Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 9149 | . 2 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | elnnuz 9535 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ≥‘1)) | |
3 | 2 | biimpi 120 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ≥‘1)) |
4 | fvi 5565 | . . . . . . . 8 ⊢ (𝑓 ∈ (ℤ≥‘1) → ( I ‘𝑓) = 𝑓) | |
5 | eluzelcn 9510 | . . . . . . . 8 ⊢ (𝑓 ∈ (ℤ≥‘1) → 𝑓 ∈ ℂ) | |
6 | 4, 5 | eqeltrd 2252 | . . . . . . 7 ⊢ (𝑓 ∈ (ℤ≥‘1) → ( I ‘𝑓) ∈ ℂ) |
7 | 6 | adantl 277 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑓 ∈ (ℤ≥‘1)) → ( I ‘𝑓) ∈ ℂ) |
8 | mulcl 7913 | . . . . . . 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 10430 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (seq1( · , I )‘(𝑁 + 1)) = ((seq1( · , I )‘𝑁) · ( I ‘(𝑁 + 1)))) |
11 | peano2nn 8902 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ) | |
12 | fvi 5565 | . . . . . . 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 5881 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ((seq1( · , I )‘𝑁) · ( I ‘(𝑁 + 1))) = ((seq1( · , I )‘𝑁) · (𝑁 + 1))) |
15 | 10, 14 | eqtrd 2208 | . . . 4 ⊢ (𝑁 ∈ ℕ → (seq1( · , I )‘(𝑁 + 1)) = ((seq1( · , I )‘𝑁) · (𝑁 + 1))) |
16 | facnn 10673 | . . . . 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 10673 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (!‘𝑁) = (seq1( · , I )‘𝑁)) | |
19 | 18 | oveq1d 5880 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((!‘𝑁) · (𝑁 + 1)) = ((seq1( · , I )‘𝑁) · (𝑁 + 1))) |
20 | 15, 17, 19 | 3eqtr4d 2218 | . . 3 ⊢ (𝑁 ∈ ℕ → (!‘(𝑁 + 1)) = ((!‘𝑁) · (𝑁 + 1))) |
21 | 0p1e1 9004 | . . . . . 6 ⊢ (0 + 1) = 1 | |
22 | 21 | fveq2i 5510 | . . . . 5 ⊢ (!‘(0 + 1)) = (!‘1) |
23 | fac1 10675 | . . . . 5 ⊢ (!‘1) = 1 | |
24 | 22, 23 | eqtri 2196 | . . . 4 ⊢ (!‘(0 + 1)) = 1 |
25 | fvoveq1 5888 | . . . 4 ⊢ (𝑁 = 0 → (!‘(𝑁 + 1)) = (!‘(0 + 1))) | |
26 | fveq2 5507 | . . . . . 6 ⊢ (𝑁 = 0 → (!‘𝑁) = (!‘0)) | |
27 | oveq1 5872 | . . . . . 6 ⊢ (𝑁 = 0 → (𝑁 + 1) = (0 + 1)) | |
28 | 26, 27 | oveq12d 5883 | . . . . 5 ⊢ (𝑁 = 0 → ((!‘𝑁) · (𝑁 + 1)) = ((!‘0) · (0 + 1))) |
29 | fac0 10674 | . . . . . . 7 ⊢ (!‘0) = 1 | |
30 | 29, 21 | oveq12i 5877 | . . . . . 6 ⊢ ((!‘0) · (0 + 1)) = (1 · 1) |
31 | 1t1e1 9042 | . . . . . 6 ⊢ (1 · 1) = 1 | |
32 | 30, 31 | eqtri 2196 | . . . . 5 ⊢ ((!‘0) · (0 + 1)) = 1 |
33 | 28, 32 | eqtrdi 2224 | . . . 4 ⊢ (𝑁 = 0 → ((!‘𝑁) · (𝑁 + 1)) = 1) |
34 | 24, 25, 33 | 3eqtr4a 2234 | . . 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 2146 I cid 4282 ‘cfv 5208 (class class class)co 5865 ℂcc 7784 0cc0 7786 1c1 7787 + caddc 7789 · cmul 7791 ℕcn 8890 ℕ0cn0 9147 ℤ≥cuz 9499 seqcseq 10413 !cfa 10671 |
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 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-iinf 4581 ax-cnex 7877 ax-resscn 7878 ax-1cn 7879 ax-1re 7880 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-addcom 7886 ax-mulcom 7887 ax-addass 7888 ax-mulass 7889 ax-distr 7890 ax-i2m1 7891 ax-0lt1 7892 ax-1rid 7893 ax-0id 7894 ax-rnegex 7895 ax-cnre 7897 ax-pre-ltirr 7898 ax-pre-ltwlin 7899 ax-pre-lttrn 7900 ax-pre-ltadd 7902 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-reu 2460 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-id 4287 df-iord 4360 df-on 4362 df-ilim 4363 df-suc 4365 df-iom 4584 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 df-recs 6296 df-frec 6382 df-pnf 7968 df-mnf 7969 df-xr 7970 df-ltxr 7971 df-le 7972 df-sub 8104 df-neg 8105 df-inn 8891 df-n0 9148 df-z 9225 df-uz 9500 df-seqfrec 10414 df-fac 10672 |
This theorem is referenced by: fac2 10677 fac3 10678 fac4 10679 facnn2 10680 faccl 10681 facdiv 10684 facwordi 10686 faclbnd 10687 faclbnd6 10690 facubnd 10691 bcm1k 10706 bcp1n 10707 4bc2eq6 10720 fprodfac 11589 efcllemp 11632 ef01bndlem 11730 eirraplem 11750 dvdsfac 11831 prmfac1 12117 pcfac 12313 ex-fac 14020 |
Copyright terms: Public domain | W3C validator |