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Mirrors > Home > ILE Home > Th. List > nnrecgt0 | GIF version |
Description: The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.) |
Ref | Expression |
---|---|
nnrecgt0 | ⊢ (𝐴 ∈ ℕ → 0 < (1 / 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnge1 8945 | . 2 ⊢ (𝐴 ∈ ℕ → 1 ≤ 𝐴) | |
2 | 0lt1 8087 | . . 3 ⊢ 0 < 1 | |
3 | nnre 8929 | . . . 4 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
4 | 0re 7960 | . . . . . 6 ⊢ 0 ∈ ℝ | |
5 | 1re 7959 | . . . . . 6 ⊢ 1 ∈ ℝ | |
6 | ltletr 8050 | . . . . . 6 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((0 < 1 ∧ 1 ≤ 𝐴) → 0 < 𝐴)) | |
7 | 4, 5, 6 | mp3an12 1327 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((0 < 1 ∧ 1 ≤ 𝐴) → 0 < 𝐴)) |
8 | recgt0 8810 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 0 < (1 / 𝐴)) | |
9 | 8 | ex 115 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (0 < 𝐴 → 0 < (1 / 𝐴))) |
10 | 7, 9 | syld 45 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((0 < 1 ∧ 1 ≤ 𝐴) → 0 < (1 / 𝐴))) |
11 | 3, 10 | syl 14 | . . 3 ⊢ (𝐴 ∈ ℕ → ((0 < 1 ∧ 1 ≤ 𝐴) → 0 < (1 / 𝐴))) |
12 | 2, 11 | mpani 430 | . 2 ⊢ (𝐴 ∈ ℕ → (1 ≤ 𝐴 → 0 < (1 / 𝐴))) |
13 | 1, 12 | mpd 13 | 1 ⊢ (𝐴 ∈ ℕ → 0 < (1 / 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2148 class class class wbr 4005 (class class class)co 5878 ℝcr 7813 0cc0 7814 1c1 7815 < clt 7995 ≤ cle 7996 / cdiv 8632 ℕcn 8922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7905 ax-resscn 7906 ax-1cn 7907 ax-1re 7908 ax-icn 7909 ax-addcl 7910 ax-addrcl 7911 ax-mulcl 7912 ax-mulrcl 7913 ax-addcom 7914 ax-mulcom 7915 ax-addass 7916 ax-mulass 7917 ax-distr 7918 ax-i2m1 7919 ax-0lt1 7920 ax-1rid 7921 ax-0id 7922 ax-rnegex 7923 ax-precex 7924 ax-cnre 7925 ax-pre-ltirr 7926 ax-pre-ltwlin 7927 ax-pre-lttrn 7928 ax-pre-apti 7929 ax-pre-ltadd 7930 ax-pre-mulgt0 7931 ax-pre-mulext 7932 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-br 4006 df-opab 4067 df-id 4295 df-po 4298 df-iso 4299 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-iota 5180 df-fun 5220 df-fv 5226 df-riota 5834 df-ov 5881 df-oprab 5882 df-mpo 5883 df-pnf 7997 df-mnf 7998 df-xr 7999 df-ltxr 8000 df-le 8001 df-sub 8133 df-neg 8134 df-reap 8535 df-ap 8542 df-div 8633 df-inn 8923 |
This theorem is referenced by: (None) |
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