fallfacfwd - Metamath Proof Explorer

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Theorem fallfacfwd 15971

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 11317	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴 + 1) ∈ ℂ)
2		nnnn0 12420	. . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
3		fallfacval 15944	. . . . 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 12274	. . . . . . . . 9 ⊢ (0 + 1) = 1
6	5	oveq1i 7378	. . . . . . . 8 ⊢ ((0 + 1)...(𝑁 − 1)) = (1...(𝑁 − 1))
7	6	prodeq1i 15851	. . . . . . 7 ⊢ ∏𝑘 ∈ ((0 + 1)...(𝑁 − 1))(𝐴 − (𝑘 − 1)) = ∏𝑘 ∈ (1...(𝑁 − 1))(𝐴 − (𝑘 − 1))
8	7	oveq2i 7379	. . . . . 6 ⊢ ((𝐴 − -1) · ∏𝑘 ∈ ((0 + 1)...(𝑁 − 1))(𝐴 − (𝑘 − 1))) = ((𝐴 − -1) · ∏𝑘 ∈ (1...(𝑁 − 1))(𝐴 − (𝑘 − 1)))
9		nnm1nn0 12454	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
10	9	adantl 481	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝑁 − 1) ∈ ℕ₀)
11		nn0uz 12801	. . . . . . . 8 ⊢ ℕ₀ = (ℤ_≥‘0)
12	10, 11	eleqtrdi 2847	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝑁 − 1) ∈ (ℤ_≥‘0))
13		simpll 767	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝐴 ∈ ℂ)
14		elfzelz 13452	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℤ)
15	14	adantl 481	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ ℤ)
16		peano2zm 12546	. . . . . . . . . 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 12609	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑘 − 1) ∈ ℂ)
19	13, 18	subcld 11504	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝐴 − (𝑘 − 1)) ∈ ℂ)
20		oveq1 7375	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝑘 − 1) = (0 − 1))
21		df-neg 11379	. . . . . . . . 9 ⊢ -1 = (0 − 1)
22	20, 21	eqtr4di 2790	. . . . . . . 8 ⊢ (𝑘 = 0 → (𝑘 − 1) = -1)
23	22	oveq2d 7384	. . . . . . 7 ⊢ (𝑘 = 0 → (𝐴 − (𝑘 − 1)) = (𝐴 − -1))
24	12, 19, 23	fprod1p 15903	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴 − (𝑘 − 1)) = ((𝐴 − -1) · ∏𝑘 ∈ ((0 + 1)...(𝑁 − 1))(𝐴 − (𝑘 − 1))))
25		fallfacval2 15946	. . . . . . . 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 7384	. . . . . 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 2798	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴 − (𝑘 − 1)) = ((𝐴 − -1) · (𝐴 FallFac (𝑁 − 1))))
29		elfznn0 13548	. . . . . . . . 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 12476	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ ℂ)
32		1cnd 11139	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 1 ∈ ℂ)
33	13, 31, 32	subsub3d 11534	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝐴 − (𝑘 − 1)) = ((𝐴 + 1) − 𝑘))
34	33	prodeq2dv 15857	. . . . 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 11139	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → 1 ∈ ℂ)
37	35, 36	subnegd 11511	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 − -1) = (𝐴 + 1))
38	37	oveq1d 7383	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴 − -1) · (𝐴 FallFac (𝑁 − 1))) = ((𝐴 + 1) · (𝐴 FallFac (𝑁 − 1))))
39	28, 34, 38	3eqtr3d 2780	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ∏𝑘 ∈ (0...(𝑁 − 1))((𝐴 + 1) − 𝑘) = ((𝐴 + 1) · (𝐴 FallFac (𝑁 − 1))))
40	4, 39	eqtrd 2772	. . 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 12173	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℂ)
43	42, 36	npcand 11508	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝑁 − 1) + 1) = 𝑁)
44	43	oveq2d 7384	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 FallFac ((𝑁 − 1) + 1)) = (𝐴 FallFac 𝑁))
45		fallfacp1 15965	. . . . 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 2774	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 FallFac 𝑁) = ((𝐴 FallFac (𝑁 − 1)) · (𝐴 − (𝑁 − 1))))
48	40, 47	oveq12d 7386	. 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 15951	. . . . . 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 12476	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝑁 − 1) ∈ ℂ)
52	35, 51	subcld 11504	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 − (𝑁 − 1)) ∈ ℂ)
53	50, 52	mulcomd 11165	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴 FallFac (𝑁 − 1)) · (𝐴 − (𝑁 − 1))) = ((𝐴 − (𝑁 − 1)) · (𝐴 FallFac (𝑁 − 1))))
54	53	oveq2d 7384	. . 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 11606	. . 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 11543	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴 + 1) − (𝐴 − (𝑁 − 1))) = (1 + (𝑁 − 1)))
58	36, 42	pncan3d 11507	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (1 + (𝑁 − 1)) = 𝑁)
59	57, 58	eqtrd 2772	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴 + 1) − (𝐴 − (𝑁 − 1))) = 𝑁)
60	59	oveq1d 7383	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (((𝐴 + 1) − (𝐴 − (𝑁 − 1))) · (𝐴 FallFac (𝑁 − 1))) = (𝑁 · (𝐴 FallFac (𝑁 − 1))))
61	54, 56, 60	3eqtr2d 2778	. 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 2772	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 1542 ∈ wcel 2114 ‘cfv 6500 (class class class)co 7368 ℂcc 11036 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 − cmin 11376 -cneg 11377 ℕcn 12157 ℕ₀cn0 12413 ℤcz 12500 ℤ_≥cuz 12763 ...cfz 13435 ∏cprod 15838 FallFac cfallfac 15939

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9357 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-n0 12414 df-z 12501 df-uz 12764 df-rp 12918 df-fz 13436 df-fzo 13583 df-seq 13937 df-exp 13997 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-clim 15423 df-prod 15839 df-fallfac 15942

This theorem is referenced by: (None)

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