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Mirrors > Home > ILE Home > Th. List > nnrecred | GIF version |
Description: The reciprocal of a positive integer is real. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
nnge1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℕ) |
Ref | Expression |
---|---|
nnrecred | ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnge1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℕ) | |
2 | nnrecre 8888 | . 2 ⊢ (𝐴 ∈ ℕ → (1 / 𝐴) ∈ ℝ) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2135 (class class class)co 5839 ℝcr 7746 1c1 7748 / cdiv 8562 ℕcn 8851 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4097 ax-pow 4150 ax-pr 4184 ax-un 4408 ax-setind 4511 ax-cnex 7838 ax-resscn 7839 ax-1cn 7840 ax-1re 7841 ax-icn 7842 ax-addcl 7843 ax-addrcl 7844 ax-mulcl 7845 ax-mulrcl 7846 ax-addcom 7847 ax-mulcom 7848 ax-addass 7849 ax-mulass 7850 ax-distr 7851 ax-i2m1 7852 ax-0lt1 7853 ax-1rid 7854 ax-0id 7855 ax-rnegex 7856 ax-precex 7857 ax-cnre 7858 ax-pre-ltirr 7859 ax-pre-ltwlin 7860 ax-pre-lttrn 7861 ax-pre-apti 7862 ax-pre-ltadd 7863 ax-pre-mulgt0 7864 ax-pre-mulext 7865 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2726 df-sbc 2950 df-dif 3116 df-un 3118 df-in 3120 df-ss 3127 df-pw 3558 df-sn 3579 df-pr 3580 df-op 3582 df-uni 3787 df-int 3822 df-br 3980 df-opab 4041 df-id 4268 df-po 4271 df-iso 4272 df-xp 4607 df-rel 4608 df-cnv 4609 df-co 4610 df-dm 4611 df-iota 5150 df-fun 5187 df-fv 5193 df-riota 5795 df-ov 5842 df-oprab 5843 df-mpo 5844 df-pnf 7929 df-mnf 7930 df-xr 7931 df-ltxr 7932 df-le 7933 df-sub 8065 df-neg 8066 df-reap 8467 df-ap 8474 df-div 8563 df-inn 8852 |
This theorem is referenced by: flqdiv 10250 caucvgrelemcau 10916 cvg1nlemcau 10920 resqrexlemnm 10954 trireciplem 11435 trirecip 11436 geo2sum 11449 ege2le3 11606 eftlub 11625 eirraplem 11711 sqrt2irrap 12106 cvgcmp2nlemabs 13804 trilpolemlt1 13813 nconstwlpolem0 13834 |
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