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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 7939 | . . 3 ⊢ 0 ∈ V | |
2 | 1ex 7940 | . . 3 ⊢ 1 ∈ V | |
3 | df-fac 10687 | . . . 4 ⊢ ! = ({〈0, 1〉} ∪ seq1( · , I )) | |
4 | nnuz 9549 | . . . . . . . 8 ⊢ ℕ = (ℤ≥‘1) | |
5 | dfn2 9175 | . . . . . . . 8 ⊢ ℕ = (ℕ0 ∖ {0}) | |
6 | 4, 5 | eqtr3i 2200 | . . . . . . 7 ⊢ (ℤ≥‘1) = (ℕ0 ∖ {0}) |
7 | 6 | reseq2i 4900 | . . . . . 6 ⊢ (seq1( · , I ) ↾ (ℤ≥‘1)) = (seq1( · , I ) ↾ (ℕ0 ∖ {0})) |
8 | eqid 2177 | . . . . . . . . . 10 ⊢ (ℤ≥‘1) = (ℤ≥‘1) | |
9 | 1zzd 9266 | . . . . . . . . . 10 ⊢ (⊤ → 1 ∈ ℤ) | |
10 | fvi 5569 | . . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (ℤ≥‘1) → ( I ‘𝑓) = 𝑓) | |
11 | 10 | eleq1d 2246 | . . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (ℤ≥‘1) → (( I ‘𝑓) ∈ (ℤ≥‘1) ↔ 𝑓 ∈ (ℤ≥‘1))) |
12 | 11 | ibir 177 | . . . . . . . . . . . 12 ⊢ (𝑓 ∈ (ℤ≥‘1) → ( I ‘𝑓) ∈ (ℤ≥‘1)) |
13 | eluzelcn 9525 | . . . . . . . . . . . 12 ⊢ (( I ‘𝑓) ∈ (ℤ≥‘1) → ( I ‘𝑓) ∈ ℂ) | |
14 | 12, 13 | syl 14 | . . . . . . . . . . 11 ⊢ (𝑓 ∈ (ℤ≥‘1) → ( I ‘𝑓) ∈ ℂ) |
15 | 14 | adantl 277 | . . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑓 ∈ (ℤ≥‘1)) → ( I ‘𝑓) ∈ ℂ) |
16 | mulcl 7926 | . . . . . . . . . . 11 ⊢ ((𝑓 ∈ ℂ ∧ 𝑔 ∈ ℂ) → (𝑓 · 𝑔) ∈ ℂ) | |
17 | 16 | adantl 277 | . . . . . . . . . 10 ⊢ ((⊤ ∧ (𝑓 ∈ ℂ ∧ 𝑔 ∈ ℂ)) → (𝑓 · 𝑔) ∈ ℂ) |
18 | 8, 9, 15, 17 | seqf 10444 | . . . . . . . . 9 ⊢ (⊤ → seq1( · , I ):(ℤ≥‘1)⟶ℂ) |
19 | 18 | ffnd 5362 | . . . . . . . 8 ⊢ (⊤ → seq1( · , I ) Fn (ℤ≥‘1)) |
20 | 19 | mptru 1362 | . . . . . . 7 ⊢ seq1( · , I ) Fn (ℤ≥‘1) |
21 | fnresdm 5321 | . . . . . . 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 2200 | . . . . 5 ⊢ (seq1( · , I ) ↾ (ℕ0 ∖ {0})) = seq1( · , I ) |
24 | 23 | uneq2i 3286 | . . . 4 ⊢ ({〈0, 1〉} ∪ (seq1( · , I ) ↾ (ℕ0 ∖ {0}))) = ({〈0, 1〉} ∪ seq1( · , I )) |
25 | 3, 24 | eqtr4i 2201 | . . 3 ⊢ ! = ({〈0, 1〉} ∪ (seq1( · , I ) ↾ (ℕ0 ∖ {0}))) |
26 | 1, 2, 25 | fvsnun2 5710 | . 2 ⊢ (𝑁 ∈ (ℕ0 ∖ {0}) → (!‘𝑁) = (seq1( · , I )‘𝑁)) |
27 | 26, 5 | eleq2s 2272 | 1 ⊢ (𝑁 ∈ ℕ → (!‘𝑁) = (seq1( · , I )‘𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ⊤wtru 1354 ∈ wcel 2148 ∖ cdif 3126 ∪ cun 3127 {csn 3591 〈cop 3594 I cid 4285 ↾ cres 4625 Fn wfn 5207 ‘cfv 5212 (class class class)co 5869 ℂcc 7797 0cc0 7799 1c1 7800 · cmul 7804 ℕcn 8905 ℕ0cn0 9162 ℤ≥cuz 9514 seqcseq 10428 !cfa 10686 |
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 7890 ax-resscn 7891 ax-1cn 7892 ax-1re 7893 ax-icn 7894 ax-addcl 7895 ax-addrcl 7896 ax-mulcl 7897 ax-addcom 7899 ax-addass 7901 ax-distr 7903 ax-i2m1 7904 ax-0lt1 7905 ax-0id 7907 ax-rnegex 7908 ax-cnre 7910 ax-pre-ltirr 7911 ax-pre-ltwlin 7912 ax-pre-lttrn 7913 ax-pre-ltadd 7915 |
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 7981 df-mnf 7982 df-xr 7983 df-ltxr 7984 df-le 7985 df-sub 8117 df-neg 8118 df-inn 8906 df-n0 9163 df-z 9240 df-uz 9515 df-seqfrec 10429 df-fac 10687 |
This theorem is referenced by: fac1 10690 facp1 10691 bcval5 10724 |
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