ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  prmfac1 Unicode version

Theorem prmfac1 11757
Description: The factorial of a number only contains primes less than the base. (Contributed by Mario Carneiro, 6-Mar-2014.)
Assertion
Ref Expression
prmfac1  |-  ( ( N  e.  NN0  /\  P  e.  Prime  /\  P  ||  ( ! `  N
) )  ->  P  <_  N )

Proof of Theorem prmfac1
Dummy variables  x  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5389 . . . . . 6  |-  ( x  =  0  ->  ( ! `  x )  =  ( ! ` 
0 ) )
21breq2d 3911 . . . . 5  |-  ( x  =  0  ->  ( P  ||  ( ! `  x )  <->  P  ||  ( ! `  0 )
) )
3 breq2 3903 . . . . 5  |-  ( x  =  0  ->  ( P  <_  x  <->  P  <_  0 ) )
42, 3imbi12d 233 . . . 4  |-  ( x  =  0  ->  (
( P  ||  ( ! `  x )  ->  P  <_  x )  <->  ( P  ||  ( ! `
 0 )  ->  P  <_  0 ) ) )
54imbi2d 229 . . 3  |-  ( x  =  0  ->  (
( P  e.  Prime  -> 
( P  ||  ( ! `  x )  ->  P  <_  x )
)  <->  ( P  e. 
Prime  ->  ( P  ||  ( ! `  0 )  ->  P  <_  0
) ) ) )
6 fveq2 5389 . . . . . 6  |-  ( x  =  k  ->  ( ! `  x )  =  ( ! `  k ) )
76breq2d 3911 . . . . 5  |-  ( x  =  k  ->  ( P  ||  ( ! `  x )  <->  P  ||  ( ! `  k )
) )
8 breq2 3903 . . . . 5  |-  ( x  =  k  ->  ( P  <_  x  <->  P  <_  k ) )
97, 8imbi12d 233 . . . 4  |-  ( x  =  k  ->  (
( P  ||  ( ! `  x )  ->  P  <_  x )  <->  ( P  ||  ( ! `
 k )  ->  P  <_  k ) ) )
109imbi2d 229 . . 3  |-  ( x  =  k  ->  (
( P  e.  Prime  -> 
( P  ||  ( ! `  x )  ->  P  <_  x )
)  <->  ( P  e. 
Prime  ->  ( P  ||  ( ! `  k )  ->  P  <_  k
) ) ) )
11 fveq2 5389 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( ! `  x )  =  ( ! `  ( k  +  1 ) ) )
1211breq2d 3911 . . . . 5  |-  ( x  =  ( k  +  1 )  ->  ( P  ||  ( ! `  x )  <->  P  ||  ( ! `  ( k  +  1 ) ) ) )
13 breq2 3903 . . . . 5  |-  ( x  =  ( k  +  1 )  ->  ( P  <_  x  <->  P  <_  ( k  +  1 ) ) )
1412, 13imbi12d 233 . . . 4  |-  ( x  =  ( k  +  1 )  ->  (
( P  ||  ( ! `  x )  ->  P  <_  x )  <->  ( P  ||  ( ! `
 ( k  +  1 ) )  ->  P  <_  ( k  +  1 ) ) ) )
1514imbi2d 229 . . 3  |-  ( x  =  ( k  +  1 )  ->  (
( P  e.  Prime  -> 
( P  ||  ( ! `  x )  ->  P  <_  x )
)  <->  ( P  e. 
Prime  ->  ( P  ||  ( ! `  ( k  +  1 ) )  ->  P  <_  (
k  +  1 ) ) ) ) )
16 fveq2 5389 . . . . . 6  |-  ( x  =  N  ->  ( ! `  x )  =  ( ! `  N ) )
1716breq2d 3911 . . . . 5  |-  ( x  =  N  ->  ( P  ||  ( ! `  x )  <->  P  ||  ( ! `  N )
) )
18 breq2 3903 . . . . 5  |-  ( x  =  N  ->  ( P  <_  x  <->  P  <_  N ) )
1917, 18imbi12d 233 . . . 4  |-  ( x  =  N  ->  (
( P  ||  ( ! `  x )  ->  P  <_  x )  <->  ( P  ||  ( ! `
 N )  ->  P  <_  N ) ) )
2019imbi2d 229 . . 3  |-  ( x  =  N  ->  (
( P  e.  Prime  -> 
( P  ||  ( ! `  x )  ->  P  <_  x )
)  <->  ( P  e. 
Prime  ->  ( P  ||  ( ! `  N )  ->  P  <_  N
) ) ) )
21 fac0 10442 . . . . 5  |-  ( ! `
 0 )  =  1
2221breq2i 3907 . . . 4  |-  ( P 
||  ( ! ` 
0 )  <->  P  ||  1
)
23 nprmdvds1 11747 . . . . 5  |-  ( P  e.  Prime  ->  -.  P  ||  1 )
2423pm2.21d 593 . . . 4  |-  ( P  e.  Prime  ->  ( P 
||  1  ->  P  <_  0 ) )
2522, 24syl5bi 151 . . 3  |-  ( P  e.  Prime  ->  ( P 
||  ( ! ` 
0 )  ->  P  <_  0 ) )
26 facp1 10444 . . . . . . . . . 10  |-  ( k  e.  NN0  ->  ( ! `
 ( k  +  1 ) )  =  ( ( ! `  k )  x.  (
k  +  1 ) ) )
2726adantr 274 . . . . . . . . 9  |-  ( ( k  e.  NN0  /\  P  e.  Prime )  -> 
( ! `  (
k  +  1 ) )  =  ( ( ! `  k )  x.  ( k  +  1 ) ) )
2827breq2d 3911 . . . . . . . 8  |-  ( ( k  e.  NN0  /\  P  e.  Prime )  -> 
( P  ||  ( ! `  ( k  +  1 ) )  <-> 
P  ||  ( ( ! `  k )  x.  ( k  +  1 ) ) ) )
29 simpr 109 . . . . . . . . 9  |-  ( ( k  e.  NN0  /\  P  e.  Prime )  ->  P  e.  Prime )
30 faccl 10449 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
3130adantr 274 . . . . . . . . . 10  |-  ( ( k  e.  NN0  /\  P  e.  Prime )  -> 
( ! `  k
)  e.  NN )
3231nnzd 9140 . . . . . . . . 9  |-  ( ( k  e.  NN0  /\  P  e.  Prime )  -> 
( ! `  k
)  e.  ZZ )
33 nn0p1nn 8984 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  ( k  +  1 )  e.  NN )
3433adantr 274 . . . . . . . . . 10  |-  ( ( k  e.  NN0  /\  P  e.  Prime )  -> 
( k  +  1 )  e.  NN )
3534nnzd 9140 . . . . . . . . 9  |-  ( ( k  e.  NN0  /\  P  e.  Prime )  -> 
( k  +  1 )  e.  ZZ )
36 euclemma 11751 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( ! `  k )  e.  ZZ  /\  ( k  +  1 )  e.  ZZ )  ->  ( P  ||  ( ( ! `
 k )  x.  ( k  +  1 ) )  <->  ( P  ||  ( ! `  k
)  \/  P  ||  ( k  +  1 ) ) ) )
3729, 32, 35, 36syl3anc 1201 . . . . . . . 8  |-  ( ( k  e.  NN0  /\  P  e.  Prime )  -> 
( P  ||  (
( ! `  k
)  x.  ( k  +  1 ) )  <-> 
( P  ||  ( ! `  k )  \/  P  ||  ( k  +  1 ) ) ) )
3828, 37bitrd 187 . . . . . . 7  |-  ( ( k  e.  NN0  /\  P  e.  Prime )  -> 
( P  ||  ( ! `  ( k  +  1 ) )  <-> 
( P  ||  ( ! `  k )  \/  P  ||  ( k  +  1 ) ) ) )
39 nn0re 8954 . . . . . . . . . . . . 13  |-  ( k  e.  NN0  ->  k  e.  RR )
4039adantr 274 . . . . . . . . . . . 12  |-  ( ( k  e.  NN0  /\  P  e.  Prime )  -> 
k  e.  RR )
4140lep1d 8657 . . . . . . . . . . 11  |-  ( ( k  e.  NN0  /\  P  e.  Prime )  -> 
k  <_  ( k  +  1 ) )
42 prmz 11719 . . . . . . . . . . . . . 14  |-  ( P  e.  Prime  ->  P  e.  ZZ )
4342adantl 275 . . . . . . . . . . . . 13  |-  ( ( k  e.  NN0  /\  P  e.  Prime )  ->  P  e.  ZZ )
4443zred 9141 . . . . . . . . . . . 12  |-  ( ( k  e.  NN0  /\  P  e.  Prime )  ->  P  e.  RR )
4534nnred 8701 . . . . . . . . . . . 12  |-  ( ( k  e.  NN0  /\  P  e.  Prime )  -> 
( k  +  1 )  e.  RR )
46 letr 7815 . . . . . . . . . . . 12  |-  ( ( P  e.  RR  /\  k  e.  RR  /\  (
k  +  1 )  e.  RR )  -> 
( ( P  <_ 
k  /\  k  <_  ( k  +  1 ) )  ->  P  <_  ( k  +  1 ) ) )
4744, 40, 45, 46syl3anc 1201 . . . . . . . . . . 11  |-  ( ( k  e.  NN0  /\  P  e.  Prime )  -> 
( ( P  <_ 
k  /\  k  <_  ( k  +  1 ) )  ->  P  <_  ( k  +  1 ) ) )
4841, 47mpan2d 424 . . . . . . . . . 10  |-  ( ( k  e.  NN0  /\  P  e.  Prime )  -> 
( P  <_  k  ->  P  <_  ( k  +  1 ) ) )
4948imim2d 54 . . . . . . . . 9  |-  ( ( k  e.  NN0  /\  P  e.  Prime )  -> 
( ( P  ||  ( ! `  k )  ->  P  <_  k
)  ->  ( P  ||  ( ! `  k
)  ->  P  <_  ( k  +  1 ) ) ) )
5049com23 78 . . . . . . . 8  |-  ( ( k  e.  NN0  /\  P  e.  Prime )  -> 
( P  ||  ( ! `  k )  ->  ( ( P  ||  ( ! `  k )  ->  P  <_  k
)  ->  P  <_  ( k  +  1 ) ) ) )
51 dvdsle 11469 . . . . . . . . . 10  |-  ( ( P  e.  ZZ  /\  ( k  +  1 )  e.  NN )  ->  ( P  ||  ( k  +  1 )  ->  P  <_  ( k  +  1 ) ) )
5243, 34, 51syl2anc 408 . . . . . . . . 9  |-  ( ( k  e.  NN0  /\  P  e.  Prime )  -> 
( P  ||  (
k  +  1 )  ->  P  <_  (
k  +  1 ) ) )
5352a1dd 48 . . . . . . . 8  |-  ( ( k  e.  NN0  /\  P  e.  Prime )  -> 
( P  ||  (
k  +  1 )  ->  ( ( P 
||  ( ! `  k )  ->  P  <_  k )  ->  P  <_  ( k  +  1 ) ) ) )
5450, 53jaod 691 . . . . . . 7  |-  ( ( k  e.  NN0  /\  P  e.  Prime )  -> 
( ( P  ||  ( ! `  k )  \/  P  ||  (
k  +  1 ) )  ->  ( ( P  ||  ( ! `  k )  ->  P  <_  k )  ->  P  <_  ( k  +  1 ) ) ) )
5538, 54sylbid 149 . . . . . 6  |-  ( ( k  e.  NN0  /\  P  e.  Prime )  -> 
( P  ||  ( ! `  ( k  +  1 ) )  ->  ( ( P 
||  ( ! `  k )  ->  P  <_  k )  ->  P  <_  ( k  +  1 ) ) ) )
5655com23 78 . . . . 5  |-  ( ( k  e.  NN0  /\  P  e.  Prime )  -> 
( ( P  ||  ( ! `  k )  ->  P  <_  k
)  ->  ( P  ||  ( ! `  (
k  +  1 ) )  ->  P  <_  ( k  +  1 ) ) ) )
5756ex 114 . . . 4  |-  ( k  e.  NN0  ->  ( P  e.  Prime  ->  ( ( P  ||  ( ! `
 k )  ->  P  <_  k )  -> 
( P  ||  ( ! `  ( k  +  1 ) )  ->  P  <_  (
k  +  1 ) ) ) ) )
5857a2d 26 . . 3  |-  ( k  e.  NN0  ->  ( ( P  e.  Prime  ->  ( P  ||  ( ! `
 k )  ->  P  <_  k ) )  ->  ( P  e. 
Prime  ->  ( P  ||  ( ! `  ( k  +  1 ) )  ->  P  <_  (
k  +  1 ) ) ) ) )
595, 10, 15, 20, 25, 58nn0ind 9133 . 2  |-  ( N  e.  NN0  ->  ( P  e.  Prime  ->  ( P 
||  ( ! `  N )  ->  P  <_  N ) ) )
60593imp 1160 1  |-  ( ( N  e.  NN0  /\  P  e.  Prime  /\  P  ||  ( ! `  N
) )  ->  P  <_  N )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 682    /\ w3a 947    = wceq 1316    e. wcel 1465   class class class wbr 3899   ` cfv 5093  (class class class)co 5742   RRcr 7587   0cc0 7588   1c1 7589    + caddc 7591    x. cmul 7593    <_ cle 7769   NNcn 8688   NN0cn0 8945   ZZcz 9022   !cfa 10439    || cdvds 11420   Primecprime 11715
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707  ax-caucvg 7708
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-1o 6281  df-2o 6282  df-er 6397  df-en 6603  df-sup 6839  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312  df-div 8401  df-inn 8689  df-2 8747  df-3 8748  df-4 8749  df-n0 8946  df-z 9023  df-uz 9295  df-q 9380  df-rp 9410  df-fz 9759  df-fzo 9888  df-fl 10011  df-mod 10064  df-seqfrec 10187  df-exp 10261  df-fac 10440  df-cj 10582  df-re 10583  df-im 10584  df-rsqrt 10738  df-abs 10739  df-dvds 11421  df-gcd 11563  df-prm 11716
This theorem is referenced by:  prmndvdsfaclt  11761
  Copyright terms: Public domain W3C validator