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Theorem exprmfct 10726
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 eluz2nn 8790 . 2  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  e.  NN )
2 eleq1 2145 . . . 4  |-  ( x  =  1  ->  (
x  e.  ( ZZ>= ` 
2 )  <->  1  e.  ( ZZ>= `  2 )
) )
32imbi1d 229 . . 3  |-  ( x  =  1  ->  (
( x  e.  (
ZZ>= `  2 )  ->  E. p  e.  Prime  p 
||  x )  <->  ( 1  e.  ( ZZ>= `  2
)  ->  E. p  e.  Prime  p  ||  x
) ) )
4 eleq1 2145 . . . 4  |-  ( x  =  y  ->  (
x  e.  ( ZZ>= ` 
2 )  <->  y  e.  ( ZZ>= `  2 )
) )
5 breq2 3809 . . . . 5  |-  ( x  =  y  ->  (
p  ||  x  <->  p  ||  y
) )
65rexbidv 2374 . . . 4  |-  ( x  =  y  ->  ( E. p  e.  Prime  p 
||  x  <->  E. p  e.  Prime  p  ||  y
) )
74, 6imbi12d 232 . . 3  |-  ( x  =  y  ->  (
( x  e.  (
ZZ>= `  2 )  ->  E. p  e.  Prime  p 
||  x )  <->  ( y  e.  ( ZZ>= `  2 )  ->  E. p  e.  Prime  p 
||  y ) ) )
8 eleq1 2145 . . . 4  |-  ( x  =  z  ->  (
x  e.  ( ZZ>= ` 
2 )  <->  z  e.  ( ZZ>= `  2 )
) )
9 breq2 3809 . . . . 5  |-  ( x  =  z  ->  (
p  ||  x  <->  p  ||  z
) )
109rexbidv 2374 . . . 4  |-  ( x  =  z  ->  ( E. p  e.  Prime  p 
||  x  <->  E. p  e.  Prime  p  ||  z
) )
118, 10imbi12d 232 . . 3  |-  ( x  =  z  ->  (
( x  e.  (
ZZ>= `  2 )  ->  E. p  e.  Prime  p 
||  x )  <->  ( z  e.  ( ZZ>= `  2 )  ->  E. p  e.  Prime  p 
||  z ) ) )
12 eleq1 2145 . . . 4  |-  ( x  =  ( y  x.  z )  ->  (
x  e.  ( ZZ>= ` 
2 )  <->  ( y  x.  z )  e.  (
ZZ>= `  2 ) ) )
13 breq2 3809 . . . . 5  |-  ( x  =  ( y  x.  z )  ->  (
p  ||  x  <->  p  ||  (
y  x.  z ) ) )
1413rexbidv 2374 . . . 4  |-  ( x  =  ( y  x.  z )  ->  ( E. p  e.  Prime  p 
||  x  <->  E. p  e.  Prime  p  ||  (
y  x.  z ) ) )
1512, 14imbi12d 232 . . 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 ) ) ) )
16 eleq1 2145 . . . 4  |-  ( x  =  N  ->  (
x  e.  ( ZZ>= ` 
2 )  <->  N  e.  ( ZZ>= `  2 )
) )
17 breq2 3809 . . . . 5  |-  ( x  =  N  ->  (
p  ||  x  <->  p  ||  N
) )
1817rexbidv 2374 . . . 4  |-  ( x  =  N  ->  ( E. p  e.  Prime  p 
||  x  <->  E. p  e.  Prime  p  ||  N
) )
1916, 18imbi12d 232 . . 3  |-  ( x  =  N  ->  (
( x  e.  (
ZZ>= `  2 )  ->  E. p  e.  Prime  p 
||  x )  <->  ( N  e.  ( ZZ>= `  2 )  ->  E. p  e.  Prime  p 
||  N ) ) )
20 1m1e0 8227 . . . . 5  |-  ( 1  -  1 )  =  0
21 uz2m1nn 8825 . . . . 5  |-  ( 1  e.  ( ZZ>= `  2
)  ->  ( 1  -  1 )  e.  NN )
2220, 21syl5eqelr 2170 . . . 4  |-  ( 1  e.  ( ZZ>= `  2
)  ->  0  e.  NN )
23 0nnn 8185 . . . . 5  |-  -.  0  e.  NN
2423pm2.21i 608 . . . 4  |-  ( 0  e.  NN  ->  E. p  e.  Prime  p  ||  x
)
2522, 24syl 14 . . 3  |-  ( 1  e.  ( ZZ>= `  2
)  ->  E. p  e.  Prime  p  ||  x
)
26 prmz 10700 . . . . . 6  |-  ( x  e.  Prime  ->  x  e.  ZZ )
27 iddvds 10416 . . . . . 6  |-  ( x  e.  ZZ  ->  x  ||  x )
2826, 27syl 14 . . . . 5  |-  ( x  e.  Prime  ->  x  ||  x )
29 breq1 3808 . . . . . 6  |-  ( p  =  x  ->  (
p  ||  x  <->  x  ||  x
) )
3029rspcev 2710 . . . . 5  |-  ( ( x  e.  Prime  /\  x  ||  x )  ->  E. p  e.  Prime  p  ||  x
)
3128, 30mpdan 412 . . . 4  |-  ( x  e.  Prime  ->  E. p  e.  Prime  p  ||  x
)
3231a1d 22 . . 3  |-  ( x  e.  Prime  ->  ( x  e.  ( ZZ>= `  2
)  ->  E. p  e.  Prime  p  ||  x
) )
33 simpl 107 . . . . . 6  |-  ( ( y  e.  ( ZZ>= ` 
2 )  /\  z  e.  ( ZZ>= `  2 )
)  ->  y  e.  ( ZZ>= `  2 )
)
34 eluzelz 8761 . . . . . . . . . 10  |-  ( y  e.  ( ZZ>= `  2
)  ->  y  e.  ZZ )
3534ad2antrr 472 . . . . . . . . 9  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  -> 
y  e.  ZZ )
36 eluzelz 8761 . . . . . . . . . 10  |-  ( z  e.  ( ZZ>= `  2
)  ->  z  e.  ZZ )
3736ad2antlr 473 . . . . . . . . 9  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  -> 
z  e.  ZZ )
38 dvdsmul1 10425 . . . . . . . . 9  |-  ( ( y  e.  ZZ  /\  z  e.  ZZ )  ->  y  ||  ( y  x.  z ) )
3935, 37, 38syl2anc 403 . . . . . . . 8  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  -> 
y  ||  ( y  x.  z ) )
40 prmz 10700 . . . . . . . . . 10  |-  ( p  e.  Prime  ->  p  e.  ZZ )
4140adantl 271 . . . . . . . . 9  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  ->  p  e.  ZZ )
4235, 37zmulcld 8608 . . . . . . . . 9  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  -> 
( y  x.  z
)  e.  ZZ )
43 dvdstr 10440 . . . . . . . . 9  |-  ( ( p  e.  ZZ  /\  y  e.  ZZ  /\  (
y  x.  z )  e.  ZZ )  -> 
( ( p  ||  y  /\  y  ||  (
y  x.  z ) )  ->  p  ||  (
y  x.  z ) ) )
4441, 35, 42, 43syl3anc 1170 . . . . . . . 8  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  -> 
( ( p  ||  y  /\  y  ||  (
y  x.  z ) )  ->  p  ||  (
y  x.  z ) ) )
4539, 44mpan2d 419 . . . . . . 7  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  -> 
( p  ||  y  ->  p  ||  ( y  x.  z ) ) )
4645reximdva 2468 . . . . . 6  |-  ( ( y  e.  ( ZZ>= ` 
2 )  /\  z  e.  ( ZZ>= `  2 )
)  ->  ( E. p  e.  Prime  p  ||  y  ->  E. p  e.  Prime  p 
||  ( y  x.  z ) ) )
4733, 46embantd 55 . . . . 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 ) ) )
4847a1dd 47 . . . 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 ) ) ) )
4948adantrd 273 . . 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 ) ) ) )
503, 7, 11, 15, 19, 25, 32, 49prmind 10710 . 2  |-  ( N  e.  NN  ->  ( N  e.  ( ZZ>= ` 
2 )  ->  E. p  e.  Prime  p  ||  N
) )
511, 50mpcom 36 1  |-  ( N  e.  ( ZZ>= `  2
)  ->  E. p  e.  Prime  p  ||  N
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   E.wrex 2354   class class class wbr 3805   ` cfv 4952  (class class class)co 5563   0cc0 7095   1c1 7096    x. cmul 7100    - cmin 7398   NNcn 8158   2c2 8208   ZZcz 8484   ZZ>=cuz 8752    || cdvds 10403   Primecprime 10696
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-mulrcl 7189  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-precex 7200  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-apti 7205  ax-pre-ltadd 7206  ax-pre-mulgt0 7207  ax-pre-mulext 7208  ax-arch 7209  ax-caucvg 7210
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-if 3369  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-frec 6060  df-1o 6085  df-2o 6086  df-er 6193  df-en 6309  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-reap 7794  df-ap 7801  df-div 7880  df-inn 8159  df-2 8217  df-3 8218  df-4 8219  df-n0 8408  df-z 8485  df-uz 8753  df-q 8838  df-rp 8868  df-fz 9158  df-fzo 9282  df-fl 9404  df-mod 9457  df-iseq 9574  df-iexp 9625  df-cj 9930  df-re 9931  df-im 9932  df-rsqrt 10085  df-abs 10086  df-dvds 10404  df-prm 10697
This theorem is referenced by:  prmdvdsfz  10727  rpexp  10739
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