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Mirrors > Home > MPE Home > Th. List > fallfacfac | Structured version Visualization version GIF version |
Description: Relate falling factorial to factorial. (Contributed by Scott Fenton, 5-Jan-2018.) |
Ref | Expression |
---|---|
fallfacfac | ⊢ (𝑁 ∈ ℕ0 → (𝑁 FallFac 𝑁) = (!‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0fz0 13596 | . . 3 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (0...𝑁)) | |
2 | fallfacval4 15984 | . . 3 ⊢ (𝑁 ∈ (0...𝑁) → (𝑁 FallFac 𝑁) = ((!‘𝑁) / (!‘(𝑁 − 𝑁)))) | |
3 | 1, 2 | sylbi 216 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝑁 FallFac 𝑁) = ((!‘𝑁) / (!‘(𝑁 − 𝑁)))) |
4 | nn0cn 12479 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ) | |
5 | 4 | subidd 11556 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 − 𝑁) = 0) |
6 | 5 | fveq2d 6893 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (!‘(𝑁 − 𝑁)) = (!‘0)) |
7 | fac0 14233 | . . . 4 ⊢ (!‘0) = 1 | |
8 | 6, 7 | eqtrdi 2789 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (!‘(𝑁 − 𝑁)) = 1) |
9 | 8 | oveq2d 7422 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((!‘𝑁) / (!‘(𝑁 − 𝑁))) = ((!‘𝑁) / 1)) |
10 | faccl 14240 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℕ) | |
11 | 10 | nncnd 12225 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℂ) |
12 | 11 | div1d 11979 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((!‘𝑁) / 1) = (!‘𝑁)) |
13 | 3, 9, 12 | 3eqtrd 2777 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝑁 FallFac 𝑁) = (!‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ‘cfv 6541 (class class class)co 7406 0cc0 11107 1c1 11108 − cmin 11441 / cdiv 11868 ℕ0cn0 12469 ...cfz 13481 !cfa 14230 FallFac cfallfac 15945 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-inf2 9633 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-oi 9502 df-card 9931 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-n0 12470 df-z 12556 df-uz 12820 df-rp 12972 df-fz 13482 df-fzo 13625 df-seq 13964 df-exp 14025 df-fac 14231 df-hash 14288 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-clim 15429 df-prod 15847 df-fallfac 15948 |
This theorem is referenced by: (None) |
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