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Theorem ncoprmgcdne1b 12091
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 8980 . . . . . . 7  |-  2  =  ( 1  +  1 )
2 2re 8991 . . . . . . . . 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 9539 . . . . . . . . . 10  |-  ( i  e.  ( ZZ>= `  2
)  ->  i  e.  ZZ )
54ad2antlr 489 . . . . . . . . 9  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
i  e.  ZZ )
65zred 9377 . . . . . . . 8  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
i  e.  RR )
7 simplll 533 . . . . . . . . . 10  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  ->  A  e.  NN )
8 simpllr 534 . . . . . . . . . 10  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  ->  B  e.  NN )
9 gcdnncl 11970 . . . . . . . . . 10  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  gcd  B
)  e.  NN )
107, 8, 9syl2anc 411 . . . . . . . . 9  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
( A  gcd  B
)  e.  NN )
1110nnred 8934 . . . . . . . 8  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
( A  gcd  B
)  e.  RR )
12 eluzle 9542 . . . . . . . . 9  |-  ( i  e.  ( ZZ>= `  2
)  ->  2  <_  i )
1312ad2antlr 489 . . . . . . . 8  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
2  <_  i )
14 simpr 110 . . . . . . . . . 10  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
( i  ||  A  /\  i  ||  B ) )
157nnzd 9376 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  ->  A  e.  ZZ )
168nnzd 9376 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  ->  B  e.  ZZ )
17 dvdsgcd 12015 . . . . . . . . . . 11  |-  ( ( i  e.  ZZ  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( i  ||  A  /\  i  ||  B )  ->  i  ||  ( A  gcd  B ) ) )
185, 15, 16, 17syl3anc 1238 . . . . . . . . . 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 11852 . . . . . . . . . 10  |-  ( ( i  e.  ZZ  /\  ( A  gcd  B )  e.  NN )  -> 
( i  ||  ( A  gcd  B )  -> 
i  <_  ( A  gcd  B ) ) )
215, 10, 20syl2anc 411 . . . . . . . . 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 8083 . . . . . . 7  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
2  <_  ( A  gcd  B ) )
241, 23eqbrtrrid 4041 . . . . . 6  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
( 1  +  1 )  <_  ( A  gcd  B ) )
25 1nn 8932 . . . . . . . 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 9315 . . . . . . 7  |-  ( ( 1  e.  NN  /\  ( A  gcd  B )  e.  NN )  -> 
( 1  <  ( A  gcd  B )  <->  ( 1  +  1 )  <_ 
( A  gcd  B
) ) )
2826, 10, 27syl2anc 411 . . . . . 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 167 . . . . 5  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
1  <  ( A  gcd  B ) )
30 nngt1ne1 8956 . . . . . 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 147 . . . 4  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
( A  gcd  B
)  =/=  1 )
3332ex 115 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  (
ZZ>= `  2 ) )  ->  ( ( i 
||  A  /\  i  ||  B )  ->  ( A  gcd  B )  =/=  1 ) )
3433rexlimdva 2594 . 2  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( E. i  e.  ( ZZ>= `  2 )
( i  ||  A  /\  i  ||  B )  ->  ( A  gcd  B )  =/=  1 ) )
359adantr 276 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =/=  1 )  ->  ( A  gcd  B )  e.  NN )
36 simpr 110 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =/=  1 )  ->  ( A  gcd  B )  =/=  1 )
37 eluz2b3 9606 . . . . 5  |-  ( ( A  gcd  B )  e.  ( ZZ>= `  2
)  <->  ( ( A  gcd  B )  e.  NN  /\  ( A  gcd  B )  =/=  1 ) )
3835, 36, 37sylanbrc 417 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =/=  1 )  ->  ( A  gcd  B )  e.  ( ZZ>= ` 
2 ) )
39 simpll 527 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =/=  1 )  ->  A  e.  NN )
4039nnzd 9376 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =/=  1 )  ->  A  e.  ZZ )
41 simplr 528 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =/=  1 )  ->  B  e.  NN )
4241nnzd 9376 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =/=  1 )  ->  B  e.  ZZ )
43 gcddvds 11966 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
4440, 42, 43syl2anc 411 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =/=  1 )  ->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B )  ||  B ) )
45 breq1 4008 . . . . . 6  |-  ( i  =  ( A  gcd  B )  ->  ( i  ||  A  <->  ( A  gcd  B )  ||  A ) )
46 breq1 4008 . . . . . 6  |-  ( i  =  ( A  gcd  B )  ->  ( i  ||  B  <->  ( A  gcd  B )  ||  B ) )
4745, 46anbi12d 473 . . . . 5  |-  ( i  =  ( A  gcd  B )  ->  ( (
i  ||  A  /\  i  ||  B )  <->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B )  ||  B ) ) )
4847rspcev 2843 . . . 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 411 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =/=  1 )  ->  E. i  e.  (
ZZ>= `  2 ) ( i  ||  A  /\  i  ||  B ) )
5049ex 115 . 2  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  gcd  B )  =/=  1  ->  E. i  e.  ( ZZ>=
`  2 ) ( i  ||  A  /\  i  ||  B ) ) )
5134, 50impbid 129 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 104    <-> wb 105    = wceq 1353    e. wcel 2148    =/= wne 2347   E.wrex 2456   class class class wbr 4005   ` cfv 5218  (class class class)co 5877   RRcr 7812   1c1 7814    + caddc 7816    < clt 7994    <_ cle 7995   NNcn 8921   2c2 8972   ZZcz 9255   ZZ>=cuz 9530    || cdvds 11796    gcd cgcd 11945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-sup 6985  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-fz 10011  df-fzo 10145  df-fl 10272  df-mod 10325  df-seqfrec 10448  df-exp 10522  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-dvds 11797  df-gcd 11946
This theorem is referenced by:  ncoprmgcdgt1b  12092
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