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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 12528 | . 2 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0)) | |
| 2 | facnn 14314 | . . . 4 ⊢ (𝐴 ∈ ℕ → (!‘𝐴) = (seq1( · , I )‘𝐴)) | |
| 3 | vex 3484 | . . . . . 6 ⊢ 𝑘 ∈ V | |
| 4 | fvi 6985 | . . . . . 6 ⊢ (𝑘 ∈ V → ( I ‘𝑘) = 𝑘) | |
| 5 | 3, 4 | mp1i 13 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (1...𝐴)) → ( I ‘𝑘) = 𝑘) |
| 6 | elnnuz 12922 | . . . . . 6 ⊢ (𝐴 ∈ ℕ ↔ 𝐴 ∈ (ℤ≥‘1)) | |
| 7 | 6 | biimpi 216 | . . . . 5 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ (ℤ≥‘1)) |
| 8 | elfznn 13593 | . . . . . . 7 ⊢ (𝑘 ∈ (1...𝐴) → 𝑘 ∈ ℕ) | |
| 9 | 8 | nncnd 12282 | . . . . . 6 ⊢ (𝑘 ∈ (1...𝐴) → 𝑘 ∈ ℂ) |
| 10 | 9 | adantl 481 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (1...𝐴)) → 𝑘 ∈ ℂ) |
| 11 | 5, 7, 10 | fprodser 15985 | . . . 4 ⊢ (𝐴 ∈ ℕ → ∏𝑘 ∈ (1...𝐴)𝑘 = (seq1( · , I )‘𝐴)) |
| 12 | 2, 11 | eqtr4d 2780 | . . 3 ⊢ (𝐴 ∈ ℕ → (!‘𝐴) = ∏𝑘 ∈ (1...𝐴)𝑘) |
| 13 | prod0 15979 | . . . . 5 ⊢ ∏𝑘 ∈ ∅ 𝑘 = 1 | |
| 14 | 13 | eqcomi 2746 | . . . 4 ⊢ 1 = ∏𝑘 ∈ ∅ 𝑘 |
| 15 | fveq2 6906 | . . . . 5 ⊢ (𝐴 = 0 → (!‘𝐴) = (!‘0)) | |
| 16 | fac0 14315 | . . . . 5 ⊢ (!‘0) = 1 | |
| 17 | 15, 16 | eqtrdi 2793 | . . . 4 ⊢ (𝐴 = 0 → (!‘𝐴) = 1) |
| 18 | oveq2 7439 | . . . . . 6 ⊢ (𝐴 = 0 → (1...𝐴) = (1...0)) | |
| 19 | fz10 13585 | . . . . . 6 ⊢ (1...0) = ∅ | |
| 20 | 18, 19 | eqtrdi 2793 | . . . . 5 ⊢ (𝐴 = 0 → (1...𝐴) = ∅) |
| 21 | 20 | prodeq1d 15956 | . . . 4 ⊢ (𝐴 = 0 → ∏𝑘 ∈ (1...𝐴)𝑘 = ∏𝑘 ∈ ∅ 𝑘) |
| 22 | 14, 17, 21 | 3eqtr4a 2803 | . . 3 ⊢ (𝐴 = 0 → (!‘𝐴) = ∏𝑘 ∈ (1...𝐴)𝑘) |
| 23 | 12, 22 | jaoi 858 | . 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 848 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 I cid 5577 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 · cmul 11160 ℕcn 12266 ℕ0cn0 12526 ℤ≥cuz 12878 ...cfz 13547 seqcseq 14042 !cfa 14312 ∏cprod 15939 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-fac 14313 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-prod 15940 |
| This theorem is referenced by: risefacfac 16071 fallfacval4 16079 prmolefac 17084 gausslemma2dlem1 27410 gausslemma2dlem6 27416 bcprod 35738 etransclem41 46290 |
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