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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 12526 | . 2 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0)) | |
2 | facnn 14292 | . . . 4 ⊢ (𝐴 ∈ ℕ → (!‘𝐴) = (seq1( · , I )‘𝐴)) | |
3 | vex 3466 | . . . . . 6 ⊢ 𝑘 ∈ V | |
4 | fvi 6978 | . . . . . 6 ⊢ (𝑘 ∈ V → ( I ‘𝑘) = 𝑘) | |
5 | 3, 4 | mp1i 13 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (1...𝐴)) → ( I ‘𝑘) = 𝑘) |
6 | elnnuz 12918 | . . . . . 6 ⊢ (𝐴 ∈ ℕ ↔ 𝐴 ∈ (ℤ≥‘1)) | |
7 | 6 | biimpi 215 | . . . . 5 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ (ℤ≥‘1)) |
8 | elfznn 13584 | . . . . . . 7 ⊢ (𝑘 ∈ (1...𝐴) → 𝑘 ∈ ℕ) | |
9 | 8 | nncnd 12280 | . . . . . 6 ⊢ (𝑘 ∈ (1...𝐴) → 𝑘 ∈ ℂ) |
10 | 9 | adantl 480 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (1...𝐴)) → 𝑘 ∈ ℂ) |
11 | 5, 7, 10 | fprodser 15951 | . . . 4 ⊢ (𝐴 ∈ ℕ → ∏𝑘 ∈ (1...𝐴)𝑘 = (seq1( · , I )‘𝐴)) |
12 | 2, 11 | eqtr4d 2769 | . . 3 ⊢ (𝐴 ∈ ℕ → (!‘𝐴) = ∏𝑘 ∈ (1...𝐴)𝑘) |
13 | prod0 15945 | . . . . 5 ⊢ ∏𝑘 ∈ ∅ 𝑘 = 1 | |
14 | 13 | eqcomi 2735 | . . . 4 ⊢ 1 = ∏𝑘 ∈ ∅ 𝑘 |
15 | fveq2 6901 | . . . . 5 ⊢ (𝐴 = 0 → (!‘𝐴) = (!‘0)) | |
16 | fac0 14293 | . . . . 5 ⊢ (!‘0) = 1 | |
17 | 15, 16 | eqtrdi 2782 | . . . 4 ⊢ (𝐴 = 0 → (!‘𝐴) = 1) |
18 | oveq2 7432 | . . . . . 6 ⊢ (𝐴 = 0 → (1...𝐴) = (1...0)) | |
19 | fz10 13576 | . . . . . 6 ⊢ (1...0) = ∅ | |
20 | 18, 19 | eqtrdi 2782 | . . . . 5 ⊢ (𝐴 = 0 → (1...𝐴) = ∅) |
21 | 20 | prodeq1d 15923 | . . . 4 ⊢ (𝐴 = 0 → ∏𝑘 ∈ (1...𝐴)𝑘 = ∏𝑘 ∈ ∅ 𝑘) |
22 | 14, 17, 21 | 3eqtr4a 2792 | . . 3 ⊢ (𝐴 = 0 → (!‘𝐴) = ∏𝑘 ∈ (1...𝐴)𝑘) |
23 | 12, 22 | jaoi 855 | . 2 ⊢ ((𝐴 ∈ ℕ ∨ 𝐴 = 0) → (!‘𝐴) = ∏𝑘 ∈ (1...𝐴)𝑘) |
24 | 1, 23 | sylbi 216 | 1 ⊢ (𝐴 ∈ ℕ0 → (!‘𝐴) = ∏𝑘 ∈ (1...𝐴)𝑘) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∨ wo 845 = wceq 1534 ∈ wcel 2099 Vcvv 3462 ∅c0 4325 I cid 5579 ‘cfv 6554 (class class class)co 7424 ℂcc 11156 0cc0 11158 1c1 11159 · cmul 11163 ℕcn 12264 ℕ0cn0 12524 ℤ≥cuz 12874 ...cfz 13538 seqcseq 14021 !cfa 14290 ∏cprod 15907 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-sup 9485 df-oi 9553 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12611 df-uz 12875 df-rp 13029 df-fz 13539 df-fzo 13682 df-seq 14022 df-exp 14082 df-fac 14291 df-hash 14348 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-clim 15490 df-prod 15908 |
This theorem is referenced by: risefacfac 16037 fallfacval4 16045 prmolefac 17048 gausslemma2dlem1 27395 gausslemma2dlem6 27401 bcprod 35560 etransclem41 45896 |
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