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Mirrors > Home > ILE Home > Th. List > facp1 | GIF version |
Description: The factorial of a successor. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.) |
Ref | Expression |
---|---|
facp1 | ⊢ (𝑁 ∈ ℕ0 → (!‘(𝑁 + 1)) = ((!‘𝑁) · (𝑁 + 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 8979 | . 2 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | elnnuz 9362 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ≥‘1)) | |
3 | 2 | biimpi 119 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ≥‘1)) |
4 | fvi 5478 | . . . . . . . 8 ⊢ (𝑓 ∈ (ℤ≥‘1) → ( I ‘𝑓) = 𝑓) | |
5 | eluzelcn 9337 | . . . . . . . 8 ⊢ (𝑓 ∈ (ℤ≥‘1) → 𝑓 ∈ ℂ) | |
6 | 4, 5 | eqeltrd 2216 | . . . . . . 7 ⊢ (𝑓 ∈ (ℤ≥‘1) → ( I ‘𝑓) ∈ ℂ) |
7 | 6 | adantl 275 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑓 ∈ (ℤ≥‘1)) → ( I ‘𝑓) ∈ ℂ) |
8 | mulcl 7747 | . . . . . . 7 ⊢ ((𝑓 ∈ ℂ ∧ 𝑔 ∈ ℂ) → (𝑓 · 𝑔) ∈ ℂ) | |
9 | 8 | adantl 275 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑓 ∈ ℂ ∧ 𝑔 ∈ ℂ)) → (𝑓 · 𝑔) ∈ ℂ) |
10 | 3, 7, 9 | seq3p1 10235 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (seq1( · , I )‘(𝑁 + 1)) = ((seq1( · , I )‘𝑁) · ( I ‘(𝑁 + 1)))) |
11 | peano2nn 8732 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ) | |
12 | fvi 5478 | . . . . . . 7 ⊢ ((𝑁 + 1) ∈ ℕ → ( I ‘(𝑁 + 1)) = (𝑁 + 1)) | |
13 | 11, 12 | syl 14 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → ( I ‘(𝑁 + 1)) = (𝑁 + 1)) |
14 | 13 | oveq2d 5790 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ((seq1( · , I )‘𝑁) · ( I ‘(𝑁 + 1))) = ((seq1( · , I )‘𝑁) · (𝑁 + 1))) |
15 | 10, 14 | eqtrd 2172 | . . . 4 ⊢ (𝑁 ∈ ℕ → (seq1( · , I )‘(𝑁 + 1)) = ((seq1( · , I )‘𝑁) · (𝑁 + 1))) |
16 | facnn 10473 | . . . . 5 ⊢ ((𝑁 + 1) ∈ ℕ → (!‘(𝑁 + 1)) = (seq1( · , I )‘(𝑁 + 1))) | |
17 | 11, 16 | syl 14 | . . . 4 ⊢ (𝑁 ∈ ℕ → (!‘(𝑁 + 1)) = (seq1( · , I )‘(𝑁 + 1))) |
18 | facnn 10473 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (!‘𝑁) = (seq1( · , I )‘𝑁)) | |
19 | 18 | oveq1d 5789 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((!‘𝑁) · (𝑁 + 1)) = ((seq1( · , I )‘𝑁) · (𝑁 + 1))) |
20 | 15, 17, 19 | 3eqtr4d 2182 | . . 3 ⊢ (𝑁 ∈ ℕ → (!‘(𝑁 + 1)) = ((!‘𝑁) · (𝑁 + 1))) |
21 | 0p1e1 8834 | . . . . . 6 ⊢ (0 + 1) = 1 | |
22 | 21 | fveq2i 5424 | . . . . 5 ⊢ (!‘(0 + 1)) = (!‘1) |
23 | fac1 10475 | . . . . 5 ⊢ (!‘1) = 1 | |
24 | 22, 23 | eqtri 2160 | . . . 4 ⊢ (!‘(0 + 1)) = 1 |
25 | fvoveq1 5797 | . . . 4 ⊢ (𝑁 = 0 → (!‘(𝑁 + 1)) = (!‘(0 + 1))) | |
26 | fveq2 5421 | . . . . . 6 ⊢ (𝑁 = 0 → (!‘𝑁) = (!‘0)) | |
27 | oveq1 5781 | . . . . . 6 ⊢ (𝑁 = 0 → (𝑁 + 1) = (0 + 1)) | |
28 | 26, 27 | oveq12d 5792 | . . . . 5 ⊢ (𝑁 = 0 → ((!‘𝑁) · (𝑁 + 1)) = ((!‘0) · (0 + 1))) |
29 | fac0 10474 | . . . . . . 7 ⊢ (!‘0) = 1 | |
30 | 29, 21 | oveq12i 5786 | . . . . . 6 ⊢ ((!‘0) · (0 + 1)) = (1 · 1) |
31 | 1t1e1 8872 | . . . . . 6 ⊢ (1 · 1) = 1 | |
32 | 30, 31 | eqtri 2160 | . . . . 5 ⊢ ((!‘0) · (0 + 1)) = 1 |
33 | 28, 32 | syl6eq 2188 | . . . 4 ⊢ (𝑁 = 0 → ((!‘𝑁) · (𝑁 + 1)) = 1) |
34 | 24, 25, 33 | 3eqtr4a 2198 | . . 3 ⊢ (𝑁 = 0 → (!‘(𝑁 + 1)) = ((!‘𝑁) · (𝑁 + 1))) |
35 | 20, 34 | jaoi 705 | . 2 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (!‘(𝑁 + 1)) = ((!‘𝑁) · (𝑁 + 1))) |
36 | 1, 35 | sylbi 120 | 1 ⊢ (𝑁 ∈ ℕ0 → (!‘(𝑁 + 1)) = ((!‘𝑁) · (𝑁 + 1))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∨ wo 697 = wceq 1331 ∈ wcel 1480 I cid 4210 ‘cfv 5123 (class class class)co 5774 ℂcc 7618 0cc0 7620 1c1 7621 + caddc 7623 · cmul 7625 ℕcn 8720 ℕ0cn0 8977 ℤ≥cuz 9326 seqcseq 10218 !cfa 10471 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055 df-uz 9327 df-seqfrec 10219 df-fac 10472 |
This theorem is referenced by: fac2 10477 fac3 10478 fac4 10479 facnn2 10480 faccl 10481 facdiv 10484 facwordi 10486 faclbnd 10487 faclbnd6 10490 facubnd 10491 bcm1k 10506 bcp1n 10507 4bc2eq6 10520 efcllemp 11364 ef01bndlem 11463 eirraplem 11483 dvdsfac 11558 prmfac1 11830 ex-fac 12940 |
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