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Mirrors > Home > MPE Home > Th. List > faccld | Structured version Visualization version GIF version |
Description: Closure of the factorial function, deduction version of faccl 13070. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
faccld.1 | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
Ref | Expression |
---|---|
faccld | ⊢ (𝜑 → (!‘𝑁) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | faccld.1 | . 2 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
2 | faccl 13070 | . 2 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℕ) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (!‘𝑁) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 ‘cfv 5888 ℕcn 11020 ℕ0cn0 11292 !cfa 13060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-seq 12802 df-fac 13061 |
This theorem is referenced by: facwordi 13076 faclbnd 13077 bcp1n 13103 2expltfac 15799 tayl0 24116 wilthimp 24798 gausslemma2dlem0c 25083 gausslemma2dlem6 25097 gausslemma2dlem7 25098 mccllem 39829 dvnprodlem2 40162 etransclem14 40465 etransclem15 40466 etransclem20 40471 etransclem21 40472 etransclem22 40473 etransclem23 40474 etransclem24 40475 etransclem25 40476 etransclem28 40479 etransclem31 40482 etransclem32 40483 etransclem33 40484 etransclem34 40485 etransclem35 40486 etransclem37 40488 etransclem38 40489 etransclem41 40492 etransclem44 40495 etransclem45 40496 etransclem47 40498 etransclem48 40499 |
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