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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 12050 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
3 | bccbc.k | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℕ0) | |
4 | 2, 3 | bccval 41534 | . . . 4 ⊢ (𝜑 → (𝑁C𝑐𝐾) = ((𝑁 FallFac 𝐾) / (!‘𝐾))) |
5 | 4 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...𝑁)) → (𝑁C𝑐𝐾) = ((𝑁 FallFac 𝐾) / (!‘𝐾))) |
6 | bcfallfac 15502 | . . . 4 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((𝑁 FallFac 𝐾) / (!‘𝐾))) | |
7 | 6 | adantl 485 | . . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = ((𝑁 FallFac 𝐾) / (!‘𝐾))) |
8 | 5, 7 | eqtr4d 2777 | . 2 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...𝑁)) → (𝑁C𝑐𝐾) = (𝑁C𝐾)) |
9 | nn0split 13125 | . . . . . . . . 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 4049 | . . . . . . 7 ⊢ (𝐾 ∈ ((0...𝑁) ∪ (ℤ≥‘(𝑁 + 1))) ↔ (𝐾 ∈ (0...𝑁) ∨ 𝐾 ∈ (ℤ≥‘(𝑁 + 1)))) | |
13 | 11, 12 | sylib 221 | . . . . . 6 ⊢ (𝜑 → (𝐾 ∈ (0...𝑁) ∨ 𝐾 ∈ (ℤ≥‘(𝑁 + 1)))) |
14 | 13 | orcanai 1002 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ (ℤ≥‘(𝑁 + 1))) |
15 | eluzle 12349 | . . . . . . 7 ⊢ (𝐾 ∈ (ℤ≥‘(𝑁 + 1)) → (𝑁 + 1) ≤ 𝐾) | |
16 | 15 | adantl 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑁 + 1))) → (𝑁 + 1) ≤ 𝐾) |
17 | 1 | nn0zd 12178 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
18 | 3 | nn0zd 12178 | . . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℤ) |
19 | zltp1le 12125 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑁 < 𝐾 ↔ (𝑁 + 1) ≤ 𝐾)) | |
20 | 17, 18, 19 | syl2anc 587 | . . . . . . 7 ⊢ (𝜑 → (𝑁 < 𝐾 ↔ (𝑁 + 1) ≤ 𝐾)) |
21 | 20 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑁 + 1))) → (𝑁 < 𝐾 ↔ (𝑁 + 1) ≤ 𝐾)) |
22 | 16, 21 | mpbird 260 | . . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑁 < 𝐾) |
23 | 14, 22 | syldan 594 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...𝑁)) → 𝑁 < 𝐾) |
24 | 1 | nn0ge0d 12051 | . . . . . 6 ⊢ (𝜑 → 0 ≤ 𝑁) |
25 | 0zd 12086 | . . . . . . . . 9 ⊢ (𝜑 → 0 ∈ ℤ) | |
26 | elfzo 13143 | . . . . . . . . 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 481 | . . . . . . 7 ⊢ ((𝜑 ∧ (0 ≤ 𝑁 ∧ 𝑁 < 𝐾)) → 𝑁 ∈ (0..^𝐾)) |
29 | fzoval 13142 | . . . . . . . . . . 11 ⊢ (𝐾 ∈ ℤ → (0..^𝐾) = (0...(𝐾 − 1))) | |
30 | 18, 29 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → (0..^𝐾) = (0...(𝐾 − 1))) |
31 | 30 | eleq2d 2819 | . . . . . . . . 9 ⊢ (𝜑 → (𝑁 ∈ (0..^𝐾) ↔ 𝑁 ∈ (0...(𝐾 − 1)))) |
32 | 31 | biimpa 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ (0..^𝐾)) → 𝑁 ∈ (0...(𝐾 − 1))) |
33 | 2, 3 | bcc0 41536 | . . . . . . . . 9 ⊢ (𝜑 → ((𝑁C𝑐𝐾) = 0 ↔ 𝑁 ∈ (0...(𝐾 − 1)))) |
34 | 33 | biimpar 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ (0...(𝐾 − 1))) → (𝑁C𝑐𝐾) = 0) |
35 | 32, 34 | syldan 594 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ (0..^𝐾)) → (𝑁C𝑐𝐾) = 0) |
36 | 28, 35 | syldan 594 | . . . . . 6 ⊢ ((𝜑 ∧ (0 ≤ 𝑁 ∧ 𝑁 < 𝐾)) → (𝑁C𝑐𝐾) = 0) |
37 | 24, 36 | sylanr1 682 | . . . . 5 ⊢ ((𝜑 ∧ (𝜑 ∧ 𝑁 < 𝐾)) → (𝑁C𝑐𝐾) = 0) |
38 | 37 | anabss5 668 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 < 𝐾) → (𝑁C𝑐𝐾) = 0) |
39 | 23, 38 | syldan 594 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁C𝑐𝐾) = 0) |
40 | 1, 18 | jca 515 | . . . 4 ⊢ (𝜑 → (𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ)) |
41 | bcval3 13770 | . . . . 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 583 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = 0) |
44 | 39, 43 | eqtr4d 2777 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁C𝑐𝐾) = (𝑁C𝐾)) |
45 | 8, 44 | pm2.61dan 813 | 1 ⊢ (𝜑 → (𝑁C𝑐𝐾) = (𝑁C𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 846 = wceq 1542 ∈ wcel 2114 ∪ cun 3851 class class class wbr 5040 ‘cfv 6349 (class class class)co 7182 0cc0 10627 1c1 10628 + caddc 10630 < clt 10765 ≤ cle 10766 − cmin 10960 / cdiv 11387 ℕ0cn0 11988 ℤcz 12074 ℤ≥cuz 12336 ...cfz 12993 ..^cfzo 13136 !cfa 13737 Ccbc 13766 FallFac cfallfac 15462 C𝑐cbcc 41532 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-inf2 9189 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 ax-pre-sup 10705 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-se 5494 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-om 7612 df-1st 7726 df-2nd 7727 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-1o 8143 df-er 8332 df-en 8568 df-dom 8569 df-sdom 8570 df-fin 8571 df-sup 8991 df-oi 9059 df-card 9453 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-div 11388 df-nn 11729 df-2 11791 df-3 11792 df-n0 11989 df-z 12075 df-uz 12337 df-rp 12485 df-fz 12994 df-fzo 13137 df-seq 13473 df-exp 13534 df-fac 13738 df-bc 13767 df-hash 13795 df-cj 14560 df-re 14561 df-im 14562 df-sqrt 14696 df-abs 14697 df-clim 14947 df-prod 15364 df-fallfac 15465 df-bcc 41533 |
This theorem is referenced by: binomcxplemnn0 41545 |
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