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Theorem bcfallfac 16020

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 13627	. . . . 5 ⊢ (𝐾 ∈ (0...𝑁) → 𝑁 ∈ ℕ₀)
2	1	faccld 14275	. . . 4 ⊢ (𝐾 ∈ (0...𝑁) → (!‘𝑁) ∈ ℕ)
3	2	nncnd 12258	. . 3 ⊢ (𝐾 ∈ (0...𝑁) → (!‘𝑁) ∈ ℂ)
4		fznn0sub 13565	. . . . 5 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁 − 𝐾) ∈ ℕ₀)
5	4	faccld 14275	. . . 4 ⊢ (𝐾 ∈ (0...𝑁) → (!‘(𝑁 − 𝐾)) ∈ ℕ)
6	5	nncnd 12258	. . 3 ⊢ (𝐾 ∈ (0...𝑁) → (!‘(𝑁 − 𝐾)) ∈ ℂ)
7		elfznn0 13626	. . . . 5 ⊢ (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℕ₀)
8	7	faccld 14275	. . . 4 ⊢ (𝐾 ∈ (0...𝑁) → (!‘𝐾) ∈ ℕ)
9	8	nncnd 12258	. . 3 ⊢ (𝐾 ∈ (0...𝑁) → (!‘𝐾) ∈ ℂ)
10	5	nnne0d 12292	. . 3 ⊢ (𝐾 ∈ (0...𝑁) → (!‘(𝑁 − 𝐾)) ≠ 0)
11	8	nnne0d 12292	. . 3 ⊢ (𝐾 ∈ (0...𝑁) → (!‘𝐾) ≠ 0)
12	3, 6, 9, 10, 11	divdiv1d 12051	. 2 ⊢ (𝐾 ∈ (0...𝑁) → (((!‘𝑁) / (!‘(𝑁 − 𝐾))) / (!‘𝐾)) = ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))))
13		fallfacval4 16019	. . 3 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁 FallFac 𝐾) = ((!‘𝑁) / (!‘(𝑁 − 𝐾))))
14	13	oveq1d 7431	. 2 ⊢ (𝐾 ∈ (0...𝑁) → ((𝑁 FallFac 𝐾) / (!‘𝐾)) = (((!‘𝑁) / (!‘(𝑁 − 𝐾))) / (!‘𝐾)))
15		bcval2 14296	. 2 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))))
16	12, 14, 15	3eqtr4rd 2776	1 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((𝑁 FallFac 𝐾) / (!‘𝐾)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ‘cfv 6543 (class class class)co 7416 0cc0 11138 · cmul 11143 − cmin 11474 / cdiv 11901 ...cfz 13516 !cfa 14264 Ccbc 14293 FallFac cfallfac 15980

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-sup 9465 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-z 12589 df-uz 12853 df-rp 13007 df-fz 13517 df-fzo 13660 df-seq 13999 df-exp 14059 df-fac 14265 df-bc 14294 df-hash 14322 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-clim 15464 df-prod 15882 df-fallfac 15983

This theorem is referenced by: bccbc 43847

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