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Theorem exprmfct 14350
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 11081 . 2  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  e.  NN )
2 eleq1 2472 . . . 4  |-  ( x  =  1  ->  (
x  e.  ( ZZ>= ` 
2 )  <->  1  e.  ( ZZ>= `  2 )
) )
32imbi1d 315 . . 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 2472 . . . 4  |-  ( x  =  y  ->  (
x  e.  ( ZZ>= ` 
2 )  <->  y  e.  ( ZZ>= `  2 )
) )
5 breq2 4396 . . . . 5  |-  ( x  =  y  ->  (
p  ||  x  <->  p  ||  y
) )
65rexbidv 2915 . . . 4  |-  ( x  =  y  ->  ( E. p  e.  Prime  p 
||  x  <->  E. p  e.  Prime  p  ||  y
) )
74, 6imbi12d 318 . . 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 2472 . . . 4  |-  ( x  =  z  ->  (
x  e.  ( ZZ>= ` 
2 )  <->  z  e.  ( ZZ>= `  2 )
) )
9 breq2 4396 . . . . 5  |-  ( x  =  z  ->  (
p  ||  x  <->  p  ||  z
) )
109rexbidv 2915 . . . 4  |-  ( x  =  z  ->  ( E. p  e.  Prime  p 
||  x  <->  E. p  e.  Prime  p  ||  z
) )
118, 10imbi12d 318 . . 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 2472 . . . 4  |-  ( x  =  ( y  x.  z )  ->  (
x  e.  ( ZZ>= ` 
2 )  <->  ( y  x.  z )  e.  (
ZZ>= `  2 ) ) )
13 breq2 4396 . . . . 5  |-  ( x  =  ( y  x.  z )  ->  (
p  ||  x  <->  p  ||  (
y  x.  z ) ) )
1413rexbidv 2915 . . . 4  |-  ( x  =  ( y  x.  z )  ->  ( E. p  e.  Prime  p 
||  x  <->  E. p  e.  Prime  p  ||  (
y  x.  z ) ) )
1512, 14imbi12d 318 . . 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 2472 . . . 4  |-  ( x  =  N  ->  (
x  e.  ( ZZ>= ` 
2 )  <->  N  e.  ( ZZ>= `  2 )
) )
17 breq2 4396 . . . . 5  |-  ( x  =  N  ->  (
p  ||  x  <->  p  ||  N
) )
1817rexbidv 2915 . . . 4  |-  ( x  =  N  ->  ( E. p  e.  Prime  p 
||  x  <->  E. p  e.  Prime  p  ||  N
) )
1916, 18imbi12d 318 . . 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 10563 . . . . 5  |-  ( 1  -  1 )  =  0
21 uz2m1nn 11117 . . . . 5  |-  ( 1  e.  ( ZZ>= `  2
)  ->  ( 1  -  1 )  e.  NN )
2220, 21syl5eqelr 2493 . . . 4  |-  ( 1  e.  ( ZZ>= `  2
)  ->  0  e.  NN )
23 0nnn 10526 . . . . 5  |-  -.  0  e.  NN
2423pm2.21i 131 . . . 4  |-  ( 0  e.  NN  ->  E. p  e.  Prime  p  ||  x
)
2522, 24syl 17 . . 3  |-  ( 1  e.  ( ZZ>= `  2
)  ->  E. p  e.  Prime  p  ||  x
)
26 prmz 14320 . . . . . 6  |-  ( x  e.  Prime  ->  x  e.  ZZ )
27 iddvds 14096 . . . . . 6  |-  ( x  e.  ZZ  ->  x  ||  x )
2826, 27syl 17 . . . . 5  |-  ( x  e.  Prime  ->  x  ||  x )
29 breq1 4395 . . . . . 6  |-  ( p  =  x  ->  (
p  ||  x  <->  x  ||  x
) )
3029rspcev 3157 . . . . 5  |-  ( ( x  e.  Prime  /\  x  ||  x )  ->  E. p  e.  Prime  p  ||  x
)
3128, 30mpdan 666 . . . 4  |-  ( x  e.  Prime  ->  E. p  e.  Prime  p  ||  x
)
3231a1d 25 . . 3  |-  ( x  e.  Prime  ->  ( x  e.  ( ZZ>= `  2
)  ->  E. p  e.  Prime  p  ||  x
) )
33 simpl 455 . . . . . 6  |-  ( ( y  e.  ( ZZ>= ` 
2 )  /\  z  e.  ( ZZ>= `  2 )
)  ->  y  e.  ( ZZ>= `  2 )
)
34 eluzelz 11052 . . . . . . . . . 10  |-  ( y  e.  ( ZZ>= `  2
)  ->  y  e.  ZZ )
3534ad2antrr 724 . . . . . . . . 9  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  -> 
y  e.  ZZ )
36 eluzelz 11052 . . . . . . . . . 10  |-  ( z  e.  ( ZZ>= `  2
)  ->  z  e.  ZZ )
3736ad2antlr 725 . . . . . . . . 9  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  -> 
z  e.  ZZ )
38 dvdsmul1 14104 . . . . . . . . 9  |-  ( ( y  e.  ZZ  /\  z  e.  ZZ )  ->  y  ||  ( y  x.  z ) )
3935, 37, 38syl2anc 659 . . . . . . . 8  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  -> 
y  ||  ( y  x.  z ) )
40 prmz 14320 . . . . . . . . . 10  |-  ( p  e.  Prime  ->  p  e.  ZZ )
4140adantl 464 . . . . . . . . 9  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  ->  p  e.  ZZ )
4235, 37zmulcld 10932 . . . . . . . . 9  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  -> 
( y  x.  z
)  e.  ZZ )
43 dvdstr 14117 . . . . . . . . 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 1228 . . . . . . . 8  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  -> 
( ( p  ||  y  /\  y  ||  (
y  x.  z ) )  ->  p  ||  (
y  x.  z ) ) )
4539, 44mpan2d 672 . . . . . . 7  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  -> 
( p  ||  y  ->  p  ||  ( y  x.  z ) ) )
4645reximdva 2876 . . . . . 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 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 ) ) )
4847a1dd 44 . . . 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 466 . . 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 14328 . 2  |-  ( N  e.  NN  ->  ( N  e.  ( ZZ>= ` 
2 )  ->  E. p  e.  Prime  p  ||  N
) )
511, 50mpcom 34 1  |-  ( N  e.  ( ZZ>= `  2
)  ->  E. p  e.  Prime  p  ||  N
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1403    e. wcel 1840   E.wrex 2752   class class class wbr 4392   ` cfv 5523  (class class class)co 6232   0cc0 9440   1c1 9441    x. cmul 9445    - cmin 9759   NNcn 10494   2c2 10544   ZZcz 10823   ZZ>=cuz 11043    || cdvds 14085   Primecprime 14316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-recs 6997  df-rdg 7031  df-1o 7085  df-2o 7086  df-oadd 7089  df-er 7266  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-div 10166  df-nn 10495  df-2 10553  df-n0 10755  df-z 10824  df-uz 11044  df-rp 11182  df-fz 11642  df-dvds 14086  df-prm 14317
This theorem is referenced by:  isprm5  14352  maxprmfct  14353  rpexp  14360  pc2dvds  14501  prmunb  14531  ablfacrplem  17326  muval1  23678  musum  23738  lgsne0  23879  dchrisum0flb  23966  frgrareggt1  25415  nn0prpwlem  30533
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