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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 6672 | . . 3 ⊢ (𝑗 = 0 → (!‘𝑗) = (!‘0)) | |
2 | 1 | eleq1d 2899 | . 2 ⊢ (𝑗 = 0 → ((!‘𝑗) ∈ ℕ ↔ (!‘0) ∈ ℕ)) |
3 | fveq2 6672 | . . 3 ⊢ (𝑗 = 𝑘 → (!‘𝑗) = (!‘𝑘)) | |
4 | 3 | eleq1d 2899 | . 2 ⊢ (𝑗 = 𝑘 → ((!‘𝑗) ∈ ℕ ↔ (!‘𝑘) ∈ ℕ)) |
5 | fveq2 6672 | . . 3 ⊢ (𝑗 = (𝑘 + 1) → (!‘𝑗) = (!‘(𝑘 + 1))) | |
6 | 5 | eleq1d 2899 | . 2 ⊢ (𝑗 = (𝑘 + 1) → ((!‘𝑗) ∈ ℕ ↔ (!‘(𝑘 + 1)) ∈ ℕ)) |
7 | fveq2 6672 | . . 3 ⊢ (𝑗 = 𝑁 → (!‘𝑗) = (!‘𝑁)) | |
8 | 7 | eleq1d 2899 | . 2 ⊢ (𝑗 = 𝑁 → ((!‘𝑗) ∈ ℕ ↔ (!‘𝑁) ∈ ℕ)) |
9 | fac0 13639 | . . 3 ⊢ (!‘0) = 1 | |
10 | 1nn 11651 | . . 3 ⊢ 1 ∈ ℕ | |
11 | 9, 10 | eqeltri 2911 | . 2 ⊢ (!‘0) ∈ ℕ |
12 | facp1 13641 | . . . . 5 ⊢ (𝑘 ∈ ℕ0 → (!‘(𝑘 + 1)) = ((!‘𝑘) · (𝑘 + 1))) | |
13 | 12 | adantl 484 | . . . 4 ⊢ (((!‘𝑘) ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (!‘(𝑘 + 1)) = ((!‘𝑘) · (𝑘 + 1))) |
14 | nn0p1nn 11939 | . . . . 5 ⊢ (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ) | |
15 | nnmulcl 11664 | . . . . 5 ⊢ (((!‘𝑘) ∈ ℕ ∧ (𝑘 + 1) ∈ ℕ) → ((!‘𝑘) · (𝑘 + 1)) ∈ ℕ) | |
16 | 14, 15 | sylan2 594 | . . . 4 ⊢ (((!‘𝑘) ∈ ℕ ∧ 𝑘 ∈ ℕ0) → ((!‘𝑘) · (𝑘 + 1)) ∈ ℕ) |
17 | 13, 16 | eqeltrd 2915 | . . 3 ⊢ (((!‘𝑘) ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (!‘(𝑘 + 1)) ∈ ℕ) |
18 | 17 | expcom 416 | . 2 ⊢ (𝑘 ∈ ℕ0 → ((!‘𝑘) ∈ ℕ → (!‘(𝑘 + 1)) ∈ ℕ)) |
19 | 2, 4, 6, 8, 11, 18 | nn0ind 12080 | 1 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 0cc0 10539 1c1 10540 + caddc 10542 · cmul 10544 ℕcn 11640 ℕ0cn0 11900 !cfa 13636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-seq 13373 df-fac 13637 |
This theorem is referenced by: faccld 13647 facne0 13649 facdiv 13650 facndiv 13651 facwordi 13652 faclbnd 13653 faclbnd2 13654 faclbnd3 13655 faclbnd4lem1 13656 faclbnd5 13661 faclbnd6 13662 facubnd 13663 facavg 13664 bcrpcl 13671 bcn0 13673 bcm1k 13678 bcval5 13681 permnn 13689 4bc2eq6 13692 fallfacfac 15401 eftcl 15429 reeftcl 15430 eftabs 15431 ef0lem 15434 ege2le3 15445 efcj 15447 efaddlem 15448 effsumlt 15466 eflegeo 15476 ef01bndlem 15539 eirrlem 15559 prmfac1 16065 pcfac 16237 prmunb 16252 aaliou3lem7 24940 aaliou3lem9 24941 advlogexp 25240 wilth 25650 logfacrlim 25802 logexprlim 25803 bcmono 25855 vmadivsum 26060 subfacval2 32436 subfaclim 32437 subfacval3 32438 bcprod 32972 faclim2 32982 fac2xp3 39101 facp2 39102 factwoffsmonot 39105 bcccl 40678 bcc0 40679 bccp1k 40680 binomcxplemwb 40687 dvnxpaek 42234 wallispi2lem2 42364 stirlinglem2 42367 stirlinglem3 42368 stirlinglem4 42369 stirlinglem13 42378 stirlinglem14 42379 stirlinglem15 42380 stirlingr 42382 pgrple2abl 44420 |
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