0fallfac - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > 0fallfac		Structured version Visualization version GIF version

Theorem 0fallfac 15962

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 11126	. . 3 ⊢ 0 ∈ ℂ
2		nnnn0 12409	. . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
3		fallfacval 15934	. . 3 ⊢ ((0 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → (0 FallFac 𝑁) = ∏𝑘 ∈ (0...(𝑁 − 1))(0 − 𝑘))
4	1, 2, 3	sylancr 587	. 2 ⊢ (𝑁 ∈ ℕ → (0 FallFac 𝑁) = ∏𝑘 ∈ (0...(𝑁 − 1))(0 − 𝑘))
5		nnm1nn0 12443	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
6		nn0uz 12795	. . . 4 ⊢ ℕ₀ = (ℤ_≥‘0)
7	5, 6	eleqtrdi 2838	. . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ (ℤ_≥‘0))
8		elfzelz 13445	. . . . . 6 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℤ)
9	8	zcnd 12599	. . . . 5 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℂ)
10		subcl 11380	. . . . 5 ⊢ ((0 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (0 − 𝑘) ∈ ℂ)
11	1, 9, 10	sylancr 587	. . . 4 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → (0 − 𝑘) ∈ ℂ)
12	11	adantl 481	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (0 − 𝑘) ∈ ℂ)
13		oveq2 7361	. . . 4 ⊢ (𝑘 = 0 → (0 − 𝑘) = (0 − 0))
14		0m0e0 12261	. . . 4 ⊢ (0 − 0) = 0
15	13, 14	eqtrdi 2780	. . 3 ⊢ (𝑘 = 0 → (0 − 𝑘) = 0)
16	7, 12, 15	fprod1p 15893	. 2 ⊢ (𝑁 ∈ ℕ → ∏𝑘 ∈ (0...(𝑁 − 1))(0 − 𝑘) = (0 · ∏𝑘 ∈ ((0 + 1)...(𝑁 − 1))(0 − 𝑘)))
17		fzfid 13898	. . . 4 ⊢ (𝑁 ∈ ℕ → ((0 + 1)...(𝑁 − 1)) ∈ Fin)
18		elfzelz 13445	. . . . . . 7 ⊢ (𝑘 ∈ ((0 + 1)...(𝑁 − 1)) → 𝑘 ∈ ℤ)
19	18	zcnd 12599	. . . . . 6 ⊢ (𝑘 ∈ ((0 + 1)...(𝑁 − 1)) → 𝑘 ∈ ℂ)
20	1, 19, 10	sylancr 587	. . . . 5 ⊢ (𝑘 ∈ ((0 + 1)...(𝑁 − 1)) → (0 − 𝑘) ∈ ℂ)
21	20	adantl 481	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ ((0 + 1)...(𝑁 − 1))) → (0 − 𝑘) ∈ ℂ)
22	17, 21	fprodcl 15877	. . 3 ⊢ (𝑁 ∈ ℕ → ∏𝑘 ∈ ((0 + 1)...(𝑁 − 1))(0 − 𝑘) ∈ ℂ)
23	22	mul02d 11332	. 2 ⊢ (𝑁 ∈ ℕ → (0 · ∏𝑘 ∈ ((0 + 1)...(𝑁 − 1))(0 − 𝑘)) = 0)
24	4, 16, 23	3eqtrd 2768	1 ⊢ (𝑁 ∈ ℕ → (0 FallFac 𝑁) = 0)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 ‘cfv 6486 (class class class)co 7353 ℂcc 11026 0cc0 11028 1c1 11029 + caddc 11031 · cmul 11033 − cmin 11365 ℕcn 12146 ℕ₀cn0 12402 ℤ_≥cuz 12753 ...cfz 13428 ∏cprod 15828 FallFac cfallfac 15929

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-inf2 9556 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8632 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-sup 9351 df-oi 9421 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-n0 12403 df-z 12490 df-uz 12754 df-rp 12912 df-fz 13429 df-fzo 13576 df-seq 13927 df-exp 13987 df-hash 14256 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-clim 15413 df-prod 15829 df-fallfac 15932

This theorem is referenced by: 0risefac 15963

W3C validator