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Theorem ncoprmgcdne1b 11682
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 8743 . . . . . . 7  |-  2  =  ( 1  +  1 )
2 2re 8754 . . . . . . . . 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 9291 . . . . . . . . . 10  |-  ( i  e.  ( ZZ>= `  2
)  ->  i  e.  ZZ )
54ad2antlr 480 . . . . . . . . 9  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
i  e.  ZZ )
65zred 9131 . . . . . . . 8  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
i  e.  RR )
7 simplll 507 . . . . . . . . . 10  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  ->  A  e.  NN )
8 simpllr 508 . . . . . . . . . 10  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  ->  B  e.  NN )
9 gcdnncl 11568 . . . . . . . . . 10  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  gcd  B
)  e.  NN )
107, 8, 9syl2anc 408 . . . . . . . . 9  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
( A  gcd  B
)  e.  NN )
1110nnred 8697 . . . . . . . 8  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
( A  gcd  B
)  e.  RR )
12 eluzle 9294 . . . . . . . . 9  |-  ( i  e.  ( ZZ>= `  2
)  ->  2  <_  i )
1312ad2antlr 480 . . . . . . . 8  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
2  <_  i )
14 simpr 109 . . . . . . . . . 10  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
( i  ||  A  /\  i  ||  B ) )
157nnzd 9130 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  ->  A  e.  ZZ )
168nnzd 9130 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  ->  B  e.  ZZ )
17 dvdsgcd 11612 . . . . . . . . . . 11  |-  ( ( i  e.  ZZ  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( i  ||  A  /\  i  ||  B )  ->  i  ||  ( A  gcd  B ) ) )
185, 15, 16, 17syl3anc 1201 . . . . . . . . . 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 11454 . . . . . . . . . 10  |-  ( ( i  e.  ZZ  /\  ( A  gcd  B )  e.  NN )  -> 
( i  ||  ( A  gcd  B )  -> 
i  <_  ( A  gcd  B ) ) )
215, 10, 20syl2anc 408 . . . . . . . . 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 7854 . . . . . . 7  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
2  <_  ( A  gcd  B ) )
241, 23eqbrtrrid 3934 . . . . . 6  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
( 1  +  1 )  <_  ( A  gcd  B ) )
25 1nn 8695 . . . . . . . 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 9072 . . . . . . 7  |-  ( ( 1  e.  NN  /\  ( A  gcd  B )  e.  NN )  -> 
( 1  <  ( A  gcd  B )  <->  ( 1  +  1 )  <_ 
( A  gcd  B
) ) )
2826, 10, 27syl2anc 408 . . . . . 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 166 . . . . 5  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
1  <  ( A  gcd  B ) )
30 nngt1ne1 8719 . . . . . 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 146 . . . 4  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  ( ZZ>= `  2 )
)  /\  ( i  ||  A  /\  i  ||  B ) )  -> 
( A  gcd  B
)  =/=  1 )
3332ex 114 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  i  e.  (
ZZ>= `  2 ) )  ->  ( ( i 
||  A  /\  i  ||  B )  ->  ( A  gcd  B )  =/=  1 ) )
3433rexlimdva 2526 . 2  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( E. i  e.  ( ZZ>= `  2 )
( i  ||  A  /\  i  ||  B )  ->  ( A  gcd  B )  =/=  1 ) )
359adantr 274 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =/=  1 )  ->  ( A  gcd  B )  e.  NN )
36 simpr 109 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =/=  1 )  ->  ( A  gcd  B )  =/=  1 )
37 eluz2b3 9354 . . . . 5  |-  ( ( A  gcd  B )  e.  ( ZZ>= `  2
)  <->  ( ( A  gcd  B )  e.  NN  /\  ( A  gcd  B )  =/=  1 ) )
3835, 36, 37sylanbrc 413 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =/=  1 )  ->  ( A  gcd  B )  e.  ( ZZ>= ` 
2 ) )
39 simpll 503 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =/=  1 )  ->  A  e.  NN )
4039nnzd 9130 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =/=  1 )  ->  A  e.  ZZ )
41 simplr 504 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =/=  1 )  ->  B  e.  NN )
4241nnzd 9130 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =/=  1 )  ->  B  e.  ZZ )
43 gcddvds 11564 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
4440, 42, 43syl2anc 408 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =/=  1 )  ->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B )  ||  B ) )
45 breq1 3902 . . . . . 6  |-  ( i  =  ( A  gcd  B )  ->  ( i  ||  A  <->  ( A  gcd  B )  ||  A ) )
46 breq1 3902 . . . . . 6  |-  ( i  =  ( A  gcd  B )  ->  ( i  ||  B  <->  ( A  gcd  B )  ||  B ) )
4745, 46anbi12d 464 . . . . 5  |-  ( i  =  ( A  gcd  B )  ->  ( (
i  ||  A  /\  i  ||  B )  <->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B )  ||  B ) ) )
4847rspcev 2763 . . . 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 408 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =/=  1 )  ->  E. i  e.  (
ZZ>= `  2 ) ( i  ||  A  /\  i  ||  B ) )
5049ex 114 . 2  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  gcd  B )  =/=  1  ->  E. i  e.  ( ZZ>=
`  2 ) ( i  ||  A  /\  i  ||  B ) ) )
5134, 50impbid 128 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 103    <-> wb 104    = wceq 1316    e. wcel 1465    =/= wne 2285   E.wrex 2394   class class class wbr 3899   ` cfv 5093  (class class class)co 5742   RRcr 7587   1c1 7589    + caddc 7591    < clt 7768    <_ cle 7769   NNcn 8684   2c2 8735   ZZcz 9012   ZZ>=cuz 9282    || cdvds 11405    gcd cgcd 11547
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707  ax-caucvg 7708
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-sup 6839  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8304  df-ap 8311  df-div 8400  df-inn 8685  df-2 8743  df-3 8744  df-4 8745  df-n0 8936  df-z 9013  df-uz 9283  df-q 9368  df-rp 9398  df-fz 9746  df-fzo 9875  df-fl 9998  df-mod 10051  df-seqfrec 10174  df-exp 10248  df-cj 10569  df-re 10570  df-im 10571  df-rsqrt 10725  df-abs 10726  df-dvds 11406  df-gcd 11548
This theorem is referenced by:  ncoprmgcdgt1b  11683
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