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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 7728 | . . 3 ⊢ 0 ∈ V | |
2 | 1ex 7729 | . . 3 ⊢ 1 ∈ V | |
3 | df-fac 10440 | . . . 4 ⊢ ! = ({〈0, 1〉} ∪ seq1( · , I )) | |
4 | nnuz 9329 | . . . . . . . 8 ⊢ ℕ = (ℤ≥‘1) | |
5 | dfn2 8958 | . . . . . . . 8 ⊢ ℕ = (ℕ0 ∖ {0}) | |
6 | 4, 5 | eqtr3i 2140 | . . . . . . 7 ⊢ (ℤ≥‘1) = (ℕ0 ∖ {0}) |
7 | 6 | reseq2i 4786 | . . . . . 6 ⊢ (seq1( · , I ) ↾ (ℤ≥‘1)) = (seq1( · , I ) ↾ (ℕ0 ∖ {0})) |
8 | eqid 2117 | . . . . . . . . . 10 ⊢ (ℤ≥‘1) = (ℤ≥‘1) | |
9 | 1zzd 9049 | . . . . . . . . . 10 ⊢ (⊤ → 1 ∈ ℤ) | |
10 | fvi 5446 | . . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (ℤ≥‘1) → ( I ‘𝑓) = 𝑓) | |
11 | 10 | eleq1d 2186 | . . . . . . . . . . . . 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 9305 | . . . . . . . . . . . 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 7715 | . . . . . . . . . . 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 10202 | . . . . . . . . 9 ⊢ (⊤ → seq1( · , I ):(ℤ≥‘1)⟶ℂ) |
19 | 18 | ffnd 5243 | . . . . . . . 8 ⊢ (⊤ → seq1( · , I ) Fn (ℤ≥‘1)) |
20 | 19 | mptru 1325 | . . . . . . 7 ⊢ seq1( · , I ) Fn (ℤ≥‘1) |
21 | fnresdm 5202 | . . . . . . 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 2140 | . . . . 5 ⊢ (seq1( · , I ) ↾ (ℕ0 ∖ {0})) = seq1( · , I ) |
24 | 23 | uneq2i 3197 | . . . 4 ⊢ ({〈0, 1〉} ∪ (seq1( · , I ) ↾ (ℕ0 ∖ {0}))) = ({〈0, 1〉} ∪ seq1( · , I )) |
25 | 3, 24 | eqtr4i 2141 | . . 3 ⊢ ! = ({〈0, 1〉} ∪ (seq1( · , I ) ↾ (ℕ0 ∖ {0}))) |
26 | 1, 2, 25 | fvsnun2 5586 | . 2 ⊢ (𝑁 ∈ (ℕ0 ∖ {0}) → (!‘𝑁) = (seq1( · , I )‘𝑁)) |
27 | 26, 5 | eleq2s 2212 | 1 ⊢ (𝑁 ∈ ℕ → (!‘𝑁) = (seq1( · , I )‘𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1316 ⊤wtru 1317 ∈ wcel 1465 ∖ cdif 3038 ∪ cun 3039 {csn 3497 〈cop 3500 I cid 4180 ↾ cres 4511 Fn wfn 5088 ‘cfv 5093 (class class class)co 5742 ℂcc 7586 0cc0 7588 1c1 7589 · cmul 7593 ℕcn 8688 ℕ0cn0 8945 ℤ≥cuz 9294 seqcseq 10186 !cfa 10439 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-ilim 4261 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-frec 6256 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-inn 8689 df-n0 8946 df-z 9023 df-uz 9295 df-seqfrec 10187 df-fac 10440 |
This theorem is referenced by: fac1 10443 facp1 10444 bcval5 10477 |
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