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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 7784 | . . 3 ⊢ 0 ∈ V | |
2 | 1ex 7785 | . . 3 ⊢ 1 ∈ V | |
3 | df-fac 10504 | . . . 4 ⊢ ! = ({〈0, 1〉} ∪ seq1( · , I )) | |
4 | nnuz 9385 | . . . . . . . 8 ⊢ ℕ = (ℤ≥‘1) | |
5 | dfn2 9014 | . . . . . . . 8 ⊢ ℕ = (ℕ0 ∖ {0}) | |
6 | 4, 5 | eqtr3i 2163 | . . . . . . 7 ⊢ (ℤ≥‘1) = (ℕ0 ∖ {0}) |
7 | 6 | reseq2i 4824 | . . . . . 6 ⊢ (seq1( · , I ) ↾ (ℤ≥‘1)) = (seq1( · , I ) ↾ (ℕ0 ∖ {0})) |
8 | eqid 2140 | . . . . . . . . . 10 ⊢ (ℤ≥‘1) = (ℤ≥‘1) | |
9 | 1zzd 9105 | . . . . . . . . . 10 ⊢ (⊤ → 1 ∈ ℤ) | |
10 | fvi 5486 | . . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (ℤ≥‘1) → ( I ‘𝑓) = 𝑓) | |
11 | 10 | eleq1d 2209 | . . . . . . . . . . . . 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 9361 | . . . . . . . . . . . 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 7771 | . . . . . . . . . . 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 10265 | . . . . . . . . 9 ⊢ (⊤ → seq1( · , I ):(ℤ≥‘1)⟶ℂ) |
19 | 18 | ffnd 5281 | . . . . . . . 8 ⊢ (⊤ → seq1( · , I ) Fn (ℤ≥‘1)) |
20 | 19 | mptru 1341 | . . . . . . 7 ⊢ seq1( · , I ) Fn (ℤ≥‘1) |
21 | fnresdm 5240 | . . . . . . 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 2163 | . . . . 5 ⊢ (seq1( · , I ) ↾ (ℕ0 ∖ {0})) = seq1( · , I ) |
24 | 23 | uneq2i 3232 | . . . 4 ⊢ ({〈0, 1〉} ∪ (seq1( · , I ) ↾ (ℕ0 ∖ {0}))) = ({〈0, 1〉} ∪ seq1( · , I )) |
25 | 3, 24 | eqtr4i 2164 | . . 3 ⊢ ! = ({〈0, 1〉} ∪ (seq1( · , I ) ↾ (ℕ0 ∖ {0}))) |
26 | 1, 2, 25 | fvsnun2 5626 | . 2 ⊢ (𝑁 ∈ (ℕ0 ∖ {0}) → (!‘𝑁) = (seq1( · , I )‘𝑁)) |
27 | 26, 5 | eleq2s 2235 | 1 ⊢ (𝑁 ∈ ℕ → (!‘𝑁) = (seq1( · , I )‘𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ⊤wtru 1333 ∈ wcel 1481 ∖ cdif 3073 ∪ cun 3074 {csn 3532 〈cop 3535 I cid 4218 ↾ cres 4549 Fn wfn 5126 ‘cfv 5131 (class class class)co 5782 ℂcc 7642 0cc0 7644 1c1 7645 · cmul 7649 ℕcn 8744 ℕ0cn0 9001 ℤ≥cuz 9350 seqcseq 10249 !cfa 10503 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-frec 6296 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-inn 8745 df-n0 9002 df-z 9079 df-uz 9351 df-seqfrec 10250 df-fac 10504 |
This theorem is referenced by: fac1 10507 facp1 10508 bcval5 10541 |
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