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Mirrors > Home > MPE Home > Th. List > 0fallfac | Structured version Visualization version GIF version |
Description: The value of the zero falling factorial at natural 𝑁. (Contributed by Scott Fenton, 17-Feb-2018.) |
Ref | Expression |
---|---|
0fallfac | ⊢ (𝑁 ∈ ℕ → (0 FallFac 𝑁) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 11251 | . . 3 ⊢ 0 ∈ ℂ | |
2 | nnnn0 12531 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
3 | fallfacval 16042 | . . 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 12565 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) | |
6 | nn0uz 12918 | . . . 4 ⊢ ℕ0 = (ℤ≥‘0) | |
7 | 5, 6 | eleqtrdi 2849 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ (ℤ≥‘0)) |
8 | elfzelz 13561 | . . . . . 6 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℤ) | |
9 | 8 | zcnd 12721 | . . . . 5 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℂ) |
10 | subcl 11505 | . . . . 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 7439 | . . . 4 ⊢ (𝑘 = 0 → (0 − 𝑘) = (0 − 0)) | |
14 | 0m0e0 12384 | . . . 4 ⊢ (0 − 0) = 0 | |
15 | 13, 14 | eqtrdi 2791 | . . 3 ⊢ (𝑘 = 0 → (0 − 𝑘) = 0) |
16 | 7, 12, 15 | fprod1p 16001 | . 2 ⊢ (𝑁 ∈ ℕ → ∏𝑘 ∈ (0...(𝑁 − 1))(0 − 𝑘) = (0 · ∏𝑘 ∈ ((0 + 1)...(𝑁 − 1))(0 − 𝑘))) |
17 | fzfid 14011 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((0 + 1)...(𝑁 − 1)) ∈ Fin) | |
18 | elfzelz 13561 | . . . . . . 7 ⊢ (𝑘 ∈ ((0 + 1)...(𝑁 − 1)) → 𝑘 ∈ ℤ) | |
19 | 18 | zcnd 12721 | . . . . . 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 15985 | . . 3 ⊢ (𝑁 ∈ ℕ → ∏𝑘 ∈ ((0 + 1)...(𝑁 − 1))(0 − 𝑘) ∈ ℂ) |
23 | 22 | mul02d 11457 | . 2 ⊢ (𝑁 ∈ ℕ → (0 · ∏𝑘 ∈ ((0 + 1)...(𝑁 − 1))(0 − 𝑘)) = 0) |
24 | 4, 16, 23 | 3eqtrd 2779 | 1 ⊢ (𝑁 ∈ ℕ → (0 FallFac 𝑁) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 − cmin 11490 ℕcn 12264 ℕ0cn0 12524 ℤ≥cuz 12876 ...cfz 13544 ∏cprod 15936 FallFac cfallfac 16037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-prod 15937 df-fallfac 16040 |
This theorem is referenced by: 0risefac 16071 |
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