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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 9207 | . 2 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | elnnuz 9593 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ≥‘1)) | |
3 | 2 | biimpi 120 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ≥‘1)) |
4 | fvi 5593 | . . . . . . . 8 ⊢ (𝑓 ∈ (ℤ≥‘1) → ( I ‘𝑓) = 𝑓) | |
5 | eluzelcn 9568 | . . . . . . . 8 ⊢ (𝑓 ∈ (ℤ≥‘1) → 𝑓 ∈ ℂ) | |
6 | 4, 5 | eqeltrd 2266 | . . . . . . 7 ⊢ (𝑓 ∈ (ℤ≥‘1) → ( I ‘𝑓) ∈ ℂ) |
7 | 6 | adantl 277 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑓 ∈ (ℤ≥‘1)) → ( I ‘𝑓) ∈ ℂ) |
8 | mulcl 7967 | . . . . . . 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 10492 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (seq1( · , I )‘(𝑁 + 1)) = ((seq1( · , I )‘𝑁) · ( I ‘(𝑁 + 1)))) |
11 | peano2nn 8960 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ) | |
12 | fvi 5593 | . . . . . . 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 5911 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ((seq1( · , I )‘𝑁) · ( I ‘(𝑁 + 1))) = ((seq1( · , I )‘𝑁) · (𝑁 + 1))) |
15 | 10, 14 | eqtrd 2222 | . . . 4 ⊢ (𝑁 ∈ ℕ → (seq1( · , I )‘(𝑁 + 1)) = ((seq1( · , I )‘𝑁) · (𝑁 + 1))) |
16 | facnn 10738 | . . . . 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 10738 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (!‘𝑁) = (seq1( · , I )‘𝑁)) | |
19 | 18 | oveq1d 5910 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((!‘𝑁) · (𝑁 + 1)) = ((seq1( · , I )‘𝑁) · (𝑁 + 1))) |
20 | 15, 17, 19 | 3eqtr4d 2232 | . . 3 ⊢ (𝑁 ∈ ℕ → (!‘(𝑁 + 1)) = ((!‘𝑁) · (𝑁 + 1))) |
21 | 0p1e1 9062 | . . . . . 6 ⊢ (0 + 1) = 1 | |
22 | 21 | fveq2i 5537 | . . . . 5 ⊢ (!‘(0 + 1)) = (!‘1) |
23 | fac1 10740 | . . . . 5 ⊢ (!‘1) = 1 | |
24 | 22, 23 | eqtri 2210 | . . . 4 ⊢ (!‘(0 + 1)) = 1 |
25 | fvoveq1 5918 | . . . 4 ⊢ (𝑁 = 0 → (!‘(𝑁 + 1)) = (!‘(0 + 1))) | |
26 | fveq2 5534 | . . . . . 6 ⊢ (𝑁 = 0 → (!‘𝑁) = (!‘0)) | |
27 | oveq1 5902 | . . . . . 6 ⊢ (𝑁 = 0 → (𝑁 + 1) = (0 + 1)) | |
28 | 26, 27 | oveq12d 5913 | . . . . 5 ⊢ (𝑁 = 0 → ((!‘𝑁) · (𝑁 + 1)) = ((!‘0) · (0 + 1))) |
29 | fac0 10739 | . . . . . . 7 ⊢ (!‘0) = 1 | |
30 | 29, 21 | oveq12i 5907 | . . . . . 6 ⊢ ((!‘0) · (0 + 1)) = (1 · 1) |
31 | 1t1e1 9100 | . . . . . 6 ⊢ (1 · 1) = 1 | |
32 | 30, 31 | eqtri 2210 | . . . . 5 ⊢ ((!‘0) · (0 + 1)) = 1 |
33 | 28, 32 | eqtrdi 2238 | . . . 4 ⊢ (𝑁 = 0 → ((!‘𝑁) · (𝑁 + 1)) = 1) |
34 | 24, 25, 33 | 3eqtr4a 2248 | . . 3 ⊢ (𝑁 = 0 → (!‘(𝑁 + 1)) = ((!‘𝑁) · (𝑁 + 1))) |
35 | 20, 34 | jaoi 717 | . 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 709 = wceq 1364 ∈ wcel 2160 I cid 4306 ‘cfv 5235 (class class class)co 5895 ℂcc 7838 0cc0 7840 1c1 7841 + caddc 7843 · cmul 7845 ℕcn 8948 ℕ0cn0 9205 ℤ≥cuz 9557 seqcseq 10475 !cfa 10736 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 ax-cnex 7931 ax-resscn 7932 ax-1cn 7933 ax-1re 7934 ax-icn 7935 ax-addcl 7936 ax-addrcl 7937 ax-mulcl 7938 ax-addcom 7940 ax-mulcom 7941 ax-addass 7942 ax-mulass 7943 ax-distr 7944 ax-i2m1 7945 ax-0lt1 7946 ax-1rid 7947 ax-0id 7948 ax-rnegex 7949 ax-cnre 7951 ax-pre-ltirr 7952 ax-pre-ltwlin 7953 ax-pre-lttrn 7954 ax-pre-ltadd 7956 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-iord 4384 df-on 4386 df-ilim 4387 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-riota 5851 df-ov 5898 df-oprab 5899 df-mpo 5900 df-1st 6164 df-2nd 6165 df-recs 6329 df-frec 6415 df-pnf 8023 df-mnf 8024 df-xr 8025 df-ltxr 8026 df-le 8027 df-sub 8159 df-neg 8160 df-inn 8949 df-n0 9206 df-z 9283 df-uz 9558 df-seqfrec 10476 df-fac 10737 |
This theorem is referenced by: fac2 10742 fac3 10743 fac4 10744 facnn2 10745 faccl 10746 facdiv 10749 facwordi 10751 faclbnd 10752 faclbnd6 10755 facubnd 10756 bcm1k 10771 bcp1n 10772 4bc2eq6 10785 fprodfac 11654 efcllemp 11697 ef01bndlem 11795 eirraplem 11815 dvdsfac 11897 prmfac1 12183 pcfac 12381 ex-fac 14933 |
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