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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 8981 | . 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 2160 (class class class)co 5892 ℝcr 7835 1c1 7837 / cdiv 8654 ℕcn 8944 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7927 ax-resscn 7928 ax-1cn 7929 ax-1re 7930 ax-icn 7931 ax-addcl 7932 ax-addrcl 7933 ax-mulcl 7934 ax-mulrcl 7935 ax-addcom 7936 ax-mulcom 7937 ax-addass 7938 ax-mulass 7939 ax-distr 7940 ax-i2m1 7941 ax-0lt1 7942 ax-1rid 7943 ax-0id 7944 ax-rnegex 7945 ax-precex 7946 ax-cnre 7947 ax-pre-ltirr 7948 ax-pre-ltwlin 7949 ax-pre-lttrn 7950 ax-pre-apti 7951 ax-pre-ltadd 7952 ax-pre-mulgt0 7953 ax-pre-mulext 7954 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-id 4308 df-po 4311 df-iso 4312 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-iota 5193 df-fun 5234 df-fv 5240 df-riota 5848 df-ov 5895 df-oprab 5896 df-mpo 5897 df-pnf 8019 df-mnf 8020 df-xr 8021 df-ltxr 8022 df-le 8023 df-sub 8155 df-neg 8156 df-reap 8557 df-ap 8564 df-div 8655 df-inn 8945 |
This theorem is referenced by: flqdiv 10347 caucvgrelemcau 11016 cvg1nlemcau 11020 resqrexlemnm 11054 trireciplem 11535 trirecip 11536 geo2sum 11549 ege2le3 11706 eftlub 11725 eirraplem 11811 sqrt2irrap 12207 cvgcmp2nlemabs 15218 trilpolemlt1 15227 nconstwlpolem0 15249 |
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