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Mirrors > Home > ILE Home > Th. List > faccl | Unicode version |
Description: Closure of the factorial function. (Contributed by NM, 2-Dec-2004.) |
Ref | Expression |
---|---|
faccl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 5481 | . . 3 | |
2 | 1 | eleq1d 2233 | . 2 |
3 | fveq2 5481 | . . 3 | |
4 | 3 | eleq1d 2233 | . 2 |
5 | fveq2 5481 | . . 3 | |
6 | 5 | eleq1d 2233 | . 2 |
7 | fveq2 5481 | . . 3 | |
8 | 7 | eleq1d 2233 | . 2 |
9 | fac0 10631 | . . 3 | |
10 | 1nn 8860 | . . 3 | |
11 | 9, 10 | eqeltri 2237 | . 2 |
12 | facp1 10633 | . . . . 5 | |
13 | 12 | adantl 275 | . . . 4 |
14 | nn0p1nn 9145 | . . . . 5 | |
15 | nnmulcl 8870 | . . . . 5 | |
16 | 14, 15 | sylan2 284 | . . . 4 |
17 | 13, 16 | eqeltrd 2241 | . . 3 |
18 | 17 | expcom 115 | . 2 |
19 | 2, 4, 6, 8, 11, 18 | nn0ind 9297 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 wcel 2135 cfv 5183 (class class class)co 5837 cc0 7745 c1 7746 caddc 7748 cmul 7750 cn 8849 cn0 9106 cfa 10628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4092 ax-sep 4095 ax-nul 4103 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-setind 4509 ax-iinf 4560 ax-cnex 7836 ax-resscn 7837 ax-1cn 7838 ax-1re 7839 ax-icn 7840 ax-addcl 7841 ax-addrcl 7842 ax-mulcl 7843 ax-addcom 7845 ax-mulcom 7846 ax-addass 7847 ax-mulass 7848 ax-distr 7849 ax-i2m1 7850 ax-0lt1 7851 ax-1rid 7852 ax-0id 7853 ax-rnegex 7854 ax-cnre 7856 ax-pre-ltirr 7857 ax-pre-ltwlin 7858 ax-pre-lttrn 7859 ax-pre-ltadd 7861 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2724 df-sbc 2948 df-csb 3042 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-nul 3406 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-int 3820 df-iun 3863 df-br 3978 df-opab 4039 df-mpt 4040 df-tr 4076 df-id 4266 df-iord 4339 df-on 4341 df-ilim 4342 df-suc 4344 df-iom 4563 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-rn 4610 df-res 4611 df-ima 4612 df-iota 5148 df-fun 5185 df-fn 5186 df-f 5187 df-f1 5188 df-fo 5189 df-f1o 5190 df-fv 5191 df-riota 5793 df-ov 5840 df-oprab 5841 df-mpo 5842 df-1st 6101 df-2nd 6102 df-recs 6265 df-frec 6351 df-pnf 7927 df-mnf 7928 df-xr 7929 df-ltxr 7930 df-le 7931 df-sub 8063 df-neg 8064 df-inn 8850 df-n0 9107 df-z 9184 df-uz 9459 df-seqfrec 10372 df-fac 10629 |
This theorem is referenced by: faccld 10639 facne0 10640 facdiv 10641 facndiv 10642 facwordi 10643 faclbnd 10644 faclbnd2 10645 faclbnd3 10646 faclbnd6 10647 facubnd 10648 facavg 10649 bcrpcl 10656 bcn0 10658 bcm1k 10663 permnn 10674 4bc2eq6 10677 eftcl 11585 reeftcl 11586 eftabs 11587 ef0lem 11591 ege2le3 11602 efcj 11604 efaddlem 11605 effsumlt 11623 eflegeo 11632 ef01bndlem 11687 eirraplem 11707 dvdsfac 11787 prmfac1 12073 pcfac 12269 prmunb 12281 |
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