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Mirrors > Home > ILE Home > Th. List > faccl | GIF version |
Description: Closure of the factorial function. (Contributed by NM, 2-Dec-2004.) |
Ref | Expression |
---|---|
faccl | ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 5514 | . . 3 ⊢ (𝑗 = 0 → (!‘𝑗) = (!‘0)) | |
2 | 1 | eleq1d 2246 | . 2 ⊢ (𝑗 = 0 → ((!‘𝑗) ∈ ℕ ↔ (!‘0) ∈ ℕ)) |
3 | fveq2 5514 | . . 3 ⊢ (𝑗 = 𝑘 → (!‘𝑗) = (!‘𝑘)) | |
4 | 3 | eleq1d 2246 | . 2 ⊢ (𝑗 = 𝑘 → ((!‘𝑗) ∈ ℕ ↔ (!‘𝑘) ∈ ℕ)) |
5 | fveq2 5514 | . . 3 ⊢ (𝑗 = (𝑘 + 1) → (!‘𝑗) = (!‘(𝑘 + 1))) | |
6 | 5 | eleq1d 2246 | . 2 ⊢ (𝑗 = (𝑘 + 1) → ((!‘𝑗) ∈ ℕ ↔ (!‘(𝑘 + 1)) ∈ ℕ)) |
7 | fveq2 5514 | . . 3 ⊢ (𝑗 = 𝑁 → (!‘𝑗) = (!‘𝑁)) | |
8 | 7 | eleq1d 2246 | . 2 ⊢ (𝑗 = 𝑁 → ((!‘𝑗) ∈ ℕ ↔ (!‘𝑁) ∈ ℕ)) |
9 | fac0 10701 | . . 3 ⊢ (!‘0) = 1 | |
10 | 1nn 8926 | . . 3 ⊢ 1 ∈ ℕ | |
11 | 9, 10 | eqeltri 2250 | . 2 ⊢ (!‘0) ∈ ℕ |
12 | facp1 10703 | . . . . 5 ⊢ (𝑘 ∈ ℕ0 → (!‘(𝑘 + 1)) = ((!‘𝑘) · (𝑘 + 1))) | |
13 | 12 | adantl 277 | . . . 4 ⊢ (((!‘𝑘) ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (!‘(𝑘 + 1)) = ((!‘𝑘) · (𝑘 + 1))) |
14 | nn0p1nn 9211 | . . . . 5 ⊢ (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ) | |
15 | nnmulcl 8936 | . . . . 5 ⊢ (((!‘𝑘) ∈ ℕ ∧ (𝑘 + 1) ∈ ℕ) → ((!‘𝑘) · (𝑘 + 1)) ∈ ℕ) | |
16 | 14, 15 | sylan2 286 | . . . 4 ⊢ (((!‘𝑘) ∈ ℕ ∧ 𝑘 ∈ ℕ0) → ((!‘𝑘) · (𝑘 + 1)) ∈ ℕ) |
17 | 13, 16 | eqeltrd 2254 | . . 3 ⊢ (((!‘𝑘) ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (!‘(𝑘 + 1)) ∈ ℕ) |
18 | 17 | expcom 116 | . 2 ⊢ (𝑘 ∈ ℕ0 → ((!‘𝑘) ∈ ℕ → (!‘(𝑘 + 1)) ∈ ℕ)) |
19 | 2, 4, 6, 8, 11, 18 | nn0ind 9363 | 1 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℕ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ‘cfv 5215 (class class class)co 5872 0cc0 7808 1c1 7809 + caddc 7811 · cmul 7813 ℕcn 8915 ℕ0cn0 9172 !cfa 10698 |
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 4117 ax-sep 4120 ax-nul 4128 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-iinf 4586 ax-cnex 7899 ax-resscn 7900 ax-1cn 7901 ax-1re 7902 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-addcom 7908 ax-mulcom 7909 ax-addass 7910 ax-mulass 7911 ax-distr 7912 ax-i2m1 7913 ax-0lt1 7914 ax-1rid 7915 ax-0id 7916 ax-rnegex 7917 ax-cnre 7919 ax-pre-ltirr 7920 ax-pre-ltwlin 7921 ax-pre-lttrn 7922 ax-pre-ltadd 7924 |
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 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4003 df-opab 4064 df-mpt 4065 df-tr 4101 df-id 4292 df-iord 4365 df-on 4367 df-ilim 4368 df-suc 4370 df-iom 4589 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5177 df-fun 5217 df-fn 5218 df-f 5219 df-f1 5220 df-fo 5221 df-f1o 5222 df-fv 5223 df-riota 5828 df-ov 5875 df-oprab 5876 df-mpo 5877 df-1st 6138 df-2nd 6139 df-recs 6303 df-frec 6389 df-pnf 7990 df-mnf 7991 df-xr 7992 df-ltxr 7993 df-le 7994 df-sub 8126 df-neg 8127 df-inn 8916 df-n0 9173 df-z 9250 df-uz 9525 df-seqfrec 10441 df-fac 10699 |
This theorem is referenced by: faccld 10709 facne0 10710 facdiv 10711 facndiv 10712 facwordi 10713 faclbnd 10714 faclbnd2 10715 faclbnd3 10716 faclbnd6 10717 facubnd 10718 facavg 10719 bcrpcl 10726 bcn0 10728 bcm1k 10733 permnn 10744 4bc2eq6 10747 eftcl 11655 reeftcl 11656 eftabs 11657 ef0lem 11661 ege2le3 11672 efcj 11674 efaddlem 11675 effsumlt 11693 eflegeo 11702 ef01bndlem 11757 eirraplem 11777 dvdsfac 11858 prmfac1 12144 pcfac 12340 prmunb 12352 |
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