Mathbox for Scott Fenton |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iprodfac | Structured version Visualization version GIF version |
Description: An infinite product expression for factorial. (Contributed by Scott Fenton, 15-Dec-2017.) |
Ref | Expression |
---|---|
iprodfac | ⊢ (𝐴 ∈ ℕ0 → (!‘𝐴) = ∏𝑘 ∈ ℕ (((1 + (1 / 𝑘))↑𝐴) / (1 + (𝐴 / 𝑘)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnuz 11723 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
2 | 1zzd 11408 | . . 3 ⊢ (𝐴 ∈ ℕ0 → 1 ∈ ℤ) | |
3 | facne0 13073 | . . 3 ⊢ (𝐴 ∈ ℕ0 → (!‘𝐴) ≠ 0) | |
4 | eqid 2622 | . . . 4 ⊢ (𝑥 ∈ ℕ ↦ (((1 + (1 / 𝑥))↑𝐴) / (1 + (𝐴 / 𝑥)))) = (𝑥 ∈ ℕ ↦ (((1 + (1 / 𝑥))↑𝐴) / (1 + (𝐴 / 𝑥)))) | |
5 | 4 | faclim 31632 | . . 3 ⊢ (𝐴 ∈ ℕ0 → seq1( · , (𝑥 ∈ ℕ ↦ (((1 + (1 / 𝑥))↑𝐴) / (1 + (𝐴 / 𝑥))))) ⇝ (!‘𝐴)) |
6 | oveq2 6658 | . . . . . . . 8 ⊢ (𝑥 = 𝑘 → (1 / 𝑥) = (1 / 𝑘)) | |
7 | 6 | oveq2d 6666 | . . . . . . 7 ⊢ (𝑥 = 𝑘 → (1 + (1 / 𝑥)) = (1 + (1 / 𝑘))) |
8 | 7 | oveq1d 6665 | . . . . . 6 ⊢ (𝑥 = 𝑘 → ((1 + (1 / 𝑥))↑𝐴) = ((1 + (1 / 𝑘))↑𝐴)) |
9 | oveq2 6658 | . . . . . . 7 ⊢ (𝑥 = 𝑘 → (𝐴 / 𝑥) = (𝐴 / 𝑘)) | |
10 | 9 | oveq2d 6666 | . . . . . 6 ⊢ (𝑥 = 𝑘 → (1 + (𝐴 / 𝑥)) = (1 + (𝐴 / 𝑘))) |
11 | 8, 10 | oveq12d 6668 | . . . . 5 ⊢ (𝑥 = 𝑘 → (((1 + (1 / 𝑥))↑𝐴) / (1 + (𝐴 / 𝑥))) = (((1 + (1 / 𝑘))↑𝐴) / (1 + (𝐴 / 𝑘)))) |
12 | ovex 6678 | . . . . 5 ⊢ (((1 + (1 / 𝑘))↑𝐴) / (1 + (𝐴 / 𝑘))) ∈ V | |
13 | 11, 4, 12 | fvmpt 6282 | . . . 4 ⊢ (𝑘 ∈ ℕ → ((𝑥 ∈ ℕ ↦ (((1 + (1 / 𝑥))↑𝐴) / (1 + (𝐴 / 𝑥))))‘𝑘) = (((1 + (1 / 𝑘))↑𝐴) / (1 + (𝐴 / 𝑘)))) |
14 | 13 | adantl 482 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ (((1 + (1 / 𝑥))↑𝐴) / (1 + (𝐴 / 𝑥))))‘𝑘) = (((1 + (1 / 𝑘))↑𝐴) / (1 + (𝐴 / 𝑘)))) |
15 | 1rp 11836 | . . . . . . . 8 ⊢ 1 ∈ ℝ+ | |
16 | 15 | a1i 11 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑘 ∈ ℕ) → 1 ∈ ℝ+) |
17 | simpr 477 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ) | |
18 | 17 | nnrpd 11870 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ+) |
19 | 18 | rpreccld 11882 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ+) |
20 | 16, 19 | rpaddcld 11887 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑘 ∈ ℕ) → (1 + (1 / 𝑘)) ∈ ℝ+) |
21 | nn0z 11400 | . . . . . . 7 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ) | |
22 | 21 | adantr 481 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℤ) |
23 | 20, 22 | rpexpcld 13032 | . . . . 5 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑘 ∈ ℕ) → ((1 + (1 / 𝑘))↑𝐴) ∈ ℝ+) |
24 | 1cnd 10056 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑘 ∈ ℕ) → 1 ∈ ℂ) | |
25 | nn0nndivcl 11362 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑘 ∈ ℕ) → (𝐴 / 𝑘) ∈ ℝ) | |
26 | 25 | recnd 10068 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑘 ∈ ℕ) → (𝐴 / 𝑘) ∈ ℂ) |
27 | 24, 26 | addcomd 10238 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑘 ∈ ℕ) → (1 + (𝐴 / 𝑘)) = ((𝐴 / 𝑘) + 1)) |
28 | nn0ge0div 11446 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝐴 / 𝑘)) | |
29 | 25, 28 | ge0p1rpd 11902 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑘 ∈ ℕ) → ((𝐴 / 𝑘) + 1) ∈ ℝ+) |
30 | 27, 29 | eqeltrd 2701 | . . . . 5 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑘 ∈ ℕ) → (1 + (𝐴 / 𝑘)) ∈ ℝ+) |
31 | 23, 30 | rpdivcld 11889 | . . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑘 ∈ ℕ) → (((1 + (1 / 𝑘))↑𝐴) / (1 + (𝐴 / 𝑘))) ∈ ℝ+) |
32 | 31 | rpcnd 11874 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑘 ∈ ℕ) → (((1 + (1 / 𝑘))↑𝐴) / (1 + (𝐴 / 𝑘))) ∈ ℂ) |
33 | 1, 2, 3, 5, 14, 32 | iprodn0 14670 | . 2 ⊢ (𝐴 ∈ ℕ0 → ∏𝑘 ∈ ℕ (((1 + (1 / 𝑘))↑𝐴) / (1 + (𝐴 / 𝑘))) = (!‘𝐴)) |
34 | 33 | eqcomd 2628 | 1 ⊢ (𝐴 ∈ ℕ0 → (!‘𝐴) = ∏𝑘 ∈ ℕ (((1 + (1 / 𝑘))↑𝐴) / (1 + (𝐴 / 𝑘)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 1c1 9937 + caddc 9939 / cdiv 10684 ℕcn 11020 ℕ0cn0 11292 ℤcz 11377 ℝ+crp 11832 ↑cexp 12860 !cfa 13060 ∏cprod 14635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-fac 13061 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-rlim 14220 df-prod 14636 |
This theorem is referenced by: (None) |
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