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Theorem bcfallfac 15754

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 13350	. . . . 5 ⊢ (𝐾 ∈ (0...𝑁) → 𝑁 ∈ ℕ₀)
2	1	faccld 13998	. . . 4 ⊢ (𝐾 ∈ (0...𝑁) → (!‘𝑁) ∈ ℕ)
3	2	nncnd 11989	. . 3 ⊢ (𝐾 ∈ (0...𝑁) → (!‘𝑁) ∈ ℂ)
4		fznn0sub 13288	. . . . 5 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁 − 𝐾) ∈ ℕ₀)
5	4	faccld 13998	. . . 4 ⊢ (𝐾 ∈ (0...𝑁) → (!‘(𝑁 − 𝐾)) ∈ ℕ)
6	5	nncnd 11989	. . 3 ⊢ (𝐾 ∈ (0...𝑁) → (!‘(𝑁 − 𝐾)) ∈ ℂ)
7		elfznn0 13349	. . . . 5 ⊢ (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℕ₀)
8	7	faccld 13998	. . . 4 ⊢ (𝐾 ∈ (0...𝑁) → (!‘𝐾) ∈ ℕ)
9	8	nncnd 11989	. . 3 ⊢ (𝐾 ∈ (0...𝑁) → (!‘𝐾) ∈ ℂ)
10	5	nnne0d 12023	. . 3 ⊢ (𝐾 ∈ (0...𝑁) → (!‘(𝑁 − 𝐾)) ≠ 0)
11	8	nnne0d 12023	. . 3 ⊢ (𝐾 ∈ (0...𝑁) → (!‘𝐾) ≠ 0)
12	3, 6, 9, 10, 11	divdiv1d 11782	. 2 ⊢ (𝐾 ∈ (0...𝑁) → (((!‘𝑁) / (!‘(𝑁 − 𝐾))) / (!‘𝐾)) = ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))))
13		fallfacval4 15753	. . 3 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁 FallFac 𝐾) = ((!‘𝑁) / (!‘(𝑁 − 𝐾))))
14	13	oveq1d 7290	. 2 ⊢ (𝐾 ∈ (0...𝑁) → ((𝑁 FallFac 𝐾) / (!‘𝐾)) = (((!‘𝑁) / (!‘(𝑁 − 𝐾))) / (!‘𝐾)))
15		bcval2 14019	. 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 2106 ‘cfv 6433 (class class class)co 7275 0cc0 10871 · cmul 10876 − cmin 11205 / cdiv 11632 ...cfz 13239 !cfa 13987 Ccbc 14016 FallFac cfallfac 15714

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-fac 13988 df-bc 14017 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-prod 15616 df-fallfac 15717

This theorem is referenced by: bccbc 41963

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