | mulcomli 10639 | . . . . . . . 8 ⊢ ((1 / 𝐴) · 𝐴) = 1 |
22 | 21 | negeqi 10868 | . . . . . . 7 ⊢ -((1 / 𝐴) · 𝐴) = -1 |
23 | 19, 22 | eqtri 2821 | . . . . . 6 ⊢ (-(1 / 𝐴) · 𝐴) = -1 |
24 | 17, 23 | breqtrdi 5071 | . . . . 5 ⊢ (0 < -(1 / 𝐴) → 0 < -1) |
25 | lt0neg1 11135 | . . . . . 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 11135 | . . . . . 6 ⊢ (1 ∈ ℝ → (1 < 0 ↔ 0 < -1)) | |
28 | 11, 27 | ax-mp 5 | . . . . 5 ⊢ (1 < 0 ↔ 0 < -1) |
29 | 24, 26, 28 | 3imtr4i 295 | . . . 4 ⊢ ((1 / 𝐴) < 0 → 1 < 0) |
30 | 13, 29 | mto 200 | . . 3 ⊢ ¬ (1 / 𝐴) < 0 |
31 | 8, 30 | pm3.2ni 878 | . 2 ⊢ ¬ (0 = (1 / 𝐴) ∨ (1 / 𝐴) < 0) |
32 | axlttri 10701 | . . 3 ⊢ ((0 ∈ ℝ ∧ (1 / 𝐴) ∈ ℝ) → (0 < (1 / 𝐴) ↔ ¬ (0 = (1 / 𝐴) ∨ (1 / 𝐴) < 0))) | |
33 | 10, 14, 32 | mp2an 691 | . 2 ⊢ (0 < (1 / 𝐴) ↔ ¬ (0 = (1 / 𝐴) ∨ (1 / 𝐴) < 0)) |
34 | 31, 33 | mpbir 234 | 1 ⊢ 0 < (1 / 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∨ wo 844 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 (class class class)co 7135 ℝcr 10525 0cc0 10526 1c1 10527 · cmul 10531 < clt 10664 -cneg 10860 / cdiv 11286 |
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 |
This theorem is referenced by: halfgt0 11841 0.999... 15229 sincos2sgn 15539 rpnnen2lem3 15561 rpnnen2lem4 15562 rpnnen2lem9 15567 pcoass 23629 log2tlbnd 25531 iccioo01 34741 stoweidlem34 42676 stoweidlem59 42701 |
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