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Mirrors > Home > MPE Home > Th. List > Mathboxes > bccbc | Structured version Visualization version GIF version |
Description: The binomial coefficient and generalized binomial coefficient are equal when their arguments are nonnegative integers. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
Ref | Expression |
---|---|
bccbc.c | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
bccbc.k | ⊢ (𝜑 → 𝐾 ∈ ℕ0) |
Ref | Expression |
---|---|
bccbc | ⊢ (𝜑 → (𝑁C𝑐𝐾) = (𝑁C𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bccbc.c | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
2 | 1 | nn0cnd 12483 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
3 | bccbc.k | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℕ0) | |
4 | 2, 3 | bccval 42710 | . . . 4 ⊢ (𝜑 → (𝑁C𝑐𝐾) = ((𝑁 FallFac 𝐾) / (!‘𝐾))) |
5 | 4 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...𝑁)) → (𝑁C𝑐𝐾) = ((𝑁 FallFac 𝐾) / (!‘𝐾))) |
6 | bcfallfac 15935 | . . . 4 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((𝑁 FallFac 𝐾) / (!‘𝐾))) | |
7 | 6 | adantl 483 | . . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = ((𝑁 FallFac 𝐾) / (!‘𝐾))) |
8 | 5, 7 | eqtr4d 2776 | . 2 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...𝑁)) → (𝑁C𝑐𝐾) = (𝑁C𝐾)) |
9 | nn0split 13565 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → ℕ0 = ((0...𝑁) ∪ (ℤ≥‘(𝑁 + 1)))) | |
10 | 1, 9 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → ℕ0 = ((0...𝑁) ∪ (ℤ≥‘(𝑁 + 1)))) |
11 | 3, 10 | eleqtrd 2836 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ((0...𝑁) ∪ (ℤ≥‘(𝑁 + 1)))) |
12 | elun 4112 | . . . . . . 7 ⊢ (𝐾 ∈ ((0...𝑁) ∪ (ℤ≥‘(𝑁 + 1))) ↔ (𝐾 ∈ (0...𝑁) ∨ 𝐾 ∈ (ℤ≥‘(𝑁 + 1)))) | |
13 | 11, 12 | sylib 217 | . . . . . 6 ⊢ (𝜑 → (𝐾 ∈ (0...𝑁) ∨ 𝐾 ∈ (ℤ≥‘(𝑁 + 1)))) |
14 | 13 | orcanai 1002 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ (ℤ≥‘(𝑁 + 1))) |
15 | eluzle 12784 | . . . . . . 7 ⊢ (𝐾 ∈ (ℤ≥‘(𝑁 + 1)) → (𝑁 + 1) ≤ 𝐾) | |
16 | 15 | adantl 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑁 + 1))) → (𝑁 + 1) ≤ 𝐾) |
17 | 1 | nn0zd 12533 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
18 | 3 | nn0zd 12533 | . . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℤ) |
19 | zltp1le 12561 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑁 < 𝐾 ↔ (𝑁 + 1) ≤ 𝐾)) | |
20 | 17, 18, 19 | syl2anc 585 | . . . . . . 7 ⊢ (𝜑 → (𝑁 < 𝐾 ↔ (𝑁 + 1) ≤ 𝐾)) |
21 | 20 | adantr 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑁 + 1))) → (𝑁 < 𝐾 ↔ (𝑁 + 1) ≤ 𝐾)) |
22 | 16, 21 | mpbird 257 | . . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑁 < 𝐾) |
23 | 14, 22 | syldan 592 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...𝑁)) → 𝑁 < 𝐾) |
24 | 1 | nn0ge0d 12484 | . . . . . 6 ⊢ (𝜑 → 0 ≤ 𝑁) |
25 | 0zd 12519 | . . . . . . . . 9 ⊢ (𝜑 → 0 ∈ ℤ) | |
26 | elfzo 13583 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑁 ∈ (0..^𝐾) ↔ (0 ≤ 𝑁 ∧ 𝑁 < 𝐾))) | |
27 | 17, 25, 18, 26 | syl3anc 1372 | . . . . . . . 8 ⊢ (𝜑 → (𝑁 ∈ (0..^𝐾) ↔ (0 ≤ 𝑁 ∧ 𝑁 < 𝐾))) |
28 | 27 | biimpar 479 | . . . . . . 7 ⊢ ((𝜑 ∧ (0 ≤ 𝑁 ∧ 𝑁 < 𝐾)) → 𝑁 ∈ (0..^𝐾)) |
29 | fzoval 13582 | . . . . . . . . . . 11 ⊢ (𝐾 ∈ ℤ → (0..^𝐾) = (0...(𝐾 − 1))) | |
30 | 18, 29 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → (0..^𝐾) = (0...(𝐾 − 1))) |
31 | 30 | eleq2d 2820 | . . . . . . . . 9 ⊢ (𝜑 → (𝑁 ∈ (0..^𝐾) ↔ 𝑁 ∈ (0...(𝐾 − 1)))) |
32 | 31 | biimpa 478 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ (0..^𝐾)) → 𝑁 ∈ (0...(𝐾 − 1))) |
33 | 2, 3 | bcc0 42712 | . . . . . . . . 9 ⊢ (𝜑 → ((𝑁C𝑐𝐾) = 0 ↔ 𝑁 ∈ (0...(𝐾 − 1)))) |
34 | 33 | biimpar 479 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ (0...(𝐾 − 1))) → (𝑁C𝑐𝐾) = 0) |
35 | 32, 34 | syldan 592 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ (0..^𝐾)) → (𝑁C𝑐𝐾) = 0) |
36 | 28, 35 | syldan 592 | . . . . . 6 ⊢ ((𝜑 ∧ (0 ≤ 𝑁 ∧ 𝑁 < 𝐾)) → (𝑁C𝑐𝐾) = 0) |
37 | 24, 36 | sylanr1 681 | . . . . 5 ⊢ ((𝜑 ∧ (𝜑 ∧ 𝑁 < 𝐾)) → (𝑁C𝑐𝐾) = 0) |
38 | 37 | anabss5 667 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 < 𝐾) → (𝑁C𝑐𝐾) = 0) |
39 | 23, 38 | syldan 592 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁C𝑐𝐾) = 0) |
40 | 1, 18 | jca 513 | . . . 4 ⊢ (𝜑 → (𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ)) |
41 | bcval3 14215 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = 0) | |
42 | 41 | 3expa 1119 | . . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = 0) |
43 | 40, 42 | sylan 581 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = 0) |
44 | 39, 43 | eqtr4d 2776 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁C𝑐𝐾) = (𝑁C𝐾)) |
45 | 8, 44 | pm2.61dan 812 | 1 ⊢ (𝜑 → (𝑁C𝑐𝐾) = (𝑁C𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ∪ cun 3912 class class class wbr 5109 ‘cfv 6500 (class class class)co 7361 0cc0 11059 1c1 11060 + caddc 11062 < clt 11197 ≤ cle 11198 − cmin 11393 / cdiv 11820 ℕ0cn0 12421 ℤcz 12507 ℤ≥cuz 12771 ...cfz 13433 ..^cfzo 13576 !cfa 14182 Ccbc 14211 FallFac cfallfac 15895 C𝑐cbcc 42708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-inf2 9585 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-sup 9386 df-oi 9454 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-n0 12422 df-z 12508 df-uz 12772 df-rp 12924 df-fz 13434 df-fzo 13577 df-seq 13916 df-exp 13977 df-fac 14183 df-bc 14212 df-hash 14240 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-clim 15379 df-prod 15797 df-fallfac 15898 df-bcc 42709 |
This theorem is referenced by: binomcxplemnn0 42721 |
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