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Theorem bcfallfac 15397

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 13000	. . . . 5 ⊢ (𝐾 ∈ (0...𝑁) → 𝑁 ∈ ℕ₀)
2	1	faccld 13643	. . . 4 ⊢ (𝐾 ∈ (0...𝑁) → (!‘𝑁) ∈ ℕ)
3	2	nncnd 11653	. . 3 ⊢ (𝐾 ∈ (0...𝑁) → (!‘𝑁) ∈ ℂ)
4		fznn0sub 12938	. . . . 5 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁 − 𝐾) ∈ ℕ₀)
5	4	faccld 13643	. . . 4 ⊢ (𝐾 ∈ (0...𝑁) → (!‘(𝑁 − 𝐾)) ∈ ℕ)
6	5	nncnd 11653	. . 3 ⊢ (𝐾 ∈ (0...𝑁) → (!‘(𝑁 − 𝐾)) ∈ ℂ)
7		elfznn0 12999	. . . . 5 ⊢ (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℕ₀)
8	7	faccld 13643	. . . 4 ⊢ (𝐾 ∈ (0...𝑁) → (!‘𝐾) ∈ ℕ)
9	8	nncnd 11653	. . 3 ⊢ (𝐾 ∈ (0...𝑁) → (!‘𝐾) ∈ ℂ)
10	5	nnne0d 11686	. . 3 ⊢ (𝐾 ∈ (0...𝑁) → (!‘(𝑁 − 𝐾)) ≠ 0)
11	8	nnne0d 11686	. . 3 ⊢ (𝐾 ∈ (0...𝑁) → (!‘𝐾) ≠ 0)
12	3, 6, 9, 10, 11	divdiv1d 11446	. 2 ⊢ (𝐾 ∈ (0...𝑁) → (((!‘𝑁) / (!‘(𝑁 − 𝐾))) / (!‘𝐾)) = ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))))
13		fallfacval4 15396	. . 3 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁 FallFac 𝐾) = ((!‘𝑁) / (!‘(𝑁 − 𝐾))))
14	13	oveq1d 7170	. 2 ⊢ (𝐾 ∈ (0...𝑁) → ((𝑁 FallFac 𝐾) / (!‘𝐾)) = (((!‘𝑁) / (!‘(𝑁 − 𝐾))) / (!‘𝐾)))
15		bcval2 13664	. 2 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))))
16	12, 14, 15	3eqtr4rd 2867	1 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((𝑁 FallFac 𝐾) / (!‘𝐾)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ‘cfv 6354 (class class class)co 7155 0cc0 10536 · cmul 10541 − cmin 10869 / cdiv 11296 ...cfz 12891 !cfa 13632 Ccbc 13661 FallFac cfallfac 15357

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-sup 8905 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-n0 11897 df-z 11981 df-uz 12243 df-rp 12389 df-fz 12892 df-fzo 13033 df-seq 13369 df-exp 13429 df-fac 13633 df-bc 13662 df-hash 13690 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-clim 14844 df-prod 15259 df-fallfac 15360

This theorem is referenced by: bccbc 40675

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