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Theorem 0fallfac 16050

Description: The value of the zero falling factorial at natural 𝑁. (Contributed by Scott Fenton, 17-Feb-2018.)

**Assertion**
Ref	Expression
0fallfac	⊢ (𝑁 ∈ ℕ → (0 FallFac 𝑁) = 0)

Proof of Theorem 0fallfac Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0cn 11168	. . 3 ⊢ 0 ∈ ℂ
2		nnnn0 12485	. . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
3		fallfacval 16022	. . 3 ⊢ ((0 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → (0 FallFac 𝑁) = ∏𝑘 ∈ (0...(𝑁 − 1))(0 − 𝑘))
4	1, 2, 3	sylancr 596	. 2 ⊢ (𝑁 ∈ ℕ → (0 FallFac 𝑁) = ∏𝑘 ∈ (0...(𝑁 − 1))(0 − 𝑘))
5		nnm1nn0 12519	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
6		nn0uz 12874	. . . 4 ⊢ ℕ₀ = (ℤ_≥‘0)
7	5, 6	eleqtrdi 2871	. . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ (ℤ_≥‘0))
8		elfzelz 13526	. . . . . 6 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℤ)
9	8	zcnd 12675	. . . . 5 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℂ)
10		subcl 11426	. . . . 5 ⊢ ((0 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (0 − 𝑘) ∈ ℂ)
11	1, 9, 10	sylancr 596	. . . 4 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → (0 − 𝑘) ∈ ℂ)
12	11	adantl 485	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (0 − 𝑘) ∈ ℂ)
13		oveq2 7400	. . . 4 ⊢ (𝑘 = 0 → (0 − 𝑘) = (0 − 0))
14		0m0e0 12333	. . . 4 ⊢ (0 − 0) = 0
15	13, 14	eqtrdi 2812	. . 3 ⊢ (𝑘 = 0 → (0 − 𝑘) = 0)
16	7, 12, 15	fprod1p 15981	. 2 ⊢ (𝑁 ∈ ℕ → ∏𝑘 ∈ (0...(𝑁 − 1))(0 − 𝑘) = (0 · ∏𝑘 ∈ ((0 + 1)...(𝑁 − 1))(0 − 𝑘)))
17		fzfid 13983	. . . 4 ⊢ (𝑁 ∈ ℕ → ((0 + 1)...(𝑁 − 1)) ∈ Fin)
18		elfzelz 13526	. . . . . . 7 ⊢ (𝑘 ∈ ((0 + 1)...(𝑁 − 1)) → 𝑘 ∈ ℤ)
19	18	zcnd 12675	. . . . . 6 ⊢ (𝑘 ∈ ((0 + 1)...(𝑁 − 1)) → 𝑘 ∈ ℂ)
20	1, 19, 10	sylancr 596	. . . . 5 ⊢ (𝑘 ∈ ((0 + 1)...(𝑁 − 1)) → (0 − 𝑘) ∈ ℂ)
21	20	adantl 485	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ ((0 + 1)...(𝑁 − 1))) → (0 − 𝑘) ∈ ℂ)
22	17, 21	fprodcl 15965	. . 3 ⊢ (𝑁 ∈ ℕ → ∏𝑘 ∈ ((0 + 1)...(𝑁 − 1))(0 − 𝑘) ∈ ℂ)
23	22	mul02d 11378	. 2 ⊢ (𝑁 ∈ ℕ → (0 · ∏𝑘 ∈ ((0 + 1)...(𝑁 − 1))(0 − 𝑘)) = 0)
24	4, 16, 23	3eqtrd 2800	1 ⊢ (𝑁 ∈ ℕ → (0 FallFac 𝑁) = 0)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1559 ∈ wcel 2141 ‘cfv 6517 (class class class)co 7392 ℂcc 11068 0cc0 11070 1c1 11071 + caddc 11073 · cmul 11075 − cmin 11411 ℕcn 12207 ℕ₀cn0 12478 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: 0risefac 16051

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