fallfacfwd - Metamath Proof Explorer

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Theorem fallfacfwd 16049

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 11352	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴 + 1) ∈ ℂ)
2		nnnn0 12485	. . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
3		fallfacval 16022	. . . . 5 ⊢ (((𝐴 + 1) ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 + 1) FallFac 𝑁) = ∏𝑘 ∈ (0...(𝑁 − 1))((𝐴 + 1) − 𝑘))
4	1, 2, 3	syl2an 605	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴 + 1) FallFac 𝑁) = ∏𝑘 ∈ (0...(𝑁 − 1))((𝐴 + 1) − 𝑘))
5		0p1e1 12335	. . . . . . . . 9 ⊢ (0 + 1) = 1
6	5	oveq1i 7402	. . . . . . . 8 ⊢ ((0 + 1)...(𝑁 − 1)) = (1...(𝑁 − 1))
7	6	prodeq1i 15929	. . . . . . 7 ⊢ ∏𝑘 ∈ ((0 + 1)...(𝑁 − 1))(𝐴 − (𝑘 − 1)) = ∏𝑘 ∈ (1...(𝑁 − 1))(𝐴 − (𝑘 − 1))
8	7	oveq2i 7403	. . . . . 6 ⊢ ((𝐴 − -1) · ∏𝑘 ∈ ((0 + 1)...(𝑁 − 1))(𝐴 − (𝑘 − 1))) = ((𝐴 − -1) · ∏𝑘 ∈ (1...(𝑁 − 1))(𝐴 − (𝑘 − 1)))
9		nnm1nn0 12519	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
10	9	adantl 485	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝑁 − 1) ∈ ℕ₀)
11		nn0uz 12874	. . . . . . . 8 ⊢ ℕ₀ = (ℤ_≥‘0)
12	10, 11	eleqtrdi 2871	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝑁 − 1) ∈ (ℤ_≥‘0))
13		simpll 776	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝐴 ∈ ℂ)
14		elfzelz 13526	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℤ)
15	14	adantl 485	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ ℤ)
16		peano2zm 12611	. . . . . . . . . 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 12675	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑘 − 1) ∈ ℂ)
19	13, 18	subcld 11539	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝐴 − (𝑘 − 1)) ∈ ℂ)
20		oveq1 7399	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝑘 − 1) = (0 − 1))
21		df-neg 11414	. . . . . . . . 9 ⊢ -1 = (0 − 1)
22	20, 21	eqtr4di 2814	. . . . . . . 8 ⊢ (𝑘 = 0 → (𝑘 − 1) = -1)
23	22	oveq2d 7408	. . . . . . 7 ⊢ (𝑘 = 0 → (𝐴 − (𝑘 − 1)) = (𝐴 − -1))
24	12, 19, 23	fprod1p 15981	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴 − (𝑘 − 1)) = ((𝐴 − -1) · ∏𝑘 ∈ ((0 + 1)...(𝑁 − 1))(𝐴 − (𝑘 − 1))))
25		fallfacval2 16024	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (𝑁 − 1) ∈ ℕ₀) → (𝐴 FallFac (𝑁 − 1)) = ∏𝑘 ∈ (1...(𝑁 − 1))(𝐴 − (𝑘 − 1)))
26	9, 25	sylan2 602	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 FallFac (𝑁 − 1)) = ∏𝑘 ∈ (1...(𝑁 − 1))(𝐴 − (𝑘 − 1)))
27	26	oveq2d 7408	. . . . . 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 2822	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴 − (𝑘 − 1)) = ((𝐴 − -1) · (𝐴 FallFac (𝑁 − 1))))
29		elfznn0 13622	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℕ₀)
30	29	adantl 485	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ ℕ₀)
31	30	nn0cnd 12541	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ ℂ)
32		1cnd 11172	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 1 ∈ ℂ)
33	13, 31, 32	subsub3d 11569	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝐴 − (𝑘 − 1)) = ((𝐴 + 1) − 𝑘))
34	33	prodeq2dv 15935	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴 − (𝑘 − 1)) = ∏𝑘 ∈ (0...(𝑁 − 1))((𝐴 + 1) − 𝑘))
35		simpl 486	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → 𝐴 ∈ ℂ)
36		1cnd 11172	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → 1 ∈ ℂ)
37	35, 36	subnegd 11546	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 − -1) = (𝐴 + 1))
38	37	oveq1d 7407	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴 − -1) · (𝐴 FallFac (𝑁 − 1))) = ((𝐴 + 1) · (𝐴 FallFac (𝑁 − 1))))
39	28, 34, 38	3eqtr3d 2804	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ∏𝑘 ∈ (0...(𝑁 − 1))((𝐴 + 1) − 𝑘) = ((𝐴 + 1) · (𝐴 FallFac (𝑁 − 1))))
40	4, 39	eqtrd 2796	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴 + 1) FallFac 𝑁) = ((𝐴 + 1) · (𝐴 FallFac (𝑁 − 1))))
41		simpr 488	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ)
42	41	nncnd 12223	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℂ)
43	42, 36	npcand 11543	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝑁 − 1) + 1) = 𝑁)
44	43	oveq2d 7408	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 FallFac ((𝑁 − 1) + 1)) = (𝐴 FallFac 𝑁))
45		fallfacp1 16043	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝑁 − 1) ∈ ℕ₀) → (𝐴 FallFac ((𝑁 − 1) + 1)) = ((𝐴 FallFac (𝑁 − 1)) · (𝐴 − (𝑁 − 1))))
46	9, 45	sylan2 602	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 FallFac ((𝑁 − 1) + 1)) = ((𝐴 FallFac (𝑁 − 1)) · (𝐴 − (𝑁 − 1))))
47	44, 46	eqtr3d 2798	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 FallFac 𝑁) = ((𝐴 FallFac (𝑁 − 1)) · (𝐴 − (𝑁 − 1))))
48	40, 47	oveq12d 7410	. 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 16029	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝑁 − 1) ∈ ℕ₀) → (𝐴 FallFac (𝑁 − 1)) ∈ ℂ)
50	9, 49	sylan2 602	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 FallFac (𝑁 − 1)) ∈ ℂ)
51	10	nn0cnd 12541	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝑁 − 1) ∈ ℂ)
52	35, 51	subcld 11539	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 − (𝑁 − 1)) ∈ ℂ)
53	50, 52	mulcomd 11200	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴 FallFac (𝑁 − 1)) · (𝐴 − (𝑁 − 1))) = ((𝐴 − (𝑁 − 1)) · (𝐴 FallFac (𝑁 − 1))))
54	53	oveq2d 7408	. . 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 484	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 + 1) ∈ ℂ)
56	55, 52, 50	subdird 11641	. . 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 11578	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴 + 1) − (𝐴 − (𝑁 − 1))) = (1 + (𝑁 − 1)))
58	36, 42	pncan3d 11542	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (1 + (𝑁 − 1)) = 𝑁)
59	57, 58	eqtrd 2796	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴 + 1) − (𝐴 − (𝑁 − 1))) = 𝑁)
60	59	oveq1d 7407	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (((𝐴 + 1) − (𝐴 − (𝑁 − 1))) · (𝐴 FallFac (𝑁 − 1))) = (𝑁 · (𝐴 FallFac (𝑁 − 1))))
61	54, 56, 60	3eqtr2d 2802	. 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 2796	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 399 = wceq 1559 ∈ wcel 2141 ‘cfv 6517 (class class class)co 7392 ℂcc 11068 0cc0 11070 1c1 11071 + caddc 11073 · cmul 11075 − cmin 11411 -cneg 11412 ℕcn 12207 ℕ₀cn0 12478 ℤcz 12565 ℤ_≥cuz 12836 ...cfz 13509 ∏cprod 15916 FallFac cfallfac 16017

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-inf2 9593 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147 ax-pre-sup 11148

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-int 4905 df-iun 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-se 5599 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-isom 6526 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-om 7843 df-1st 7966 df-2nd 7967 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-er 8673 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-sup 9385 df-oi 9455 df-card 9894 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12208 df-2 12277 df-3 12278 df-n0 12479 df-z 12566 df-uz 12837 df-rp 12991 df-fz 13510 df-fzo 13657 df-seq 14012 df-exp 14072 df-hash 14341 df-cj 15109 df-re 15110 df-im 15111 df-sqrt 15245 df-abs 15246 df-clim 15498 df-prod 15917 df-fallfac 16020

This theorem is referenced by: (None)

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