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Theorem bcfallfac 15682

Description: Binomial coefficient in terms of falling factorials. (Contributed by Scott Fenton, 20-Mar-2018.)

**Assertion**
Ref	Expression
bcfallfac	⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((𝑁 FallFac 𝐾) / (!‘𝐾)))

**Proof of Theorem bcfallfac**
Step	Hyp	Ref	Expression
1		elfz3nn0 13279	. . . . 5 ⊢ (𝐾 ∈ (0...𝑁) → 𝑁 ∈ ℕ₀)
2	1	faccld 13926	. . . 4 ⊢ (𝐾 ∈ (0...𝑁) → (!‘𝑁) ∈ ℕ)
3	2	nncnd 11919	. . 3 ⊢ (𝐾 ∈ (0...𝑁) → (!‘𝑁) ∈ ℂ)
4		fznn0sub 13217	. . . . 5 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁 − 𝐾) ∈ ℕ₀)
5	4	faccld 13926	. . . 4 ⊢ (𝐾 ∈ (0...𝑁) → (!‘(𝑁 − 𝐾)) ∈ ℕ)
6	5	nncnd 11919	. . 3 ⊢ (𝐾 ∈ (0...𝑁) → (!‘(𝑁 − 𝐾)) ∈ ℂ)
7		elfznn0 13278	. . . . 5 ⊢ (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℕ₀)
8	7	faccld 13926	. . . 4 ⊢ (𝐾 ∈ (0...𝑁) → (!‘𝐾) ∈ ℕ)
9	8	nncnd 11919	. . 3 ⊢ (𝐾 ∈ (0...𝑁) → (!‘𝐾) ∈ ℂ)
10	5	nnne0d 11953	. . 3 ⊢ (𝐾 ∈ (0...𝑁) → (!‘(𝑁 − 𝐾)) ≠ 0)
11	8	nnne0d 11953	. . 3 ⊢ (𝐾 ∈ (0...𝑁) → (!‘𝐾) ≠ 0)
12	3, 6, 9, 10, 11	divdiv1d 11712	. 2 ⊢ (𝐾 ∈ (0...𝑁) → (((!‘𝑁) / (!‘(𝑁 − 𝐾))) / (!‘𝐾)) = ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))))
13		fallfacval4 15681	. . 3 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁 FallFac 𝐾) = ((!‘𝑁) / (!‘(𝑁 − 𝐾))))
14	13	oveq1d 7270	. 2 ⊢ (𝐾 ∈ (0...𝑁) → ((𝑁 FallFac 𝐾) / (!‘𝐾)) = (((!‘𝑁) / (!‘(𝑁 − 𝐾))) / (!‘𝐾)))
15		bcval2 13947	. 2 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))))
16	12, 14, 15	3eqtr4rd 2789	1 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((𝑁 FallFac 𝐾) / (!‘𝐾)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2108 ‘cfv 6418 (class class class)co 7255 0cc0 10802 · cmul 10807 − cmin 11135 / cdiv 11562 ...cfz 13168 !cfa 13915 Ccbc 13944 FallFac cfallfac 15642

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-z 12250 df-uz 12512 df-rp 12660 df-fz 13169 df-fzo 13312 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-prod 15544 df-fallfac 15645

This theorem is referenced by: bccbc 41852

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