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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 9167 | . 2 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | elnnuz 9553 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ≥‘1)) | |
3 | 2 | biimpi 120 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ≥‘1)) |
4 | fvi 5569 | . . . . . . . 8 ⊢ (𝑓 ∈ (ℤ≥‘1) → ( I ‘𝑓) = 𝑓) | |
5 | eluzelcn 9528 | . . . . . . . 8 ⊢ (𝑓 ∈ (ℤ≥‘1) → 𝑓 ∈ ℂ) | |
6 | 4, 5 | eqeltrd 2254 | . . . . . . 7 ⊢ (𝑓 ∈ (ℤ≥‘1) → ( I ‘𝑓) ∈ ℂ) |
7 | 6 | adantl 277 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑓 ∈ (ℤ≥‘1)) → ( I ‘𝑓) ∈ ℂ) |
8 | mulcl 7929 | . . . . . . 7 ⊢ ((𝑓 ∈ ℂ ∧ 𝑔 ∈ ℂ) → (𝑓 · 𝑔) ∈ ℂ) | |
9 | 8 | adantl 277 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑓 ∈ ℂ ∧ 𝑔 ∈ ℂ)) → (𝑓 · 𝑔) ∈ ℂ) |
10 | 3, 7, 9 | seq3p1 10448 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (seq1( · , I )‘(𝑁 + 1)) = ((seq1( · , I )‘𝑁) · ( I ‘(𝑁 + 1)))) |
11 | peano2nn 8920 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ) | |
12 | fvi 5569 | . . . . . . 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 5885 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ((seq1( · , I )‘𝑁) · ( I ‘(𝑁 + 1))) = ((seq1( · , I )‘𝑁) · (𝑁 + 1))) |
15 | 10, 14 | eqtrd 2210 | . . . 4 ⊢ (𝑁 ∈ ℕ → (seq1( · , I )‘(𝑁 + 1)) = ((seq1( · , I )‘𝑁) · (𝑁 + 1))) |
16 | facnn 10691 | . . . . 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 10691 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (!‘𝑁) = (seq1( · , I )‘𝑁)) | |
19 | 18 | oveq1d 5884 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((!‘𝑁) · (𝑁 + 1)) = ((seq1( · , I )‘𝑁) · (𝑁 + 1))) |
20 | 15, 17, 19 | 3eqtr4d 2220 | . . 3 ⊢ (𝑁 ∈ ℕ → (!‘(𝑁 + 1)) = ((!‘𝑁) · (𝑁 + 1))) |
21 | 0p1e1 9022 | . . . . . 6 ⊢ (0 + 1) = 1 | |
22 | 21 | fveq2i 5514 | . . . . 5 ⊢ (!‘(0 + 1)) = (!‘1) |
23 | fac1 10693 | . . . . 5 ⊢ (!‘1) = 1 | |
24 | 22, 23 | eqtri 2198 | . . . 4 ⊢ (!‘(0 + 1)) = 1 |
25 | fvoveq1 5892 | . . . 4 ⊢ (𝑁 = 0 → (!‘(𝑁 + 1)) = (!‘(0 + 1))) | |
26 | fveq2 5511 | . . . . . 6 ⊢ (𝑁 = 0 → (!‘𝑁) = (!‘0)) | |
27 | oveq1 5876 | . . . . . 6 ⊢ (𝑁 = 0 → (𝑁 + 1) = (0 + 1)) | |
28 | 26, 27 | oveq12d 5887 | . . . . 5 ⊢ (𝑁 = 0 → ((!‘𝑁) · (𝑁 + 1)) = ((!‘0) · (0 + 1))) |
29 | fac0 10692 | . . . . . . 7 ⊢ (!‘0) = 1 | |
30 | 29, 21 | oveq12i 5881 | . . . . . 6 ⊢ ((!‘0) · (0 + 1)) = (1 · 1) |
31 | 1t1e1 9060 | . . . . . 6 ⊢ (1 · 1) = 1 | |
32 | 30, 31 | eqtri 2198 | . . . . 5 ⊢ ((!‘0) · (0 + 1)) = 1 |
33 | 28, 32 | eqtrdi 2226 | . . . 4 ⊢ (𝑁 = 0 → ((!‘𝑁) · (𝑁 + 1)) = 1) |
34 | 24, 25, 33 | 3eqtr4a 2236 | . . 3 ⊢ (𝑁 = 0 → (!‘(𝑁 + 1)) = ((!‘𝑁) · (𝑁 + 1))) |
35 | 20, 34 | jaoi 716 | . 2 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (!‘(𝑁 + 1)) = ((!‘𝑁) · (𝑁 + 1))) |
36 | 1, 35 | sylbi 121 | 1 ⊢ (𝑁 ∈ ℕ0 → (!‘(𝑁 + 1)) = ((!‘𝑁) · (𝑁 + 1))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∨ wo 708 = wceq 1353 ∈ wcel 2148 I cid 4285 ‘cfv 5212 (class class class)co 5869 ℂcc 7800 0cc0 7802 1c1 7803 + caddc 7805 · cmul 7807 ℕcn 8908 ℕ0cn0 9165 ℤ≥cuz 9517 seqcseq 10431 !cfa 10689 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-ltadd 7918 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-iord 4363 df-on 4365 df-ilim 4366 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-frec 6386 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-inn 8909 df-n0 9166 df-z 9243 df-uz 9518 df-seqfrec 10432 df-fac 10690 |
This theorem is referenced by: fac2 10695 fac3 10696 fac4 10697 facnn2 10698 faccl 10699 facdiv 10702 facwordi 10704 faclbnd 10705 faclbnd6 10708 facubnd 10709 bcm1k 10724 bcp1n 10725 4bc2eq6 10738 fprodfac 11607 efcllemp 11650 ef01bndlem 11748 eirraplem 11768 dvdsfac 11849 prmfac1 12135 pcfac 12331 ex-fac 14136 |
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