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Theorem ncoprmgcdne1b 10696
Description: Two positive integers are not coprime, i.e. there is an integer greater than 1 which divides both integers, iff their greatest common divisor is not 1. (Contributed by AV, 9-Aug-2020.)
Assertion
Ref Expression
ncoprmgcdne1b  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( E. i  e.  ( ZZ>= `  2 )
( i  ||  A  /\  i  ||  B )  <-> 
( A  gcd  B
)  =/=  1 ) )
Distinct variable groups:    A, i    B, i

Proof of Theorem ncoprmgcdne1b
StepHypRef Expression
1 df-2 8235 . . . . . . 7  |-  2  =  ( 1  +  1 )
2 2re 8246 . . . . . . . . 9  |-  2  e.  RR
32a1i 9 . . . . . . . 8  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
2  e.  RR )
4 eluzelz 8779 . . . . . . . . . 10  |-  ( i  e.  ( ZZ>= `  2
)  ->  i  e.  ZZ )
54ad2antlr 473 . . . . . . . . 9  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
i  e.  ZZ )
65zred 8620 . . . . . . . 8  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
i  e.  RR )
7 simplll 500 . . . . . . . . . 10  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  ->  A  e.  NN )
8 simpllr 501 . . . . . . . . . 10  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  ->  B  e.  NN )
9 gcdnncl 10584 . . . . . . . . . 10  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  gcd  B
)  e.  NN )
107, 8, 9syl2anc 403 . . . . . . . . 9  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
( A  gcd  B
)  e.  NN )
1110nnred 8189 . . . . . . . 8  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
( A  gcd  B
)  e.  RR )
12 eluzle 8782 . . . . . . . . 9  |-  ( i  e.  ( ZZ>= `  2
)  ->  2  <_  i )
1312ad2antlr 473 . . . . . . . 8  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
2  <_  i )
14 simpr 108 . . . . . . . . . 10  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
( i  ||  A  /\  i  ||  B ) )
157nnzd 8619 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  ->  A  e.  ZZ )
168nnzd 8619 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  ->  B  e.  ZZ )
17 dvdsgcd 10626 . . . . . . . . . . 11  |-  ( ( i  e.  ZZ  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( i  ||  A  /\  i  ||  B )  ->  i  ||  ( A  gcd  B ) ) )
185, 15, 16, 17syl3anc 1170 . . . . . . . . . 10  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
( ( i  ||  A  /\  i  ||  B
)  ->  i  ||  ( A  gcd  B ) ) )
1914, 18mpd 13 . . . . . . . . 9  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
i  ||  ( A  gcd  B ) )
20 dvdsle 10470 . . . . . . . . . 10  |-  ( ( i  e.  ZZ  /\  ( A  gcd  B )  e.  NN )  -> 
( i  ||  ( A  gcd  B )  -> 
i  <_  ( A  gcd  B ) ) )
215, 10, 20syl2anc 403 . . . . . . . . 9  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
( i  ||  ( A  gcd  B )  -> 
i  <_  ( A  gcd  B ) ) )
2219, 21mpd 13 . . . . . . . 8  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
i  <_  ( A  gcd  B ) )
233, 6, 11, 13, 22letrd 7370 . . . . . . 7  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
2  <_  ( A  gcd  B ) )
241, 23syl5eqbrr 3839 . . . . . 6  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
( 1  +  1 )  <_  ( A  gcd  B ) )
25 1nn 8187 . . . . . . . 8  |-  1  e.  NN
2625a1i 9 . . . . . . 7  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
1  e.  NN )
27 nnltp1le 8562 . . . . . . 7  |-  ( ( 1  e.  NN  /\  ( A  gcd  B )  e.  NN )  -> 
( 1  <  ( A  gcd  B )  <->  ( 1  +  1 )  <_ 
( A  gcd  B
) ) )
2826, 10, 27syl2anc 403 . . . . . 6  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
( 1  <  ( A  gcd  B )  <->  ( 1  +  1 )  <_ 
( A  gcd  B
) ) )
2924, 28mpbird 165 . . . . 5  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
1  <  ( A  gcd  B ) )
30 nngt1ne1 8210 . . . . . 6  |-  ( ( A  gcd  B )  e.  NN  ->  (
1  <  ( A  gcd  B )  <->  ( A  gcd  B )  =/=  1
) )
3110, 30syl 14 . . . . 5  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
( 1  <  ( A  gcd  B )  <->  ( A  gcd  B )  =/=  1
) )
3229, 31mpbid 145 . . . 4  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
( A  gcd  B
)  =/=  1 )
3332ex 113 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  (
ZZ>= `  2 ) )  ->  ( ( i 
||  A  /\  i  ||  B )  ->  ( A  gcd  B )  =/=  1 ) )
3433rexlimdva 2482 . 2  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( E. i  e.  ( ZZ>= `  2 )
( i  ||  A  /\  i  ||  B )  ->  ( A  gcd  B )  =/=  1 ) )
359adantr 270 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =/=  1 )  ->  ( A  gcd  B )  e.  NN )
36 simpr 108 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =/=  1 )  ->  ( A  gcd  B )  =/=  1 )
37 eluz2b3 8842 . . . . 5  |-  ( ( A  gcd  B )  e.  ( ZZ>= `  2
)  <->  ( ( A  gcd  B )  e.  NN  /\  ( A  gcd  B )  =/=  1 ) )
3835, 36, 37sylanbrc 408 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =/=  1 )  ->  ( A  gcd  B )  e.  ( ZZ>= ` 
2 ) )
39 simpll 496 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =/=  1 )  ->  A  e.  NN )
4039nnzd 8619 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =/=  1 )  ->  A  e.  ZZ )
41 simplr 497 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =/=  1 )  ->  B  e.  NN )
4241nnzd 8619 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =/=  1 )  ->  B  e.  ZZ )
43 gcddvds 10580 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
4440, 42, 43syl2anc 403 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =/=  1 )  ->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B )  ||  B ) )
45 breq1 3808 . . . . . 6  |-  ( i  =  ( A  gcd  B )  ->  ( i  ||  A  <->  ( A  gcd  B )  ||  A ) )
46 breq1 3808 . . . . . 6  |-  ( i  =  ( A  gcd  B )  ->  ( i  ||  B  <->  ( A  gcd  B )  ||  B ) )
4745, 46anbi12d 457 . . . . 5  |-  ( i  =  ( A  gcd  B )  ->  ( (
i  ||  A  /\  i  ||  B )  <->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B )  ||  B ) ) )
4847rspcev 2710 . . . 4  |-  ( ( ( A  gcd  B
)  e.  ( ZZ>= ` 
2 )  /\  (
( A  gcd  B
)  ||  A  /\  ( A  gcd  B ) 
||  B ) )  ->  E. i  e.  (
ZZ>= `  2 ) ( i  ||  A  /\  i  ||  B ) )
4938, 44, 48syl2anc 403 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =/=  1 )  ->  E. i  e.  (
ZZ>= `  2 ) ( i  ||  A  /\  i  ||  B ) )
5049ex 113 . 2  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  gcd  B )  =/=  1  ->  E. i  e.  ( ZZ>=
`  2 ) ( i  ||  A  /\  i  ||  B ) ) )
5134, 50impbid 127 1  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( E. i  e.  ( ZZ>= `  2 )
( i  ||  A  /\  i  ||  B )  <-> 
( A  gcd  B
)  =/=  1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434    =/= wne 2249   E.wrex 2354   class class class wbr 3805   ` cfv 4952  (class class class)co 5564   RRcr 7112   1c1 7114    + caddc 7116    < clt 7285    <_ cle 7286   NNcn 8176   2c2 8226   ZZcz 8502   ZZ>=cuz 8770    || cdvds 10421    gcd cgcd 10563
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 7199  ax-resscn 7200  ax-1cn 7201  ax-1re 7202  ax-icn 7203  ax-addcl 7204  ax-addrcl 7205  ax-mulcl 7206  ax-mulrcl 7207  ax-addcom 7208  ax-mulcom 7209  ax-addass 7210  ax-mulass 7211  ax-distr 7212  ax-i2m1 7213  ax-0lt1 7214  ax-1rid 7215  ax-0id 7216  ax-rnegex 7217  ax-precex 7218  ax-cnre 7219  ax-pre-ltirr 7220  ax-pre-ltwlin 7221  ax-pre-lttrn 7222  ax-pre-apti 7223  ax-pre-ltadd 7224  ax-pre-mulgt0 7225  ax-pre-mulext 7226  ax-arch 7227  ax-caucvg 7228
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 5520  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-frec 6061  df-sup 6492  df-pnf 7287  df-mnf 7288  df-xr 7289  df-ltxr 7290  df-le 7291  df-sub 7418  df-neg 7419  df-reap 7812  df-ap 7819  df-div 7898  df-inn 8177  df-2 8235  df-3 8236  df-4 8237  df-n0 8426  df-z 8503  df-uz 8771  df-q 8856  df-rp 8886  df-fz 9176  df-fzo 9300  df-fl 9422  df-mod 9475  df-iseq 9592  df-iexp 9643  df-cj 9948  df-re 9949  df-im 9950  df-rsqrt 10103  df-abs 10104  df-dvds 10422  df-gcd 10564
This theorem is referenced by:  ncoprmgcdgt1b  10697
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