fallfacfwd - Metamath Proof Explorer

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Theorem fallfacfwd 16091

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 11383	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴 + 1) ∈ ℂ)
2		nnnn0 12512	. . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
3		fallfacval 16065	. . . . 5 ⊢ (((𝐴 + 1) ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 + 1) FallFac 𝑁) = ∏𝑘 ∈ (0...(𝑁 − 1))((𝐴 + 1) − 𝑘))
4	1, 2, 3	syl2an 607	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴 + 1) FallFac 𝑁) = ∏𝑘 ∈ (0...(𝑁 − 1))((𝐴 + 1) − 𝑘))
5		0p1e1 12362	. . . . . . . . 9 ⊢ (0 + 1) = 1
6	5	oveq1i 7422	. . . . . . . 8 ⊢ ((0 + 1)...(𝑁 − 1)) = (1...(𝑁 − 1))
7	6	prodeq1i 15972	. . . . . . 7 ⊢ ∏𝑘 ∈ ((0 + 1)...(𝑁 − 1))(𝐴 − (𝑘 − 1)) = ∏𝑘 ∈ (1...(𝑁 − 1))(𝐴 − (𝑘 − 1))
8	7	oveq2i 7423	. . . . . 6 ⊢ ((𝐴 − -1) · ∏𝑘 ∈ ((0 + 1)...(𝑁 − 1))(𝐴 − (𝑘 − 1))) = ((𝐴 − -1) · ∏𝑘 ∈ (1...(𝑁 − 1))(𝐴 − (𝑘 − 1)))
9		nnm1nn0 12546	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
10	9	adantl 486	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝑁 − 1) ∈ ℕ₀)
11		nn0uz 12901	. . . . . . . 8 ⊢ ℕ₀ = (ℤ_≥‘0)
12	10, 11	eleqtrdi 2873	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝑁 − 1) ∈ (ℤ_≥‘0))
13		simpll 778	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝐴 ∈ ℂ)
14		elfzelz 13553	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℤ)
15	14	adantl 486	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ ℤ)
16		peano2zm 12638	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℤ → (𝑘 − 1) ∈ ℤ)
17	15, 16	syl 18	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑘 − 1) ∈ ℤ)
18	17	zcnd 12702	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑘 − 1) ∈ ℂ)
19	13, 18	subcld 11570	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝐴 − (𝑘 − 1)) ∈ ℂ)
20		oveq1 7419	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝑘 − 1) = (0 − 1))
21		df-neg 11445	. . . . . . . . 9 ⊢ -1 = (0 − 1)
22	20, 21	eqtr4di 2816	. . . . . . . 8 ⊢ (𝑘 = 0 → (𝑘 − 1) = -1)
23	22	oveq2d 7428	. . . . . . 7 ⊢ (𝑘 = 0 → (𝐴 − (𝑘 − 1)) = (𝐴 − -1))
24	12, 19, 23	fprod1p 16024	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴 − (𝑘 − 1)) = ((𝐴 − -1) · ∏𝑘 ∈ ((0 + 1)...(𝑁 − 1))(𝐴 − (𝑘 − 1))))
25		fallfacval2 16067	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (𝑁 − 1) ∈ ℕ₀) → (𝐴 FallFac (𝑁 − 1)) = ∏𝑘 ∈ (1...(𝑁 − 1))(𝐴 − (𝑘 − 1)))
26	9, 25	sylan2 604	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 FallFac (𝑁 − 1)) = ∏𝑘 ∈ (1...(𝑁 − 1))(𝐴 − (𝑘 − 1)))
27	26	oveq2d 7428	. . . . . 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 2824	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴 − (𝑘 − 1)) = ((𝐴 − -1) · (𝐴 FallFac (𝑁 − 1))))
29		elfznn0 13650	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℕ₀)
30	29	adantl 486	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ ℕ₀)
31	30	nn0cnd 12568	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ ℂ)
32		1cnd 11203	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 1 ∈ ℂ)
33	13, 31, 32	subsub3d 11600	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝐴 − (𝑘 − 1)) = ((𝐴 + 1) − 𝑘))
34	33	prodeq2dv 15978	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴 − (𝑘 − 1)) = ∏𝑘 ∈ (0...(𝑁 − 1))((𝐴 + 1) − 𝑘))
35		simpl 487	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → 𝐴 ∈ ℂ)
36		1cnd 11203	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → 1 ∈ ℂ)
37	35, 36	subnegd 11577	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 − -1) = (𝐴 + 1))
38	37	oveq1d 7427	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴 − -1) · (𝐴 FallFac (𝑁 − 1))) = ((𝐴 + 1) · (𝐴 FallFac (𝑁 − 1))))
39	28, 34, 38	3eqtr3d 2806	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ∏𝑘 ∈ (0...(𝑁 − 1))((𝐴 + 1) − 𝑘) = ((𝐴 + 1) · (𝐴 FallFac (𝑁 − 1))))
40	4, 39	eqtrd 2798	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴 + 1) FallFac 𝑁) = ((𝐴 + 1) · (𝐴 FallFac (𝑁 − 1))))
41		simpr 489	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ)
42	41	nncnd 12250	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℂ)
43	42, 36	npcand 11574	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝑁 − 1) + 1) = 𝑁)
44	43	oveq2d 7428	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 FallFac ((𝑁 − 1) + 1)) = (𝐴 FallFac 𝑁))
45		fallfacp1 16085	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝑁 − 1) ∈ ℕ₀) → (𝐴 FallFac ((𝑁 − 1) + 1)) = ((𝐴 FallFac (𝑁 − 1)) · (𝐴 − (𝑁 − 1))))
46	9, 45	sylan2 604	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 FallFac ((𝑁 − 1) + 1)) = ((𝐴 FallFac (𝑁 − 1)) · (𝐴 − (𝑁 − 1))))
47	44, 46	eqtr3d 2800	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 FallFac 𝑁) = ((𝐴 FallFac (𝑁 − 1)) · (𝐴 − (𝑁 − 1))))
48	40, 47	oveq12d 7430	. 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 16072	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝑁 − 1) ∈ ℕ₀) → (𝐴 FallFac (𝑁 − 1)) ∈ ℂ)
50	9, 49	sylan2 604	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 FallFac (𝑁 − 1)) ∈ ℂ)
51	10	nn0cnd 12568	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝑁 − 1) ∈ ℂ)
52	35, 51	subcld 11570	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 − (𝑁 − 1)) ∈ ℂ)
53	50, 52	mulcomd 11231	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴 FallFac (𝑁 − 1)) · (𝐴 − (𝑁 − 1))) = ((𝐴 − (𝑁 − 1)) · (𝐴 FallFac (𝑁 − 1))))
54	53	oveq2d 7428	. . 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 485	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 + 1) ∈ ℂ)
56	55, 52, 50	subdird 11672	. . 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 11609	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴 + 1) − (𝐴 − (𝑁 − 1))) = (1 + (𝑁 − 1)))
58	36, 42	pncan3d 11573	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (1 + (𝑁 − 1)) = 𝑁)
59	57, 58	eqtrd 2798	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴 + 1) − (𝐴 − (𝑁 − 1))) = 𝑁)
60	59	oveq1d 7427	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (((𝐴 + 1) − (𝐴 − (𝑁 − 1))) · (𝐴 FallFac (𝑁 − 1))) = (𝑁 · (𝐴 FallFac (𝑁 − 1))))
61	54, 56, 60	3eqtr2d 2804	. 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 2798	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 400 = wceq 1570 ∈ wcel 2143 ‘cfv 6538 (class class class)co 7412 ℂcc 11099 0cc0 11101 1c1 11102 + caddc 11104 · cmul 11106 − cmin 11442 -cneg 11443 ℕcn 12234 ℕ₀cn0 12505 ℤcz 12592 ℤ_≥cuz 12863 ...cfz 13536 ∏cprod 15959 FallFac cfallfac 16060

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-inf2 9611 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-er 8695 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-sup 9403 df-oi 9473 df-card 9926 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-div 11873 df-nn 12235 df-2 12304 df-3 12305 df-n0 12506 df-z 12593 df-uz 12864 df-rp 13018 df-fz 13537 df-fzo 13685 df-seq 14040 df-exp 14100 df-hash 14369 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-clim 15541 df-prod 15960 df-fallfac 16063

This theorem is referenced by: (None)

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