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Theorem exprmfct 13115
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 10553 . . 3  |-  ( N  e.  ( ZZ>= `  2
)  <->  ( N  e.  NN  /\  1  < 
N ) )
21simplbi 448 . 2  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  e.  NN )
3 eleq1 2498 . . . 4  |-  ( x  =  1  ->  (
x  e.  ( ZZ>= ` 
2 )  <->  1  e.  ( ZZ>= `  2 )
) )
43imbi1d 310 . . 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 2498 . . . 4  |-  ( x  =  y  ->  (
x  e.  ( ZZ>= ` 
2 )  <->  y  e.  ( ZZ>= `  2 )
) )
6 breq2 4219 . . . . 5  |-  ( x  =  y  ->  (
p  ||  x  <->  p  ||  y
) )
76rexbidv 2728 . . . 4  |-  ( x  =  y  ->  ( E. p  e.  Prime  p 
||  x  <->  E. p  e.  Prime  p  ||  y
) )
85, 7imbi12d 313 . . 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 2498 . . . 4  |-  ( x  =  z  ->  (
x  e.  ( ZZ>= ` 
2 )  <->  z  e.  ( ZZ>= `  2 )
) )
10 breq2 4219 . . . . 5  |-  ( x  =  z  ->  (
p  ||  x  <->  p  ||  z
) )
1110rexbidv 2728 . . . 4  |-  ( x  =  z  ->  ( E. p  e.  Prime  p 
||  x  <->  E. p  e.  Prime  p  ||  z
) )
129, 11imbi12d 313 . . 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 2498 . . . 4  |-  ( x  =  ( y  x.  z )  ->  (
x  e.  ( ZZ>= ` 
2 )  <->  ( y  x.  z )  e.  (
ZZ>= `  2 ) ) )
14 breq2 4219 . . . . 5  |-  ( x  =  ( y  x.  z )  ->  (
p  ||  x  <->  p  ||  (
y  x.  z ) ) )
1514rexbidv 2728 . . . 4  |-  ( x  =  ( y  x.  z )  ->  ( E. p  e.  Prime  p 
||  x  <->  E. p  e.  Prime  p  ||  (
y  x.  z ) ) )
1613, 15imbi12d 313 . . 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 2498 . . . 4  |-  ( x  =  N  ->  (
x  e.  ( ZZ>= ` 
2 )  <->  N  e.  ( ZZ>= `  2 )
) )
18 breq2 4219 . . . . 5  |-  ( x  =  N  ->  (
p  ||  x  <->  p  ||  N
) )
1918rexbidv 2728 . . . 4  |-  ( x  =  N  ->  ( E. p  e.  Prime  p 
||  x  <->  E. p  e.  Prime  p  ||  N
) )
2017, 19imbi12d 313 . . 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 10073 . . . . 5  |-  ( 1  -  1 )  =  0
22 uz2m1nn 10555 . . . . 5  |-  ( 1  e.  ( ZZ>= `  2
)  ->  ( 1  -  1 )  e.  NN )
2321, 22syl5eqelr 2523 . . . 4  |-  ( 1  e.  ( ZZ>= `  2
)  ->  0  e.  NN )
24 0nnn 10036 . . . . 5  |-  -.  0  e.  NN
2524pm2.21i 126 . . . 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 13088 . . . . . 6  |-  ( x  e.  Prime  ->  x  e.  ZZ )
28 iddvds 12868 . . . . . 6  |-  ( x  e.  ZZ  ->  x  ||  x )
2927, 28syl 16 . . . . 5  |-  ( x  e.  Prime  ->  x  ||  x )
30 breq1 4218 . . . . . 6  |-  ( p  =  x  ->  (
p  ||  x  <->  x  ||  x
) )
3130rspcev 3054 . . . . 5  |-  ( ( x  e.  Prime  /\  x  ||  x )  ->  E. p  e.  Prime  p  ||  x
)
3229, 31mpdan 651 . . . 4  |-  ( x  e.  Prime  ->  E. p  e.  Prime  p  ||  x
)
3332a1d 24 . . 3  |-  ( x  e.  Prime  ->  ( x  e.  ( ZZ>= `  2
)  ->  E. p  e.  Prime  p  ||  x
) )
34 simpl 445 . . . . . 6  |-  ( ( y  e.  ( ZZ>= ` 
2 )  /\  z  e.  ( ZZ>= `  2 )
)  ->  y  e.  ( ZZ>= `  2 )
)
35 eluzelz 10501 . . . . . . . . . 10  |-  ( y  e.  ( ZZ>= `  2
)  ->  y  e.  ZZ )
3635ad2antrr 708 . . . . . . . . 9  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  -> 
y  e.  ZZ )
37 eluzelz 10501 . . . . . . . . . 10  |-  ( z  e.  ( ZZ>= `  2
)  ->  z  e.  ZZ )
3837ad2antlr 709 . . . . . . . . 9  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  -> 
z  e.  ZZ )
39 dvdsmul1 12876 . . . . . . . . 9  |-  ( ( y  e.  ZZ  /\  z  e.  ZZ )  ->  y  ||  ( y  x.  z ) )
4036, 38, 39syl2anc 644 . . . . . . . 8  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  -> 
y  ||  ( y  x.  z ) )
41 prmz 13088 . . . . . . . . . 10  |-  ( p  e.  Prime  ->  p  e.  ZZ )
4241adantl 454 . . . . . . . . 9  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  ->  p  e.  ZZ )
4336, 38zmulcld 10386 . . . . . . . . 9  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  -> 
( y  x.  z
)  e.  ZZ )
44 dvdstr 12888 . . . . . . . . 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 1185 . . . . . . . 8  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  -> 
( ( p  ||  y  /\  y  ||  (
y  x.  z ) )  ->  p  ||  (
y  x.  z ) ) )
4640, 45mpan2d 657 . . . . . . 7  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  -> 
( p  ||  y  ->  p  ||  ( y  x.  z ) ) )
4746reximdva 2820 . . . . . 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 53 . . . . 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 45 . . . 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 456 . . 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 13096 . 2  |-  ( N  e.  NN  ->  ( N  e.  ( ZZ>= ` 
2 )  ->  E. p  e.  Prime  p  ||  N
) )
522, 51mpcom 35 1  |-  ( N  e.  ( ZZ>= `  2
)  ->  E. p  e.  Prime  p  ||  N
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2708   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   0cc0 8995   1c1 8996    x. cmul 9000    < clt 9125    - cmin 9296   NNcn 10005   2c2 10054   ZZcz 10287   ZZ>=cuz 10493    || cdivides 12857   Primecprime 13084
This theorem is referenced by:  isprm5  13117  maxprmfct  13118  rpexp  13125  pc2dvds  13257  prmunb  13287  ablfacrplem  15628  muval1  20921  musum  20981  lgsne0  21122  dchrisum0flb  21209  nn0prpwlem  26339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-fz 11049  df-dvds 12858  df-prm 13085
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