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Mirrors > Home > MPE Home > Th. List > fallfacp1 | Structured version Visualization version GIF version |
Description: The value of the falling factorial at a successor. (Contributed by Scott Fenton, 5-Jan-2018.) |
Ref | Expression |
---|---|
fallfacp1 | ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴 FallFac (𝑁 + 1)) = ((𝐴 FallFac 𝑁) · (𝐴 − 𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0cn 12420 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ) | |
2 | 1 | adantl 482 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℂ) |
3 | 1cnd 11147 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → 1 ∈ ℂ) | |
4 | 2, 3 | pncand 11510 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((𝑁 + 1) − 1) = 𝑁) |
5 | 4 | oveq2d 7370 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (0...((𝑁 + 1) − 1)) = (0...𝑁)) |
6 | 5 | prodeq1d 15801 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ∏𝑘 ∈ (0...((𝑁 + 1) − 1))(𝐴 − 𝑘) = ∏𝑘 ∈ (0...𝑁)(𝐴 − 𝑘)) |
7 | elnn0uz 12805 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (ℤ≥‘0)) | |
8 | 7 | biimpi 215 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (ℤ≥‘0)) |
9 | 8 | adantl 482 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ (ℤ≥‘0)) |
10 | elfznn0 13531 | . . . . . . 7 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ0) | |
11 | 10 | nn0cnd 12472 | . . . . . 6 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℂ) |
12 | subcl 11397 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝐴 − 𝑘) ∈ ℂ) | |
13 | 11, 12 | sylan2 593 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ (0...𝑁)) → (𝐴 − 𝑘) ∈ ℂ) |
14 | 13 | adantlr 713 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑁)) → (𝐴 − 𝑘) ∈ ℂ) |
15 | oveq2 7362 | . . . 4 ⊢ (𝑘 = 𝑁 → (𝐴 − 𝑘) = (𝐴 − 𝑁)) | |
16 | 9, 14, 15 | fprodm1 15847 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ∏𝑘 ∈ (0...𝑁)(𝐴 − 𝑘) = (∏𝑘 ∈ (0...(𝑁 − 1))(𝐴 − 𝑘) · (𝐴 − 𝑁))) |
17 | 6, 16 | eqtrd 2776 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ∏𝑘 ∈ (0...((𝑁 + 1) − 1))(𝐴 − 𝑘) = (∏𝑘 ∈ (0...(𝑁 − 1))(𝐴 − 𝑘) · (𝐴 − 𝑁))) |
18 | peano2nn0 12450 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0) | |
19 | fallfacval 15889 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑁 + 1) ∈ ℕ0) → (𝐴 FallFac (𝑁 + 1)) = ∏𝑘 ∈ (0...((𝑁 + 1) − 1))(𝐴 − 𝑘)) | |
20 | 18, 19 | sylan2 593 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴 FallFac (𝑁 + 1)) = ∏𝑘 ∈ (0...((𝑁 + 1) − 1))(𝐴 − 𝑘)) |
21 | fallfacval 15889 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴 FallFac 𝑁) = ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴 − 𝑘)) | |
22 | 21 | oveq1d 7369 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((𝐴 FallFac 𝑁) · (𝐴 − 𝑁)) = (∏𝑘 ∈ (0...(𝑁 − 1))(𝐴 − 𝑘) · (𝐴 − 𝑁))) |
23 | 17, 20, 22 | 3eqtr4d 2786 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴 FallFac (𝑁 + 1)) = ((𝐴 FallFac 𝑁) · (𝐴 − 𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ‘cfv 6494 (class class class)co 7354 ℂcc 11046 0cc0 11048 1c1 11049 + caddc 11051 · cmul 11053 − cmin 11382 ℕ0cn0 12410 ℤ≥cuz 12760 ...cfz 13421 ∏cprod 15785 FallFac cfallfac 15884 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-inf2 9574 ax-cnex 11104 ax-resscn 11105 ax-1cn 11106 ax-icn 11107 ax-addcl 11108 ax-addrcl 11109 ax-mulcl 11110 ax-mulrcl 11111 ax-mulcom 11112 ax-addass 11113 ax-mulass 11114 ax-distr 11115 ax-i2m1 11116 ax-1ne0 11117 ax-1rid 11118 ax-rnegex 11119 ax-rrecex 11120 ax-cnre 11121 ax-pre-lttri 11122 ax-pre-lttrn 11123 ax-pre-ltadd 11124 ax-pre-mulgt0 11125 ax-pre-sup 11126 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-se 5588 df-we 5589 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 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-isom 6503 df-riota 7310 df-ov 7357 df-oprab 7358 df-mpo 7359 df-om 7800 df-1st 7918 df-2nd 7919 df-frecs 8209 df-wrecs 8240 df-recs 8314 df-rdg 8353 df-1o 8409 df-er 8645 df-en 8881 df-dom 8882 df-sdom 8883 df-fin 8884 df-sup 9375 df-oi 9443 df-card 9872 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11384 df-neg 11385 df-div 11810 df-nn 12151 df-2 12213 df-3 12214 df-n0 12411 df-z 12497 df-uz 12761 df-rp 12913 df-fz 13422 df-fzo 13565 df-seq 13904 df-exp 13965 df-hash 14228 df-cj 14981 df-re 14982 df-im 14983 df-sqrt 15117 df-abs 15118 df-clim 15367 df-prod 15786 df-fallfac 15887 |
This theorem is referenced by: fallfacp1d 15912 fallfac1 15914 fallfacfwd 15916 binomfallfaclem2 15920 bccp1k 42601 binomcxplemwb 42608 |
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