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Mirrors > Home > ILE Home > Th. List > fac0 | GIF version |
Description: The factorial of 0. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.) |
Ref | Expression |
---|---|
fac0 | ⊢ (!‘0) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | c0ex 7760 | . 2 ⊢ 0 ∈ V | |
2 | 1ex 7761 | . 2 ⊢ 1 ∈ V | |
3 | df-fac 10472 | . . 3 ⊢ ! = ({〈0, 1〉} ∪ seq1( · , I )) | |
4 | nnuz 9361 | . . . . . . 7 ⊢ ℕ = (ℤ≥‘1) | |
5 | dfn2 8990 | . . . . . . 7 ⊢ ℕ = (ℕ0 ∖ {0}) | |
6 | 4, 5 | eqtr3i 2162 | . . . . . 6 ⊢ (ℤ≥‘1) = (ℕ0 ∖ {0}) |
7 | 6 | reseq2i 4816 | . . . . 5 ⊢ (seq1( · , I ) ↾ (ℤ≥‘1)) = (seq1( · , I ) ↾ (ℕ0 ∖ {0})) |
8 | eqid 2139 | . . . . . . . . 9 ⊢ (ℤ≥‘1) = (ℤ≥‘1) | |
9 | 1zzd 9081 | . . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℤ) | |
10 | fvi 5478 | . . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (ℤ≥‘1) → ( I ‘𝑓) = 𝑓) | |
11 | 10 | eleq1d 2208 | . . . . . . . . . . . 12 ⊢ (𝑓 ∈ (ℤ≥‘1) → (( I ‘𝑓) ∈ (ℤ≥‘1) ↔ 𝑓 ∈ (ℤ≥‘1))) |
12 | 11 | ibir 176 | . . . . . . . . . . 11 ⊢ (𝑓 ∈ (ℤ≥‘1) → ( I ‘𝑓) ∈ (ℤ≥‘1)) |
13 | eluzelcn 9337 | . . . . . . . . . . 11 ⊢ (( I ‘𝑓) ∈ (ℤ≥‘1) → ( I ‘𝑓) ∈ ℂ) | |
14 | 12, 13 | syl 14 | . . . . . . . . . 10 ⊢ (𝑓 ∈ (ℤ≥‘1) → ( I ‘𝑓) ∈ ℂ) |
15 | 14 | adantl 275 | . . . . . . . . 9 ⊢ ((⊤ ∧ 𝑓 ∈ (ℤ≥‘1)) → ( I ‘𝑓) ∈ ℂ) |
16 | mulcl 7747 | . . . . . . . . . 10 ⊢ ((𝑓 ∈ ℂ ∧ 𝑔 ∈ ℂ) → (𝑓 · 𝑔) ∈ ℂ) | |
17 | 16 | adantl 275 | . . . . . . . . 9 ⊢ ((⊤ ∧ (𝑓 ∈ ℂ ∧ 𝑔 ∈ ℂ)) → (𝑓 · 𝑔) ∈ ℂ) |
18 | 8, 9, 15, 17 | seqf 10234 | . . . . . . . 8 ⊢ (⊤ → seq1( · , I ):(ℤ≥‘1)⟶ℂ) |
19 | 18 | ffnd 5273 | . . . . . . 7 ⊢ (⊤ → seq1( · , I ) Fn (ℤ≥‘1)) |
20 | 19 | mptru 1340 | . . . . . 6 ⊢ seq1( · , I ) Fn (ℤ≥‘1) |
21 | fnresdm 5232 | . . . . . 6 ⊢ (seq1( · , I ) Fn (ℤ≥‘1) → (seq1( · , I ) ↾ (ℤ≥‘1)) = seq1( · , I )) | |
22 | 20, 21 | ax-mp 5 | . . . . 5 ⊢ (seq1( · , I ) ↾ (ℤ≥‘1)) = seq1( · , I ) |
23 | 7, 22 | eqtr3i 2162 | . . . 4 ⊢ (seq1( · , I ) ↾ (ℕ0 ∖ {0})) = seq1( · , I ) |
24 | 23 | uneq2i 3227 | . . 3 ⊢ ({〈0, 1〉} ∪ (seq1( · , I ) ↾ (ℕ0 ∖ {0}))) = ({〈0, 1〉} ∪ seq1( · , I )) |
25 | 3, 24 | eqtr4i 2163 | . 2 ⊢ ! = ({〈0, 1〉} ∪ (seq1( · , I ) ↾ (ℕ0 ∖ {0}))) |
26 | 1, 2, 25 | fvsnun1 5617 | 1 ⊢ (!‘0) = 1 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1331 ⊤wtru 1332 ∈ wcel 1480 ∖ cdif 3068 ∪ cun 3069 {csn 3527 〈cop 3530 I cid 4210 ↾ cres 4541 Fn wfn 5118 ‘cfv 5123 (class class class)co 5774 ℂcc 7618 0cc0 7620 1c1 7621 · 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-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 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: facp1 10476 faccl 10481 facwordi 10486 faclbnd 10487 facubnd 10491 bcn0 10501 bcval5 10509 ef0lem 11366 ege2le3 11377 eft0val 11399 prmfac1 11830 |
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