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Mirrors > Home > ILE Home > Th. List > fzctr | GIF version |
Description: Lemma for theorems about the central binomial coefficient. (Contributed by Mario Carneiro, 8-Mar-2014.) (Revised by Mario Carneiro, 2-Aug-2014.) |
Ref | Expression |
---|---|
fzctr | ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (0...(2 · 𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0ge0 9130 | . 2 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁) | |
2 | nn0re 9114 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
3 | nn0addge1 9151 | . . . 4 ⊢ ((𝑁 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → 𝑁 ≤ (𝑁 + 𝑁)) | |
4 | 2, 3 | mpancom 419 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ≤ (𝑁 + 𝑁)) |
5 | nn0cn 9115 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ) | |
6 | 5 | 2timesd 9090 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (2 · 𝑁) = (𝑁 + 𝑁)) |
7 | 4, 6 | breqtrrd 4004 | . 2 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ≤ (2 · 𝑁)) |
8 | nn0z 9202 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
9 | 0zd 9194 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ ℤ) | |
10 | 2z 9210 | . . . 4 ⊢ 2 ∈ ℤ | |
11 | zmulcl 9235 | . . . 4 ⊢ ((2 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (2 · 𝑁) ∈ ℤ) | |
12 | 10, 8, 11 | sylancr 411 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (2 · 𝑁) ∈ ℤ) |
13 | elfz 9941 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ ∧ (2 · 𝑁) ∈ ℤ) → (𝑁 ∈ (0...(2 · 𝑁)) ↔ (0 ≤ 𝑁 ∧ 𝑁 ≤ (2 · 𝑁)))) | |
14 | 8, 9, 12, 13 | syl3anc 1227 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ (0...(2 · 𝑁)) ↔ (0 ≤ 𝑁 ∧ 𝑁 ≤ (2 · 𝑁)))) |
15 | 1, 7, 14 | mpbir2and 933 | 1 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (0...(2 · 𝑁))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 2135 class class class wbr 3976 (class class class)co 5836 ℝcr 7743 0cc0 7744 + caddc 7747 · cmul 7749 ≤ cle 7925 2c2 8899 ℕ0cn0 9105 ℤcz 9182 ...cfz 9935 |
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 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-3or 968 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-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-inn 8849 df-2 8907 df-n0 9106 df-z 9183 df-fz 9936 |
This theorem is referenced by: (None) |
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