fallfacfwd - Metamath Proof Explorer

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Theorem fallfacfwd 15392

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 10814	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴 + 1) ∈ ℂ)
2		nnnn0 11907	. . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
3		fallfacval 15365	. . . . 5 ⊢ (((𝐴 + 1) ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 + 1) FallFac 𝑁) = ∏𝑘 ∈ (0...(𝑁 − 1))((𝐴 + 1) − 𝑘))
4	1, 2, 3	syl2an 597	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴 + 1) FallFac 𝑁) = ∏𝑘 ∈ (0...(𝑁 − 1))((𝐴 + 1) − 𝑘))
5		0p1e1 11762	. . . . . . . . 9 ⊢ (0 + 1) = 1
6	5	oveq1i 7168	. . . . . . . 8 ⊢ ((0 + 1)...(𝑁 − 1)) = (1...(𝑁 − 1))
7	6	prodeq1i 15274	. . . . . . 7 ⊢ ∏𝑘 ∈ ((0 + 1)...(𝑁 − 1))(𝐴 − (𝑘 − 1)) = ∏𝑘 ∈ (1...(𝑁 − 1))(𝐴 − (𝑘 − 1))
8	7	oveq2i 7169	. . . . . 6 ⊢ ((𝐴 − -1) · ∏𝑘 ∈ ((0 + 1)...(𝑁 − 1))(𝐴 − (𝑘 − 1))) = ((𝐴 − -1) · ∏𝑘 ∈ (1...(𝑁 − 1))(𝐴 − (𝑘 − 1)))
9		nnm1nn0 11941	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
10	9	adantl 484	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝑁 − 1) ∈ ℕ₀)
11		nn0uz 12283	. . . . . . . 8 ⊢ ℕ₀ = (ℤ_≥‘0)
12	10, 11	eleqtrdi 2925	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝑁 − 1) ∈ (ℤ_≥‘0))
13		simpll 765	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝐴 ∈ ℂ)
14		elfzelz 12911	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℤ)
15	14	adantl 484	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ ℤ)
16		peano2zm 12028	. . . . . . . . . 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 12091	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑘 − 1) ∈ ℂ)
19	13, 18	subcld 10999	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝐴 − (𝑘 − 1)) ∈ ℂ)
20		oveq1 7165	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝑘 − 1) = (0 − 1))
21		df-neg 10875	. . . . . . . . 9 ⊢ -1 = (0 − 1)
22	20, 21	syl6eqr 2876	. . . . . . . 8 ⊢ (𝑘 = 0 → (𝑘 − 1) = -1)
23	22	oveq2d 7174	. . . . . . 7 ⊢ (𝑘 = 0 → (𝐴 − (𝑘 − 1)) = (𝐴 − -1))
24	12, 19, 23	fprod1p 15324	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴 − (𝑘 − 1)) = ((𝐴 − -1) · ∏𝑘 ∈ ((0 + 1)...(𝑁 − 1))(𝐴 − (𝑘 − 1))))
25		fallfacval2 15367	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (𝑁 − 1) ∈ ℕ₀) → (𝐴 FallFac (𝑁 − 1)) = ∏𝑘 ∈ (1...(𝑁 − 1))(𝐴 − (𝑘 − 1)))
26	9, 25	sylan2 594	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 FallFac (𝑁 − 1)) = ∏𝑘 ∈ (1...(𝑁 − 1))(𝐴 − (𝑘 − 1)))
27	26	oveq2d 7174	. . . . . 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 2884	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴 − (𝑘 − 1)) = ((𝐴 − -1) · (𝐴 FallFac (𝑁 − 1))))
29		elfznn0 13003	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℕ₀)
30	29	adantl 484	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ ℕ₀)
31	30	nn0cnd 11960	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ ℂ)
32		1cnd 10638	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 1 ∈ ℂ)
33	13, 31, 32	subsub3d 11029	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝐴 − (𝑘 − 1)) = ((𝐴 + 1) − 𝑘))
34	33	prodeq2dv 15279	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴 − (𝑘 − 1)) = ∏𝑘 ∈ (0...(𝑁 − 1))((𝐴 + 1) − 𝑘))
35		simpl 485	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → 𝐴 ∈ ℂ)
36		1cnd 10638	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → 1 ∈ ℂ)
37	35, 36	subnegd 11006	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 − -1) = (𝐴 + 1))
38	37	oveq1d 7173	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴 − -1) · (𝐴 FallFac (𝑁 − 1))) = ((𝐴 + 1) · (𝐴 FallFac (𝑁 − 1))))
39	28, 34, 38	3eqtr3d 2866	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ∏𝑘 ∈ (0...(𝑁 − 1))((𝐴 + 1) − 𝑘) = ((𝐴 + 1) · (𝐴 FallFac (𝑁 − 1))))
40	4, 39	eqtrd 2858	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴 + 1) FallFac 𝑁) = ((𝐴 + 1) · (𝐴 FallFac (𝑁 − 1))))
41		simpr 487	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ)
42	41	nncnd 11656	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℂ)
43	42, 36	npcand 11003	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝑁 − 1) + 1) = 𝑁)
44	43	oveq2d 7174	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 FallFac ((𝑁 − 1) + 1)) = (𝐴 FallFac 𝑁))
45		fallfacp1 15386	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝑁 − 1) ∈ ℕ₀) → (𝐴 FallFac ((𝑁 − 1) + 1)) = ((𝐴 FallFac (𝑁 − 1)) · (𝐴 − (𝑁 − 1))))
46	9, 45	sylan2 594	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 FallFac ((𝑁 − 1) + 1)) = ((𝐴 FallFac (𝑁 − 1)) · (𝐴 − (𝑁 − 1))))
47	44, 46	eqtr3d 2860	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 FallFac 𝑁) = ((𝐴 FallFac (𝑁 − 1)) · (𝐴 − (𝑁 − 1))))
48	40, 47	oveq12d 7176	. 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 15372	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝑁 − 1) ∈ ℕ₀) → (𝐴 FallFac (𝑁 − 1)) ∈ ℂ)
50	9, 49	sylan2 594	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 FallFac (𝑁 − 1)) ∈ ℂ)
51	10	nn0cnd 11960	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝑁 − 1) ∈ ℂ)
52	35, 51	subcld 10999	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 − (𝑁 − 1)) ∈ ℂ)
53	50, 52	mulcomd 10664	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴 FallFac (𝑁 − 1)) · (𝐴 − (𝑁 − 1))) = ((𝐴 − (𝑁 − 1)) · (𝐴 FallFac (𝑁 − 1))))
54	53	oveq2d 7174	. . 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 483	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 + 1) ∈ ℂ)
56	55, 52, 50	subdird 11099	. . 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 11038	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴 + 1) − (𝐴 − (𝑁 − 1))) = (1 + (𝑁 − 1)))
58	36, 42	pncan3d 11002	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (1 + (𝑁 − 1)) = 𝑁)
59	57, 58	eqtrd 2858	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴 + 1) − (𝐴 − (𝑁 − 1))) = 𝑁)
60	59	oveq1d 7173	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (((𝐴 + 1) − (𝐴 − (𝑁 − 1))) · (𝐴 FallFac (𝑁 − 1))) = (𝑁 · (𝐴 FallFac (𝑁 − 1))))
61	54, 56, 60	3eqtr2d 2864	. 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 2858	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 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 0cc0 10539 1c1 10540 + caddc 10542 · cmul 10544 − cmin 10872 -cneg 10873 ℕcn 11640 ℕ₀cn0 11900 ℤcz 11984 ℤ_≥cuz 12246 ...cfz 12895 ∏cprod 15261 FallFac cfallfac 15360

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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-prod 15262 df-fallfac 15363

This theorem is referenced by: (None)

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