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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 13493. (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 13493 | . 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 2081 ‘cfv 6225 ℕcn 11486 ℕ0cn0 11745 !cfa 13483 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-nn 11487 df-n0 11746 df-z 11830 df-uz 12094 df-seq 13220 df-fac 13484 |
This theorem is referenced by: facmapnn 13495 facwordi 13499 faclbnd 13500 faclbnd6 13509 facavg 13511 bcrpcl 13518 bccmpl 13519 bcn1 13523 bcm1k 13525 bcp1n 13526 bcval5 13528 permnn 13536 hashf1 13663 hashfac 13664 bcfallfac 15231 efcllem 15264 eftlub 15295 eirrlem 15390 dvdsfac 15509 lcmflefac 15821 pcbc 16065 infpnlem1 16075 infpnlem2 16076 prmgaplem1 16214 prmgaplem2 16215 2expltfac 16255 gexcl3 18442 aaliou3lem1 24614 aaliou3lem2 24615 aaliou3lem3 24616 aaliou3lem8 24617 aaliou3lem5 24619 aaliou3lem6 24620 taylfvallem1 24628 tayl0 24633 taylply2 24639 taylply 24640 dvtaylp 24641 taylthlem2 24645 advlogexp 24919 birthdaylem2 25212 wilthlem3 25329 wilthimp 25331 chtublem 25469 logfacubnd 25479 logfaclbnd 25480 logfacbnd3 25481 logexprlim 25483 bposlem3 25544 gausslemma2dlem0c 25616 gausslemma2dlem6 25630 gausslemma2dlem7 25631 prmdvdsbc 30216 mccllem 41439 dvnprodlem2 41793 etransclem14 42095 etransclem15 42096 etransclem20 42101 etransclem21 42102 etransclem22 42103 etransclem23 42104 etransclem24 42105 etransclem25 42106 etransclem28 42109 etransclem31 42112 etransclem32 42113 etransclem33 42114 etransclem34 42115 etransclem35 42116 etransclem37 42118 etransclem38 42119 etransclem41 42122 etransclem44 42125 etransclem45 42126 etransclem47 42128 etransclem48 42129 |
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