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Mirrors > Home > MPE Home > Th. List > faccl | Structured version Visualization version 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 6891 | . . 3 ⊢ (𝑗 = 0 → (!‘𝑗) = (!‘0)) | |
2 | 1 | eleq1d 2813 | . 2 ⊢ (𝑗 = 0 → ((!‘𝑗) ∈ ℕ ↔ (!‘0) ∈ ℕ)) |
3 | fveq2 6891 | . . 3 ⊢ (𝑗 = 𝑘 → (!‘𝑗) = (!‘𝑘)) | |
4 | 3 | eleq1d 2813 | . 2 ⊢ (𝑗 = 𝑘 → ((!‘𝑗) ∈ ℕ ↔ (!‘𝑘) ∈ ℕ)) |
5 | fveq2 6891 | . . 3 ⊢ (𝑗 = (𝑘 + 1) → (!‘𝑗) = (!‘(𝑘 + 1))) | |
6 | 5 | eleq1d 2813 | . 2 ⊢ (𝑗 = (𝑘 + 1) → ((!‘𝑗) ∈ ℕ ↔ (!‘(𝑘 + 1)) ∈ ℕ)) |
7 | fveq2 6891 | . . 3 ⊢ (𝑗 = 𝑁 → (!‘𝑗) = (!‘𝑁)) | |
8 | 7 | eleq1d 2813 | . 2 ⊢ (𝑗 = 𝑁 → ((!‘𝑗) ∈ ℕ ↔ (!‘𝑁) ∈ ℕ)) |
9 | fac0 14261 | . . 3 ⊢ (!‘0) = 1 | |
10 | 1nn 12247 | . . 3 ⊢ 1 ∈ ℕ | |
11 | 9, 10 | eqeltri 2824 | . 2 ⊢ (!‘0) ∈ ℕ |
12 | facp1 14263 | . . . . 5 ⊢ (𝑘 ∈ ℕ0 → (!‘(𝑘 + 1)) = ((!‘𝑘) · (𝑘 + 1))) | |
13 | 12 | adantl 481 | . . . 4 ⊢ (((!‘𝑘) ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (!‘(𝑘 + 1)) = ((!‘𝑘) · (𝑘 + 1))) |
14 | nn0p1nn 12535 | . . . . 5 ⊢ (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ) | |
15 | nnmulcl 12260 | . . . . 5 ⊢ (((!‘𝑘) ∈ ℕ ∧ (𝑘 + 1) ∈ ℕ) → ((!‘𝑘) · (𝑘 + 1)) ∈ ℕ) | |
16 | 14, 15 | sylan2 592 | . . . 4 ⊢ (((!‘𝑘) ∈ ℕ ∧ 𝑘 ∈ ℕ0) → ((!‘𝑘) · (𝑘 + 1)) ∈ ℕ) |
17 | 13, 16 | eqeltrd 2828 | . . 3 ⊢ (((!‘𝑘) ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (!‘(𝑘 + 1)) ∈ ℕ) |
18 | 17 | expcom 413 | . 2 ⊢ (𝑘 ∈ ℕ0 → ((!‘𝑘) ∈ ℕ → (!‘(𝑘 + 1)) ∈ ℕ)) |
19 | 2, 4, 6, 8, 11, 18 | nn0ind 12681 | 1 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ‘cfv 6542 (class class class)co 7414 0cc0 11132 1c1 11133 + caddc 11135 · cmul 11137 ℕcn 12236 ℕ0cn0 12496 !cfa 14258 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-n0 12497 df-z 12583 df-uz 12847 df-seq 13993 df-fac 14259 |
This theorem is referenced by: faccld 14269 facne0 14271 facdiv 14272 facndiv 14273 facwordi 14274 faclbnd 14275 faclbnd2 14276 faclbnd3 14277 faclbnd4lem1 14278 faclbnd5 14283 faclbnd6 14284 facubnd 14285 facavg 14286 bcrpcl 14293 bcn0 14295 bcm1k 14300 bcval5 14303 permnn 14311 4bc2eq6 14314 fallfacfac 16015 eftcl 16043 reeftcl 16044 eftabs 16045 ef0lem 16048 ege2le3 16060 efcj 16062 efaddlem 16063 effsumlt 16081 eflegeo 16091 ef01bndlem 16154 eirrlem 16174 prmfac1 16685 pcfac 16861 prmunb 16876 aaliou3lem7 26277 aaliou3lem9 26278 advlogexp 26582 wilth 26996 logfacrlim 27150 logexprlim 27151 bcmono 27203 vmadivsum 27408 subfacval2 34787 subfaclim 34788 subfacval3 34789 bcprod 35322 faclim2 35332 lcmineqlem18 41506 facp2 41599 fac2xp3 41663 factwoffsmonot 41666 bcccl 43748 bcc0 43749 bccp1k 43750 binomcxplemwb 43757 dvnxpaek 45302 wallispi2lem2 45432 stirlinglem2 45435 stirlinglem3 45436 stirlinglem4 45437 stirlinglem13 45446 stirlinglem14 45447 stirlinglem15 45448 stirlingr 45450 pgrple2abl 47401 |
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