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Theorem exprmfct 13801
Description: Every integer greater than or equal to 2 has a prime factor. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 20-Jun-2015.)
Assertion
Ref Expression
exprmfct  |-  ( N  e.  ( ZZ>= `  2
)  ->  E. p  e.  Prime  p  ||  N
)
Distinct variable group:    N, p

Proof of Theorem exprmfct
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eluz2b2 10932 . . 3  |-  ( N  e.  ( ZZ>= `  2
)  <->  ( N  e.  NN  /\  1  < 
N ) )
21simplbi 460 . 2  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  e.  NN )
3 eleq1 2503 . . . 4  |-  ( x  =  1  ->  (
x  e.  ( ZZ>= ` 
2 )  <->  1  e.  ( ZZ>= `  2 )
) )
43imbi1d 317 . . 3  |-  ( x  =  1  ->  (
( x  e.  (
ZZ>= `  2 )  ->  E. p  e.  Prime  p 
||  x )  <->  ( 1  e.  ( ZZ>= `  2
)  ->  E. p  e.  Prime  p  ||  x
) ) )
5 eleq1 2503 . . . 4  |-  ( x  =  y  ->  (
x  e.  ( ZZ>= ` 
2 )  <->  y  e.  ( ZZ>= `  2 )
) )
6 breq2 4301 . . . . 5  |-  ( x  =  y  ->  (
p  ||  x  <->  p  ||  y
) )
76rexbidv 2741 . . . 4  |-  ( x  =  y  ->  ( E. p  e.  Prime  p 
||  x  <->  E. p  e.  Prime  p  ||  y
) )
85, 7imbi12d 320 . . 3  |-  ( x  =  y  ->  (
( x  e.  (
ZZ>= `  2 )  ->  E. p  e.  Prime  p 
||  x )  <->  ( y  e.  ( ZZ>= `  2 )  ->  E. p  e.  Prime  p 
||  y ) ) )
9 eleq1 2503 . . . 4  |-  ( x  =  z  ->  (
x  e.  ( ZZ>= ` 
2 )  <->  z  e.  ( ZZ>= `  2 )
) )
10 breq2 4301 . . . . 5  |-  ( x  =  z  ->  (
p  ||  x  <->  p  ||  z
) )
1110rexbidv 2741 . . . 4  |-  ( x  =  z  ->  ( E. p  e.  Prime  p 
||  x  <->  E. p  e.  Prime  p  ||  z
) )
129, 11imbi12d 320 . . 3  |-  ( x  =  z  ->  (
( x  e.  (
ZZ>= `  2 )  ->  E. p  e.  Prime  p 
||  x )  <->  ( z  e.  ( ZZ>= `  2 )  ->  E. p  e.  Prime  p 
||  z ) ) )
13 eleq1 2503 . . . 4  |-  ( x  =  ( y  x.  z )  ->  (
x  e.  ( ZZ>= ` 
2 )  <->  ( y  x.  z )  e.  (
ZZ>= `  2 ) ) )
14 breq2 4301 . . . . 5  |-  ( x  =  ( y  x.  z )  ->  (
p  ||  x  <->  p  ||  (
y  x.  z ) ) )
1514rexbidv 2741 . . . 4  |-  ( x  =  ( y  x.  z )  ->  ( E. p  e.  Prime  p 
||  x  <->  E. p  e.  Prime  p  ||  (
y  x.  z ) ) )
1613, 15imbi12d 320 . . 3  |-  ( x  =  ( y  x.  z )  ->  (
( x  e.  (
ZZ>= `  2 )  ->  E. p  e.  Prime  p 
||  x )  <->  ( (
y  x.  z )  e.  ( ZZ>= `  2
)  ->  E. p  e.  Prime  p  ||  (
y  x.  z ) ) ) )
17 eleq1 2503 . . . 4  |-  ( x  =  N  ->  (
x  e.  ( ZZ>= ` 
2 )  <->  N  e.  ( ZZ>= `  2 )
) )
18 breq2 4301 . . . . 5  |-  ( x  =  N  ->  (
p  ||  x  <->  p  ||  N
) )
1918rexbidv 2741 . . . 4  |-  ( x  =  N  ->  ( E. p  e.  Prime  p 
||  x  <->  E. p  e.  Prime  p  ||  N
) )
2017, 19imbi12d 320 . . 3  |-  ( x  =  N  ->  (
( x  e.  (
ZZ>= `  2 )  ->  E. p  e.  Prime  p 
||  x )  <->  ( N  e.  ( ZZ>= `  2 )  ->  E. p  e.  Prime  p 
||  N ) ) )
21 1m1e0 10395 . . . . 5  |-  ( 1  -  1 )  =  0
22 uz2m1nn 10934 . . . . 5  |-  ( 1  e.  ( ZZ>= `  2
)  ->  ( 1  -  1 )  e.  NN )
2321, 22syl5eqelr 2528 . . . 4  |-  ( 1  e.  ( ZZ>= `  2
)  ->  0  e.  NN )
24 0nnn 10358 . . . . 5  |-  -.  0  e.  NN
2524pm2.21i 131 . . . 4  |-  ( 0  e.  NN  ->  E. p  e.  Prime  p  ||  x
)
2623, 25syl 16 . . 3  |-  ( 1  e.  ( ZZ>= `  2
)  ->  E. p  e.  Prime  p  ||  x
)
27 prmz 13772 . . . . . 6  |-  ( x  e.  Prime  ->  x  e.  ZZ )
28 iddvds 13551 . . . . . 6  |-  ( x  e.  ZZ  ->  x  ||  x )
2927, 28syl 16 . . . . 5  |-  ( x  e.  Prime  ->  x  ||  x )
30 breq1 4300 . . . . . 6  |-  ( p  =  x  ->  (
p  ||  x  <->  x  ||  x
) )
3130rspcev 3078 . . . . 5  |-  ( ( x  e.  Prime  /\  x  ||  x )  ->  E. p  e.  Prime  p  ||  x
)
3229, 31mpdan 668 . . . 4  |-  ( x  e.  Prime  ->  E. p  e.  Prime  p  ||  x
)
3332a1d 25 . . 3  |-  ( x  e.  Prime  ->  ( x  e.  ( ZZ>= `  2
)  ->  E. p  e.  Prime  p  ||  x
) )
34 simpl 457 . . . . . 6  |-  ( ( y  e.  ( ZZ>= ` 
2 )  /\  z  e.  ( ZZ>= `  2 )
)  ->  y  e.  ( ZZ>= `  2 )
)
35 eluzelz 10875 . . . . . . . . . 10  |-  ( y  e.  ( ZZ>= `  2
)  ->  y  e.  ZZ )
3635ad2antrr 725 . . . . . . . . 9  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  -> 
y  e.  ZZ )
37 eluzelz 10875 . . . . . . . . . 10  |-  ( z  e.  ( ZZ>= `  2
)  ->  z  e.  ZZ )
3837ad2antlr 726 . . . . . . . . 9  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  -> 
z  e.  ZZ )
39 dvdsmul1 13559 . . . . . . . . 9  |-  ( ( y  e.  ZZ  /\  z  e.  ZZ )  ->  y  ||  ( y  x.  z ) )
4036, 38, 39syl2anc 661 . . . . . . . 8  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  -> 
y  ||  ( y  x.  z ) )
41 prmz 13772 . . . . . . . . . 10  |-  ( p  e.  Prime  ->  p  e.  ZZ )
4241adantl 466 . . . . . . . . 9  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  ->  p  e.  ZZ )
4336, 38zmulcld 10758 . . . . . . . . 9  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  -> 
( y  x.  z
)  e.  ZZ )
44 dvdstr 13571 . . . . . . . . 9  |-  ( ( p  e.  ZZ  /\  y  e.  ZZ  /\  (
y  x.  z )  e.  ZZ )  -> 
( ( p  ||  y  /\  y  ||  (
y  x.  z ) )  ->  p  ||  (
y  x.  z ) ) )
4542, 36, 43, 44syl3anc 1218 . . . . . . . 8  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  -> 
( ( p  ||  y  /\  y  ||  (
y  x.  z ) )  ->  p  ||  (
y  x.  z ) ) )
4640, 45mpan2d 674 . . . . . . 7  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  -> 
( p  ||  y  ->  p  ||  ( y  x.  z ) ) )
4746reximdva 2833 . . . . . 6  |-  ( ( y  e.  ( ZZ>= ` 
2 )  /\  z  e.  ( ZZ>= `  2 )
)  ->  ( E. p  e.  Prime  p  ||  y  ->  E. p  e.  Prime  p 
||  ( y  x.  z ) ) )
4834, 47embantd 54 . . . . 5  |-  ( ( y  e.  ( ZZ>= ` 
2 )  /\  z  e.  ( ZZ>= `  2 )
)  ->  ( (
y  e.  ( ZZ>= ` 
2 )  ->  E. p  e.  Prime  p  ||  y
)  ->  E. p  e.  Prime  p  ||  (
y  x.  z ) ) )
4948a1dd 46 . . . 4  |-  ( ( y  e.  ( ZZ>= ` 
2 )  /\  z  e.  ( ZZ>= `  2 )
)  ->  ( (
y  e.  ( ZZ>= ` 
2 )  ->  E. p  e.  Prime  p  ||  y
)  ->  ( (
y  x.  z )  e.  ( ZZ>= `  2
)  ->  E. p  e.  Prime  p  ||  (
y  x.  z ) ) ) )
5049adantrd 468 . . 3  |-  ( ( y  e.  ( ZZ>= ` 
2 )  /\  z  e.  ( ZZ>= `  2 )
)  ->  ( (
( y  e.  (
ZZ>= `  2 )  ->  E. p  e.  Prime  p 
||  y )  /\  ( z  e.  (
ZZ>= `  2 )  ->  E. p  e.  Prime  p 
||  z ) )  ->  ( ( y  x.  z )  e.  ( ZZ>= `  2 )  ->  E. p  e.  Prime  p 
||  ( y  x.  z ) ) ) )
514, 8, 12, 16, 20, 26, 33, 50prmind 13780 . 2  |-  ( N  e.  NN  ->  ( N  e.  ( ZZ>= ` 
2 )  ->  E. p  e.  Prime  p  ||  N
) )
522, 51mpcom 36 1  |-  ( N  e.  ( ZZ>= `  2
)  ->  E. p  e.  Prime  p  ||  N
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2721   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   0cc0 9287   1c1 9288    x. cmul 9292    < clt 9423    - cmin 9600   NNcn 10327   2c2 10376   ZZcz 10651   ZZ>=cuz 10866    || cdivides 13540   Primecprime 13768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-n0 10585  df-z 10652  df-uz 10867  df-rp 10997  df-fz 11443  df-dvds 13541  df-prm 13769
This theorem is referenced by:  isprm5  13803  maxprmfct  13804  rpexp  13811  pc2dvds  13950  prmunb  13980  ablfacrplem  16571  muval1  22476  musum  22536  lgsne0  22677  dchrisum0flb  22764  nn0prpwlem  28522  frgrareggt1  30714
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