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Theorem exprmfct 12015
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 9477 . 2  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  e.  NN )
2 eleq1 2220 . . . 4  |-  ( x  =  1  ->  (
x  e.  ( ZZ>= ` 
2 )  <->  1  e.  ( ZZ>= `  2 )
) )
32imbi1d 230 . . 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 2220 . . . 4  |-  ( x  =  y  ->  (
x  e.  ( ZZ>= ` 
2 )  <->  y  e.  ( ZZ>= `  2 )
) )
5 breq2 3969 . . . . 5  |-  ( x  =  y  ->  (
p  ||  x  <->  p  ||  y
) )
65rexbidv 2458 . . . 4  |-  ( x  =  y  ->  ( E. p  e.  Prime  p 
||  x  <->  E. p  e.  Prime  p  ||  y
) )
74, 6imbi12d 233 . . 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 2220 . . . 4  |-  ( x  =  z  ->  (
x  e.  ( ZZ>= ` 
2 )  <->  z  e.  ( ZZ>= `  2 )
) )
9 breq2 3969 . . . . 5  |-  ( x  =  z  ->  (
p  ||  x  <->  p  ||  z
) )
109rexbidv 2458 . . . 4  |-  ( x  =  z  ->  ( E. p  e.  Prime  p 
||  x  <->  E. p  e.  Prime  p  ||  z
) )
118, 10imbi12d 233 . . 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 2220 . . . 4  |-  ( x  =  ( y  x.  z )  ->  (
x  e.  ( ZZ>= ` 
2 )  <->  ( y  x.  z )  e.  (
ZZ>= `  2 ) ) )
13 breq2 3969 . . . . 5  |-  ( x  =  ( y  x.  z )  ->  (
p  ||  x  <->  p  ||  (
y  x.  z ) ) )
1413rexbidv 2458 . . . 4  |-  ( x  =  ( y  x.  z )  ->  ( E. p  e.  Prime  p 
||  x  <->  E. p  e.  Prime  p  ||  (
y  x.  z ) ) )
1512, 14imbi12d 233 . . 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 2220 . . . 4  |-  ( x  =  N  ->  (
x  e.  ( ZZ>= ` 
2 )  <->  N  e.  ( ZZ>= `  2 )
) )
17 breq2 3969 . . . . 5  |-  ( x  =  N  ->  (
p  ||  x  <->  p  ||  N
) )
1817rexbidv 2458 . . . 4  |-  ( x  =  N  ->  ( E. p  e.  Prime  p 
||  x  <->  E. p  e.  Prime  p  ||  N
) )
1916, 18imbi12d 233 . . 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 8902 . . . . 5  |-  ( 1  -  1 )  =  0
21 uz2m1nn 9516 . . . . 5  |-  ( 1  e.  ( ZZ>= `  2
)  ->  ( 1  -  1 )  e.  NN )
2220, 21eqeltrrid 2245 . . . 4  |-  ( 1  e.  ( ZZ>= `  2
)  ->  0  e.  NN )
23 0nnn 8860 . . . . 5  |-  -.  0  e.  NN
2423pm2.21i 636 . . . 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 11988 . . . . . 6  |-  ( x  e.  Prime  ->  x  e.  ZZ )
27 iddvds 11700 . . . . . 6  |-  ( x  e.  ZZ  ->  x  ||  x )
2826, 27syl 14 . . . . 5  |-  ( x  e.  Prime  ->  x  ||  x )
29 breq1 3968 . . . . . 6  |-  ( p  =  x  ->  (
p  ||  x  <->  x  ||  x
) )
3029rspcev 2816 . . . . 5  |-  ( ( x  e.  Prime  /\  x  ||  x )  ->  E. p  e.  Prime  p  ||  x
)
3128, 30mpdan 418 . . . 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 108 . . . . . 6  |-  ( ( y  e.  ( ZZ>= ` 
2 )  /\  z  e.  ( ZZ>= `  2 )
)  ->  y  e.  ( ZZ>= `  2 )
)
34 eluzelz 9448 . . . . . . . . . 10  |-  ( y  e.  ( ZZ>= `  2
)  ->  y  e.  ZZ )
3534ad2antrr 480 . . . . . . . . 9  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  -> 
y  e.  ZZ )
36 eluzelz 9448 . . . . . . . . . 10  |-  ( z  e.  ( ZZ>= `  2
)  ->  z  e.  ZZ )
3736ad2antlr 481 . . . . . . . . 9  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  -> 
z  e.  ZZ )
38 dvdsmul1 11709 . . . . . . . . 9  |-  ( ( y  e.  ZZ  /\  z  e.  ZZ )  ->  y  ||  ( y  x.  z ) )
3935, 37, 38syl2anc 409 . . . . . . . 8  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  -> 
y  ||  ( y  x.  z ) )
40 prmz 11988 . . . . . . . . . 10  |-  ( p  e.  Prime  ->  p  e.  ZZ )
4140adantl 275 . . . . . . . . 9  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  ->  p  e.  ZZ )
4235, 37zmulcld 9292 . . . . . . . . 9  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  -> 
( y  x.  z
)  e.  ZZ )
43 dvdstr 11724 . . . . . . . . 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 1220 . . . . . . . 8  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  -> 
( ( p  ||  y  /\  y  ||  (
y  x.  z ) )  ->  p  ||  (
y  x.  z ) ) )
4539, 44mpan2d 425 . . . . . . 7  |-  ( ( ( y  e.  (
ZZ>= `  2 )  /\  z  e.  ( ZZ>= ` 
2 ) )  /\  p  e.  Prime )  -> 
( p  ||  y  ->  p  ||  ( y  x.  z ) ) )
4645reximdva 2559 . . . . . 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 56 . . . . 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 48 . . . 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 277 . . 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 11998 . 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 103    = wceq 1335    e. wcel 2128   E.wrex 2436   class class class wbr 3965   ` cfv 5170  (class class class)co 5824   0cc0 7732   1c1 7733    x. cmul 7737    - cmin 8046   NNcn 8833   2c2 8884   ZZcz 9167   ZZ>=cuz 9439    || cdvds 11683   Primecprime 11984
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849  ax-pre-mulext 7850  ax-arch 7851  ax-caucvg 7852
This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-frec 6338  df-1o 6363  df-2o 6364  df-er 6480  df-en 6686  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457  df-div 8546  df-inn 8834  df-2 8892  df-3 8893  df-4 8894  df-n0 9091  df-z 9168  df-uz 9440  df-q 9529  df-rp 9561  df-fz 9913  df-fzo 10042  df-fl 10169  df-mod 10222  df-seqfrec 10345  df-exp 10419  df-cj 10742  df-re 10743  df-im 10744  df-rsqrt 10898  df-abs 10899  df-dvds 11684  df-prm 11985
This theorem is referenced by:  prmdvdsfz  12016  rpexp  12028
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