fallfacfwd - Metamath Proof Explorer

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Theorem fallfacfwd 16002

Description: The forward difference of a falling factorial. (Contributed by Scott Fenton, 21-Jan-2018.)

**Assertion**
Ref	Expression
fallfacfwd	⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (((𝐴 + 1) FallFac 𝑁) − (𝐴 FallFac 𝑁)) = (𝑁 · (𝐴 FallFac (𝑁 − 1))))

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

Step	Hyp	Ref	Expression
1		peano2cn 11346	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴 + 1) ∈ ℂ)
2		nnnn0 12449	. . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
3		fallfacval 15975	. . . . 5 ⊢ (((𝐴 + 1) ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 + 1) FallFac 𝑁) = ∏𝑘 ∈ (0...(𝑁 − 1))((𝐴 + 1) − 𝑘))
4	1, 2, 3	syl2an 596	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴 + 1) FallFac 𝑁) = ∏𝑘 ∈ (0...(𝑁 − 1))((𝐴 + 1) − 𝑘))
5		0p1e1 12303	. . . . . . . . 9 ⊢ (0 + 1) = 1
6	5	oveq1i 7397	. . . . . . . 8 ⊢ ((0 + 1)...(𝑁 − 1)) = (1...(𝑁 − 1))
7	6	prodeq1i 15882	. . . . . . 7 ⊢ ∏𝑘 ∈ ((0 + 1)...(𝑁 − 1))(𝐴 − (𝑘 − 1)) = ∏𝑘 ∈ (1...(𝑁 − 1))(𝐴 − (𝑘 − 1))
8	7	oveq2i 7398	. . . . . 6 ⊢ ((𝐴 − -1) · ∏𝑘 ∈ ((0 + 1)...(𝑁 − 1))(𝐴 − (𝑘 − 1))) = ((𝐴 − -1) · ∏𝑘 ∈ (1...(𝑁 − 1))(𝐴 − (𝑘 − 1)))
9		nnm1nn0 12483	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
10	9	adantl 481	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝑁 − 1) ∈ ℕ₀)
11		nn0uz 12835	. . . . . . . 8 ⊢ ℕ₀ = (ℤ_≥‘0)
12	10, 11	eleqtrdi 2838	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝑁 − 1) ∈ (ℤ_≥‘0))
13		simpll 766	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝐴 ∈ ℂ)
14		elfzelz 13485	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℤ)
15	14	adantl 481	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ ℤ)
16		peano2zm 12576	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℤ → (𝑘 − 1) ∈ ℤ)
17	15, 16	syl 17	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑘 − 1) ∈ ℤ)
18	17	zcnd 12639	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑘 − 1) ∈ ℂ)
19	13, 18	subcld 11533	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝐴 − (𝑘 − 1)) ∈ ℂ)
20		oveq1 7394	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝑘 − 1) = (0 − 1))
21		df-neg 11408	. . . . . . . . 9 ⊢ -1 = (0 − 1)
22	20, 21	eqtr4di 2782	. . . . . . . 8 ⊢ (𝑘 = 0 → (𝑘 − 1) = -1)
23	22	oveq2d 7403	. . . . . . 7 ⊢ (𝑘 = 0 → (𝐴 − (𝑘 − 1)) = (𝐴 − -1))
24	12, 19, 23	fprod1p 15934	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴 − (𝑘 − 1)) = ((𝐴 − -1) · ∏𝑘 ∈ ((0 + 1)...(𝑁 − 1))(𝐴 − (𝑘 − 1))))
25		fallfacval2 15977	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (𝑁 − 1) ∈ ℕ₀) → (𝐴 FallFac (𝑁 − 1)) = ∏𝑘 ∈ (1...(𝑁 − 1))(𝐴 − (𝑘 − 1)))
26	9, 25	sylan2 593	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 FallFac (𝑁 − 1)) = ∏𝑘 ∈ (1...(𝑁 − 1))(𝐴 − (𝑘 − 1)))
27	26	oveq2d 7403	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴 − -1) · (𝐴 FallFac (𝑁 − 1))) = ((𝐴 − -1) · ∏𝑘 ∈ (1...(𝑁 − 1))(𝐴 − (𝑘 − 1))))
28	8, 24, 27	3eqtr4a 2790	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴 − (𝑘 − 1)) = ((𝐴 − -1) · (𝐴 FallFac (𝑁 − 1))))
29		elfznn0 13581	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℕ₀)
30	29	adantl 481	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ ℕ₀)
31	30	nn0cnd 12505	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ ℂ)
32		1cnd 11169	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 1 ∈ ℂ)
33	13, 31, 32	subsub3d 11563	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝐴 − (𝑘 − 1)) = ((𝐴 + 1) − 𝑘))
34	33	prodeq2dv 15888	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴 − (𝑘 − 1)) = ∏𝑘 ∈ (0...(𝑁 − 1))((𝐴 + 1) − 𝑘))
35		simpl 482	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → 𝐴 ∈ ℂ)
36		1cnd 11169	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → 1 ∈ ℂ)
37	35, 36	subnegd 11540	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 − -1) = (𝐴 + 1))
38	37	oveq1d 7402	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴 − -1) · (𝐴 FallFac (𝑁 − 1))) = ((𝐴 + 1) · (𝐴 FallFac (𝑁 − 1))))
39	28, 34, 38	3eqtr3d 2772	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ∏𝑘 ∈ (0...(𝑁 − 1))((𝐴 + 1) − 𝑘) = ((𝐴 + 1) · (𝐴 FallFac (𝑁 − 1))))
40	4, 39	eqtrd 2764	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴 + 1) FallFac 𝑁) = ((𝐴 + 1) · (𝐴 FallFac (𝑁 − 1))))
41		simpr 484	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ)
42	41	nncnd 12202	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℂ)
43	42, 36	npcand 11537	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝑁 − 1) + 1) = 𝑁)
44	43	oveq2d 7403	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 FallFac ((𝑁 − 1) + 1)) = (𝐴 FallFac 𝑁))
45		fallfacp1 15996	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝑁 − 1) ∈ ℕ₀) → (𝐴 FallFac ((𝑁 − 1) + 1)) = ((𝐴 FallFac (𝑁 − 1)) · (𝐴 − (𝑁 − 1))))
46	9, 45	sylan2 593	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 FallFac ((𝑁 − 1) + 1)) = ((𝐴 FallFac (𝑁 − 1)) · (𝐴 − (𝑁 − 1))))
47	44, 46	eqtr3d 2766	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 FallFac 𝑁) = ((𝐴 FallFac (𝑁 − 1)) · (𝐴 − (𝑁 − 1))))
48	40, 47	oveq12d 7405	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (((𝐴 + 1) FallFac 𝑁) − (𝐴 FallFac 𝑁)) = (((𝐴 + 1) · (𝐴 FallFac (𝑁 − 1))) − ((𝐴 FallFac (𝑁 − 1)) · (𝐴 − (𝑁 − 1)))))
49		fallfaccl 15982	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝑁 − 1) ∈ ℕ₀) → (𝐴 FallFac (𝑁 − 1)) ∈ ℂ)
50	9, 49	sylan2 593	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 FallFac (𝑁 − 1)) ∈ ℂ)
51	10	nn0cnd 12505	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝑁 − 1) ∈ ℂ)
52	35, 51	subcld 11533	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 − (𝑁 − 1)) ∈ ℂ)
53	50, 52	mulcomd 11195	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴 FallFac (𝑁 − 1)) · (𝐴 − (𝑁 − 1))) = ((𝐴 − (𝑁 − 1)) · (𝐴 FallFac (𝑁 − 1))))
54	53	oveq2d 7403	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (((𝐴 + 1) · (𝐴 FallFac (𝑁 − 1))) − ((𝐴 FallFac (𝑁 − 1)) · (𝐴 − (𝑁 − 1)))) = (((𝐴 + 1) · (𝐴 FallFac (𝑁 − 1))) − ((𝐴 − (𝑁 − 1)) · (𝐴 FallFac (𝑁 − 1)))))
55	1	adantr 480	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 + 1) ∈ ℂ)
56	55, 52, 50	subdird 11635	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (((𝐴 + 1) − (𝐴 − (𝑁 − 1))) · (𝐴 FallFac (𝑁 − 1))) = (((𝐴 + 1) · (𝐴 FallFac (𝑁 − 1))) − ((𝐴 − (𝑁 − 1)) · (𝐴 FallFac (𝑁 − 1)))))
57	35, 36, 51	pnncand 11572	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴 + 1) − (𝐴 − (𝑁 − 1))) = (1 + (𝑁 − 1)))
58	36, 42	pncan3d 11536	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (1 + (𝑁 − 1)) = 𝑁)
59	57, 58	eqtrd 2764	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴 + 1) − (𝐴 − (𝑁 − 1))) = 𝑁)
60	59	oveq1d 7402	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (((𝐴 + 1) − (𝐴 − (𝑁 − 1))) · (𝐴 FallFac (𝑁 − 1))) = (𝑁 · (𝐴 FallFac (𝑁 − 1))))
61	54, 56, 60	3eqtr2d 2770	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (((𝐴 + 1) · (𝐴 FallFac (𝑁 − 1))) − ((𝐴 FallFac (𝑁 − 1)) · (𝐴 − (𝑁 − 1)))) = (𝑁 · (𝐴 FallFac (𝑁 − 1))))
62	48, 61	eqtrd 2764	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (((𝐴 + 1) FallFac 𝑁) − (𝐴 FallFac 𝑁)) = (𝑁 · (𝐴 FallFac (𝑁 − 1))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ‘cfv 6511 (class class class)co 7387 ℂcc 11066 0cc0 11068 1c1 11069 + caddc 11071 · cmul 11073 − cmin 11405 -cneg 11406 ℕcn 12186 ℕ₀cn0 12442 ℤcz 12529 ℤ_≥cuz 12793 ...cfz 13468 ∏cprod 15869 FallFac cfallfac 15970

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-sup 9393 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-n0 12443 df-z 12530 df-uz 12794 df-rp 12952 df-fz 13469 df-fzo 13616 df-seq 13967 df-exp 14027 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-clim 15454 df-prod 15870 df-fallfac 15973

This theorem is referenced by: (None)

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