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Mirrors > Home > ILE Home > Th. List > facnn | GIF version |
Description: Value of the factorial function for positive integers. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.) |
Ref | Expression |
---|---|
facnn | ⊢ (𝑁 ∈ ℕ → (!‘𝑁) = (seq1( · , I )‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | c0ex 7753 | . . 3 ⊢ 0 ∈ V | |
2 | 1ex 7754 | . . 3 ⊢ 1 ∈ V | |
3 | df-fac 10465 | . . . 4 ⊢ ! = ({〈0, 1〉} ∪ seq1( · , I )) | |
4 | nnuz 9354 | . . . . . . . 8 ⊢ ℕ = (ℤ≥‘1) | |
5 | dfn2 8983 | . . . . . . . 8 ⊢ ℕ = (ℕ0 ∖ {0}) | |
6 | 4, 5 | eqtr3i 2160 | . . . . . . 7 ⊢ (ℤ≥‘1) = (ℕ0 ∖ {0}) |
7 | 6 | reseq2i 4811 | . . . . . 6 ⊢ (seq1( · , I ) ↾ (ℤ≥‘1)) = (seq1( · , I ) ↾ (ℕ0 ∖ {0})) |
8 | eqid 2137 | . . . . . . . . . 10 ⊢ (ℤ≥‘1) = (ℤ≥‘1) | |
9 | 1zzd 9074 | . . . . . . . . . 10 ⊢ (⊤ → 1 ∈ ℤ) | |
10 | fvi 5471 | . . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (ℤ≥‘1) → ( I ‘𝑓) = 𝑓) | |
11 | 10 | eleq1d 2206 | . . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (ℤ≥‘1) → (( I ‘𝑓) ∈ (ℤ≥‘1) ↔ 𝑓 ∈ (ℤ≥‘1))) |
12 | 11 | ibir 176 | . . . . . . . . . . . 12 ⊢ (𝑓 ∈ (ℤ≥‘1) → ( I ‘𝑓) ∈ (ℤ≥‘1)) |
13 | eluzelcn 9330 | . . . . . . . . . . . 12 ⊢ (( I ‘𝑓) ∈ (ℤ≥‘1) → ( I ‘𝑓) ∈ ℂ) | |
14 | 12, 13 | syl 14 | . . . . . . . . . . 11 ⊢ (𝑓 ∈ (ℤ≥‘1) → ( I ‘𝑓) ∈ ℂ) |
15 | 14 | adantl 275 | . . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑓 ∈ (ℤ≥‘1)) → ( I ‘𝑓) ∈ ℂ) |
16 | mulcl 7740 | . . . . . . . . . . 11 ⊢ ((𝑓 ∈ ℂ ∧ 𝑔 ∈ ℂ) → (𝑓 · 𝑔) ∈ ℂ) | |
17 | 16 | adantl 275 | . . . . . . . . . 10 ⊢ ((⊤ ∧ (𝑓 ∈ ℂ ∧ 𝑔 ∈ ℂ)) → (𝑓 · 𝑔) ∈ ℂ) |
18 | 8, 9, 15, 17 | seqf 10227 | . . . . . . . . 9 ⊢ (⊤ → seq1( · , I ):(ℤ≥‘1)⟶ℂ) |
19 | 18 | ffnd 5268 | . . . . . . . 8 ⊢ (⊤ → seq1( · , I ) Fn (ℤ≥‘1)) |
20 | 19 | mptru 1340 | . . . . . . 7 ⊢ seq1( · , I ) Fn (ℤ≥‘1) |
21 | fnresdm 5227 | . . . . . . 7 ⊢ (seq1( · , I ) Fn (ℤ≥‘1) → (seq1( · , I ) ↾ (ℤ≥‘1)) = seq1( · , I )) | |
22 | 20, 21 | ax-mp 5 | . . . . . 6 ⊢ (seq1( · , I ) ↾ (ℤ≥‘1)) = seq1( · , I ) |
23 | 7, 22 | eqtr3i 2160 | . . . . 5 ⊢ (seq1( · , I ) ↾ (ℕ0 ∖ {0})) = seq1( · , I ) |
24 | 23 | uneq2i 3222 | . . . 4 ⊢ ({〈0, 1〉} ∪ (seq1( · , I ) ↾ (ℕ0 ∖ {0}))) = ({〈0, 1〉} ∪ seq1( · , I )) |
25 | 3, 24 | eqtr4i 2161 | . . 3 ⊢ ! = ({〈0, 1〉} ∪ (seq1( · , I ) ↾ (ℕ0 ∖ {0}))) |
26 | 1, 2, 25 | fvsnun2 5611 | . 2 ⊢ (𝑁 ∈ (ℕ0 ∖ {0}) → (!‘𝑁) = (seq1( · , I )‘𝑁)) |
27 | 26, 5 | eleq2s 2232 | 1 ⊢ (𝑁 ∈ ℕ → (!‘𝑁) = (seq1( · , I )‘𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ⊤wtru 1332 ∈ wcel 1480 ∖ cdif 3063 ∪ cun 3064 {csn 3522 〈cop 3525 I cid 4205 ↾ cres 4536 Fn wfn 5113 ‘cfv 5118 (class class class)co 5767 ℂcc 7611 0cc0 7613 1c1 7614 · cmul 7618 ℕcn 8713 ℕ0cn0 8970 ℤ≥cuz 9319 seqcseq 10211 !cfa 10464 |
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 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-ltadd 7729 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-frec 6281 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-inn 8714 df-n0 8971 df-z 9048 df-uz 9320 df-seqfrec 10212 df-fac 10465 |
This theorem is referenced by: fac1 10468 facp1 10469 bcval5 10502 |
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