fallfacval4 - Metamath Proof Explorer

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Theorem fallfacval4 15992

Description: Represent the falling factorial via factorials when the first argument is a natural. (Contributed by Scott Fenton, 20-Mar-2018.)

**Assertion**
Ref	Expression
fallfacval4	⊢ (𝑁 ∈ (0...𝐴) → (𝐴 FallFac 𝑁) = ((!‘𝐴) / (!‘(𝐴 − 𝑁))))

Proof of Theorem fallfacval4 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzfid 13943	. . . . 5 ⊢ (𝑁 ∈ (0...𝐴) → (((𝐴 − 𝑁) + 1)...𝐴) ∈ Fin)
2		elfzelz 13506	. . . . . . 7 ⊢ (𝑘 ∈ (((𝐴 − 𝑁) + 1)...𝐴) → 𝑘 ∈ ℤ)
3	2	zcnd 12672	. . . . . 6 ⊢ (𝑘 ∈ (((𝐴 − 𝑁) + 1)...𝐴) → 𝑘 ∈ ℂ)
4	3	adantl 481	. . . . 5 ⊢ ((𝑁 ∈ (0...𝐴) ∧ 𝑘 ∈ (((𝐴 − 𝑁) + 1)...𝐴)) → 𝑘 ∈ ℂ)
5	1, 4	fprodcl 15901	. . . 4 ⊢ (𝑁 ∈ (0...𝐴) → ∏𝑘 ∈ (((𝐴 − 𝑁) + 1)...𝐴)𝑘 ∈ ℂ)
6		fzfid 13943	. . . . 5 ⊢ (𝑁 ∈ (0...𝐴) → (1...(𝐴 − 𝑁)) ∈ Fin)
7		elfznn 13535	. . . . . . 7 ⊢ (𝑘 ∈ (1...(𝐴 − 𝑁)) → 𝑘 ∈ ℕ)
8	7	adantl 481	. . . . . 6 ⊢ ((𝑁 ∈ (0...𝐴) ∧ 𝑘 ∈ (1...(𝐴 − 𝑁))) → 𝑘 ∈ ℕ)
9	8	nncnd 12233	. . . . 5 ⊢ ((𝑁 ∈ (0...𝐴) ∧ 𝑘 ∈ (1...(𝐴 − 𝑁))) → 𝑘 ∈ ℂ)
10	6, 9	fprodcl 15901	. . . 4 ⊢ (𝑁 ∈ (0...𝐴) → ∏𝑘 ∈ (1...(𝐴 − 𝑁))𝑘 ∈ ℂ)
11	8	nnne0d 12267	. . . . 5 ⊢ ((𝑁 ∈ (0...𝐴) ∧ 𝑘 ∈ (1...(𝐴 − 𝑁))) → 𝑘 ≠ 0)
12	6, 9, 11	fprodn0 15928	. . . 4 ⊢ (𝑁 ∈ (0...𝐴) → ∏𝑘 ∈ (1...(𝐴 − 𝑁))𝑘 ≠ 0)
13	5, 10, 12	divcan3d 12000	. . 3 ⊢ (𝑁 ∈ (0...𝐴) → ((∏𝑘 ∈ (1...(𝐴 − 𝑁))𝑘 · ∏𝑘 ∈ (((𝐴 − 𝑁) + 1)...𝐴)𝑘) / ∏𝑘 ∈ (1...(𝐴 − 𝑁))𝑘) = ∏𝑘 ∈ (((𝐴 − 𝑁) + 1)...𝐴)𝑘)
14		fznn0sub 13538	. . . . . . . 8 ⊢ (𝑁 ∈ (0...𝐴) → (𝐴 − 𝑁) ∈ ℕ₀)
15	14	nn0red 12538	. . . . . . 7 ⊢ (𝑁 ∈ (0...𝐴) → (𝐴 − 𝑁) ∈ ℝ)
16	15	ltp1d 12149	. . . . . 6 ⊢ (𝑁 ∈ (0...𝐴) → (𝐴 − 𝑁) < ((𝐴 − 𝑁) + 1))
17		fzdisj 13533	. . . . . 6 ⊢ ((𝐴 − 𝑁) < ((𝐴 − 𝑁) + 1) → ((1...(𝐴 − 𝑁)) ∩ (((𝐴 − 𝑁) + 1)...𝐴)) = ∅)
18	16, 17	syl 17	. . . . 5 ⊢ (𝑁 ∈ (0...𝐴) → ((1...(𝐴 − 𝑁)) ∩ (((𝐴 − 𝑁) + 1)...𝐴)) = ∅)
19		nn0p1nn 12516	. . . . . . . 8 ⊢ ((𝐴 − 𝑁) ∈ ℕ₀ → ((𝐴 − 𝑁) + 1) ∈ ℕ)
20	14, 19	syl 17	. . . . . . 7 ⊢ (𝑁 ∈ (0...𝐴) → ((𝐴 − 𝑁) + 1) ∈ ℕ)
21		nnuz 12870	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
22	20, 21	eleqtrdi 2842	. . . . . 6 ⊢ (𝑁 ∈ (0...𝐴) → ((𝐴 − 𝑁) + 1) ∈ (ℤ_≥‘1))
23	14	nn0zd 12589	. . . . . . 7 ⊢ (𝑁 ∈ (0...𝐴) → (𝐴 − 𝑁) ∈ ℤ)
24		elfzel2 13504	. . . . . . 7 ⊢ (𝑁 ∈ (0...𝐴) → 𝐴 ∈ ℤ)
25		elfzle1 13509	. . . . . . . 8 ⊢ (𝑁 ∈ (0...𝐴) → 0 ≤ 𝑁)
26	24	zred 12671	. . . . . . . . 9 ⊢ (𝑁 ∈ (0...𝐴) → 𝐴 ∈ ℝ)
27		elfzelz 13506	. . . . . . . . . 10 ⊢ (𝑁 ∈ (0...𝐴) → 𝑁 ∈ ℤ)
28	27	zred 12671	. . . . . . . . 9 ⊢ (𝑁 ∈ (0...𝐴) → 𝑁 ∈ ℝ)
29	26, 28	subge02d 11811	. . . . . . . 8 ⊢ (𝑁 ∈ (0...𝐴) → (0 ≤ 𝑁 ↔ (𝐴 − 𝑁) ≤ 𝐴))
30	25, 29	mpbid 231	. . . . . . 7 ⊢ (𝑁 ∈ (0...𝐴) → (𝐴 − 𝑁) ≤ 𝐴)
31		eluz2 12833	. . . . . . 7 ⊢ (𝐴 ∈ (ℤ_≥‘(𝐴 − 𝑁)) ↔ ((𝐴 − 𝑁) ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ (𝐴 − 𝑁) ≤ 𝐴))
32	23, 24, 30, 31	syl3anbrc 1342	. . . . . 6 ⊢ (𝑁 ∈ (0...𝐴) → 𝐴 ∈ (ℤ_≥‘(𝐴 − 𝑁)))
33		fzsplit2 13531	. . . . . 6 ⊢ ((((𝐴 − 𝑁) + 1) ∈ (ℤ_≥‘1) ∧ 𝐴 ∈ (ℤ_≥‘(𝐴 − 𝑁))) → (1...𝐴) = ((1...(𝐴 − 𝑁)) ∪ (((𝐴 − 𝑁) + 1)...𝐴)))
34	22, 32, 33	syl2anc 583	. . . . 5 ⊢ (𝑁 ∈ (0...𝐴) → (1...𝐴) = ((1...(𝐴 − 𝑁)) ∪ (((𝐴 − 𝑁) + 1)...𝐴)))
35		fzfid 13943	. . . . 5 ⊢ (𝑁 ∈ (0...𝐴) → (1...𝐴) ∈ Fin)
36		elfznn 13535	. . . . . . 7 ⊢ (𝑘 ∈ (1...𝐴) → 𝑘 ∈ ℕ)
37	36	nncnd 12233	. . . . . 6 ⊢ (𝑘 ∈ (1...𝐴) → 𝑘 ∈ ℂ)
38	37	adantl 481	. . . . 5 ⊢ ((𝑁 ∈ (0...𝐴) ∧ 𝑘 ∈ (1...𝐴)) → 𝑘 ∈ ℂ)
39	18, 34, 35, 38	fprodsplit 15915	. . . 4 ⊢ (𝑁 ∈ (0...𝐴) → ∏𝑘 ∈ (1...𝐴)𝑘 = (∏𝑘 ∈ (1...(𝐴 − 𝑁))𝑘 · ∏𝑘 ∈ (((𝐴 − 𝑁) + 1)...𝐴)𝑘))
40	39	oveq1d 7427	. . 3 ⊢ (𝑁 ∈ (0...𝐴) → (∏𝑘 ∈ (1...𝐴)𝑘 / ∏𝑘 ∈ (1...(𝐴 − 𝑁))𝑘) = ((∏𝑘 ∈ (1...(𝐴 − 𝑁))𝑘 · ∏𝑘 ∈ (((𝐴 − 𝑁) + 1)...𝐴)𝑘) / ∏𝑘 ∈ (1...(𝐴 − 𝑁))𝑘))
41	24	zcnd 12672	. . . . . 6 ⊢ (𝑁 ∈ (0...𝐴) → 𝐴 ∈ ℂ)
42	27	zcnd 12672	. . . . . 6 ⊢ (𝑁 ∈ (0...𝐴) → 𝑁 ∈ ℂ)
43		1cnd 11214	. . . . . 6 ⊢ (𝑁 ∈ (0...𝐴) → 1 ∈ ℂ)
44	41, 42, 43	subsubd 11604	. . . . 5 ⊢ (𝑁 ∈ (0...𝐴) → (𝐴 − (𝑁 − 1)) = ((𝐴 − 𝑁) + 1))
45	44	oveq1d 7427	. . . 4 ⊢ (𝑁 ∈ (0...𝐴) → ((𝐴 − (𝑁 − 1))...𝐴) = (((𝐴 − 𝑁) + 1)...𝐴))
46	45	prodeq1d 15870	. . 3 ⊢ (𝑁 ∈ (0...𝐴) → ∏𝑘 ∈ ((𝐴 − (𝑁 − 1))...𝐴)𝑘 = ∏𝑘 ∈ (((𝐴 − 𝑁) + 1)...𝐴)𝑘)
47	13, 40, 46	3eqtr4rd 2782	. 2 ⊢ (𝑁 ∈ (0...𝐴) → ∏𝑘 ∈ ((𝐴 − (𝑁 − 1))...𝐴)𝑘 = (∏𝑘 ∈ (1...𝐴)𝑘 / ∏𝑘 ∈ (1...(𝐴 − 𝑁))𝑘))
48		fallfacval3 15961	. 2 ⊢ (𝑁 ∈ (0...𝐴) → (𝐴 FallFac 𝑁) = ∏𝑘 ∈ ((𝐴 − (𝑁 − 1))...𝐴)𝑘)
49		elfz3nn0 13600	. . . 4 ⊢ (𝑁 ∈ (0...𝐴) → 𝐴 ∈ ℕ₀)
50		fprodfac 15922	. . . 4 ⊢ (𝐴 ∈ ℕ₀ → (!‘𝐴) = ∏𝑘 ∈ (1...𝐴)𝑘)
51	49, 50	syl 17	. . 3 ⊢ (𝑁 ∈ (0...𝐴) → (!‘𝐴) = ∏𝑘 ∈ (1...𝐴)𝑘)
52		fprodfac 15922	. . . 4 ⊢ ((𝐴 − 𝑁) ∈ ℕ₀ → (!‘(𝐴 − 𝑁)) = ∏𝑘 ∈ (1...(𝐴 − 𝑁))𝑘)
53	14, 52	syl 17	. . 3 ⊢ (𝑁 ∈ (0...𝐴) → (!‘(𝐴 − 𝑁)) = ∏𝑘 ∈ (1...(𝐴 − 𝑁))𝑘)
54	51, 53	oveq12d 7430	. 2 ⊢ (𝑁 ∈ (0...𝐴) → ((!‘𝐴) / (!‘(𝐴 − 𝑁))) = (∏𝑘 ∈ (1...𝐴)𝑘 / ∏𝑘 ∈ (1...(𝐴 − 𝑁))𝑘))
55	47, 48, 54	3eqtr4d 2781	1 ⊢ (𝑁 ∈ (0...𝐴) → (𝐴 FallFac 𝑁) = ((!‘𝐴) / (!‘(𝐴 − 𝑁))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∪ cun 3946 ∩ cin 3947 ∅c0 4322 class class class wbr 5148 ‘cfv 6543 (class class class)co 7412 ℂcc 11112 0cc0 11114 1c1 11115 + caddc 11117 · cmul 11119 < clt 11253 ≤ cle 11254 − cmin 11449 / cdiv 11876 ℕcn 12217 ℕ₀cn0 12477 ℤcz 12563 ℤ_≥cuz 12827 ...cfz 13489 !cfa 14238 ∏cprod 15854 FallFac cfallfac 15953

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9640 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 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 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9441 df-oi 9509 df-card 9938 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-n0 12478 df-z 12564 df-uz 12828 df-rp 12980 df-fz 13490 df-fzo 13633 df-seq 13972 df-exp 14033 df-fac 14239 df-hash 14296 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-clim 15437 df-prod 15855 df-fallfac 15956

This theorem is referenced by: bcfallfac 15993 fallfacfac 15994

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