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Theorem bcfallfac 15994

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 13601	. . . . 5 ⊢ (𝐾 ∈ (0...𝑁) → 𝑁 ∈ ℕ₀)
2	1	faccld 14249	. . . 4 ⊢ (𝐾 ∈ (0...𝑁) → (!‘𝑁) ∈ ℕ)
3	2	nncnd 12232	. . 3 ⊢ (𝐾 ∈ (0...𝑁) → (!‘𝑁) ∈ ℂ)
4		fznn0sub 13539	. . . . 5 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁 − 𝐾) ∈ ℕ₀)
5	4	faccld 14249	. . . 4 ⊢ (𝐾 ∈ (0...𝑁) → (!‘(𝑁 − 𝐾)) ∈ ℕ)
6	5	nncnd 12232	. . 3 ⊢ (𝐾 ∈ (0...𝑁) → (!‘(𝑁 − 𝐾)) ∈ ℂ)
7		elfznn0 13600	. . . . 5 ⊢ (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℕ₀)
8	7	faccld 14249	. . . 4 ⊢ (𝐾 ∈ (0...𝑁) → (!‘𝐾) ∈ ℕ)
9	8	nncnd 12232	. . 3 ⊢ (𝐾 ∈ (0...𝑁) → (!‘𝐾) ∈ ℂ)
10	5	nnne0d 12266	. . 3 ⊢ (𝐾 ∈ (0...𝑁) → (!‘(𝑁 − 𝐾)) ≠ 0)
11	8	nnne0d 12266	. . 3 ⊢ (𝐾 ∈ (0...𝑁) → (!‘𝐾) ≠ 0)
12	3, 6, 9, 10, 11	divdiv1d 12025	. 2 ⊢ (𝐾 ∈ (0...𝑁) → (((!‘𝑁) / (!‘(𝑁 − 𝐾))) / (!‘𝐾)) = ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))))
13		fallfacval4 15993	. . 3 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁 FallFac 𝐾) = ((!‘𝑁) / (!‘(𝑁 − 𝐾))))
14	13	oveq1d 7420	. 2 ⊢ (𝐾 ∈ (0...𝑁) → ((𝑁 FallFac 𝐾) / (!‘𝐾)) = (((!‘𝑁) / (!‘(𝑁 − 𝐾))) / (!‘𝐾)))
15		bcval2 14270	. 2 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))))
16	12, 14, 15	3eqtr4rd 2777	1 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((𝑁 FallFac 𝐾) / (!‘𝐾)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ‘cfv 6537 (class class class)co 7405 0cc0 11112 · cmul 11117 − cmin 11448 / cdiv 11875 ...cfz 13490 !cfa 14238 Ccbc 14267 FallFac cfallfac 15954

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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12981 df-fz 13491 df-fzo 13634 df-seq 13973 df-exp 14033 df-fac 14239 df-bc 14268 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 df-prod 15856 df-fallfac 15957

This theorem is referenced by: bccbc 43680

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