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Mirrors > Home > MPE Home > Th. List > fprodfac | Structured version Visualization version GIF version |
Description: Factorial using product notation. (Contributed by Scott Fenton, 15-Dec-2017.) |
Ref | Expression |
---|---|
fprodfac | ⊢ (𝐴 ∈ ℕ0 → (!‘𝐴) = ∏𝑘 ∈ (1...𝐴)𝑘) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 12525 | . 2 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0)) | |
2 | facnn 14310 | . . . 4 ⊢ (𝐴 ∈ ℕ → (!‘𝐴) = (seq1( · , I )‘𝐴)) | |
3 | vex 3481 | . . . . . 6 ⊢ 𝑘 ∈ V | |
4 | fvi 6984 | . . . . . 6 ⊢ (𝑘 ∈ V → ( I ‘𝑘) = 𝑘) | |
5 | 3, 4 | mp1i 13 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (1...𝐴)) → ( I ‘𝑘) = 𝑘) |
6 | elnnuz 12919 | . . . . . 6 ⊢ (𝐴 ∈ ℕ ↔ 𝐴 ∈ (ℤ≥‘1)) | |
7 | 6 | biimpi 216 | . . . . 5 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ (ℤ≥‘1)) |
8 | elfznn 13589 | . . . . . . 7 ⊢ (𝑘 ∈ (1...𝐴) → 𝑘 ∈ ℕ) | |
9 | 8 | nncnd 12279 | . . . . . 6 ⊢ (𝑘 ∈ (1...𝐴) → 𝑘 ∈ ℂ) |
10 | 9 | adantl 481 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (1...𝐴)) → 𝑘 ∈ ℂ) |
11 | 5, 7, 10 | fprodser 15981 | . . . 4 ⊢ (𝐴 ∈ ℕ → ∏𝑘 ∈ (1...𝐴)𝑘 = (seq1( · , I )‘𝐴)) |
12 | 2, 11 | eqtr4d 2777 | . . 3 ⊢ (𝐴 ∈ ℕ → (!‘𝐴) = ∏𝑘 ∈ (1...𝐴)𝑘) |
13 | prod0 15975 | . . . . 5 ⊢ ∏𝑘 ∈ ∅ 𝑘 = 1 | |
14 | 13 | eqcomi 2743 | . . . 4 ⊢ 1 = ∏𝑘 ∈ ∅ 𝑘 |
15 | fveq2 6906 | . . . . 5 ⊢ (𝐴 = 0 → (!‘𝐴) = (!‘0)) | |
16 | fac0 14311 | . . . . 5 ⊢ (!‘0) = 1 | |
17 | 15, 16 | eqtrdi 2790 | . . . 4 ⊢ (𝐴 = 0 → (!‘𝐴) = 1) |
18 | oveq2 7438 | . . . . . 6 ⊢ (𝐴 = 0 → (1...𝐴) = (1...0)) | |
19 | fz10 13581 | . . . . . 6 ⊢ (1...0) = ∅ | |
20 | 18, 19 | eqtrdi 2790 | . . . . 5 ⊢ (𝐴 = 0 → (1...𝐴) = ∅) |
21 | 20 | prodeq1d 15952 | . . . 4 ⊢ (𝐴 = 0 → ∏𝑘 ∈ (1...𝐴)𝑘 = ∏𝑘 ∈ ∅ 𝑘) |
22 | 14, 17, 21 | 3eqtr4a 2800 | . . 3 ⊢ (𝐴 = 0 → (!‘𝐴) = ∏𝑘 ∈ (1...𝐴)𝑘) |
23 | 12, 22 | jaoi 857 | . 2 ⊢ ((𝐴 ∈ ℕ ∨ 𝐴 = 0) → (!‘𝐴) = ∏𝑘 ∈ (1...𝐴)𝑘) |
24 | 1, 23 | sylbi 217 | 1 ⊢ (𝐴 ∈ ℕ0 → (!‘𝐴) = ∏𝑘 ∈ (1...𝐴)𝑘) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ∅c0 4338 I cid 5581 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 0cc0 11152 1c1 11153 · cmul 11157 ℕcn 12263 ℕ0cn0 12523 ℤ≥cuz 12875 ...cfz 13543 seqcseq 14038 !cfa 14308 ∏cprod 15935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-fz 13544 df-fzo 13691 df-seq 14039 df-exp 14099 df-fac 14309 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 df-prod 15936 |
This theorem is referenced by: risefacfac 16067 fallfacval4 16075 prmolefac 17079 gausslemma2dlem1 27424 gausslemma2dlem6 27430 bcprod 35717 etransclem41 46230 |
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