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Theorem lcmgcdeq 10672
Description: Two integers' absolute values are equal iff their least common multiple and greatest common divisor are equal. (Contributed by Steve Rodriguez, 20-Jan-2020.)
Assertion
Ref Expression
lcmgcdeq  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M lcm  N
)  =  ( M  gcd  N )  <->  ( abs `  M )  =  ( abs `  N ) ) )

Proof of Theorem lcmgcdeq
StepHypRef Expression
1 dvdslcm 10658 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  ( M lcm  N )  /\  N  ||  ( M lcm  N ) ) )
21simpld 110 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  M  ||  ( M lcm 
N ) )
32adantr 270 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  M  ||  ( M lcm  N ) )
4 gcddvds 10562 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  gcd  N )  ||  M  /\  ( M  gcd  N ) 
||  N ) )
54simprd 112 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  ||  N )
6 breq1 3808 . . . . . . 7  |-  ( ( M lcm  N )  =  ( M  gcd  N
)  ->  ( ( M lcm  N )  ||  N  <->  ( M  gcd  N ) 
||  N ) )
75, 6syl5ibrcom 155 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M lcm  N
)  =  ( M  gcd  N )  -> 
( M lcm  N ) 
||  N ) )
87imp 122 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  ( M lcm  N
)  ||  N )
9 lcmcl 10661 . . . . . . . . . . 11  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  N )  e.  NN0 )
109nn0zd 8600 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  N )  e.  ZZ )
11 dvdstr 10440 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  ( M lcm  N )  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  ||  ( M lcm  N )  /\  ( M lcm  N )  ||  N
)  ->  M  ||  N
) )
1210, 11syl3an2 1204 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  e.  ZZ )  ->  ( ( M 
||  ( M lcm  N
)  /\  ( M lcm  N )  ||  N )  ->  M  ||  N
) )
13123com12 1143 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M 
||  ( M lcm  N
)  /\  ( M lcm  N )  ||  N )  ->  M  ||  N
) )
14133expb 1140 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( ( M  ||  ( M lcm  N )  /\  ( M lcm  N ) 
||  N )  ->  M  ||  N ) )
1514anidms 389 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  ||  ( M lcm  N )  /\  ( M lcm  N ) 
||  N )  ->  M  ||  N ) )
1615adantr 270 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  ( ( M 
||  ( M lcm  N
)  /\  ( M lcm  N )  ||  N )  ->  M  ||  N
) )
173, 8, 16mp2and 424 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  M  ||  N
)
18 absdvdsb 10421 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  ( abs `  M ) 
||  N ) )
19 zabscl 10173 . . . . . . 7  |-  ( M  e.  ZZ  ->  ( abs `  M )  e.  ZZ )
20 dvdsabsb 10422 . . . . . . 7  |-  ( ( ( abs `  M
)  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  M
)  ||  N  <->  ( abs `  M )  ||  ( abs `  N ) ) )
2119, 20sylan 277 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  M
)  ||  N  <->  ( abs `  M )  ||  ( abs `  N ) ) )
2218, 21bitrd 186 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  ( abs `  M ) 
||  ( abs `  N
) ) )
2322adantr 270 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  ( M  ||  N 
<->  ( abs `  M
)  ||  ( abs `  N ) ) )
2417, 23mpbid 145 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  ( abs `  M
)  ||  ( abs `  N ) )
251simprd 112 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  N  ||  ( M lcm 
N ) )
2625adantr 270 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  N  ||  ( M lcm  N ) )
274simpld 110 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  ||  M )
28 breq1 3808 . . . . . . 7  |-  ( ( M lcm  N )  =  ( M  gcd  N
)  ->  ( ( M lcm  N )  ||  M  <->  ( M  gcd  N ) 
||  M ) )
2927, 28syl5ibrcom 155 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M lcm  N
)  =  ( M  gcd  N )  -> 
( M lcm  N ) 
||  M ) )
3029imp 122 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  ( M lcm  N
)  ||  M )
31 dvdstr 10440 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  ( M lcm  N )  e.  ZZ  /\  M  e.  ZZ )  ->  (
( N  ||  ( M lcm  N )  /\  ( M lcm  N )  ||  M
)  ->  N  ||  M
) )
3210, 31syl3an2 1204 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  e.  ZZ )  ->  ( ( N 
||  ( M lcm  N
)  /\  ( M lcm  N )  ||  M )  ->  N  ||  M
) )
33323coml 1146 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( N 
||  ( M lcm  N
)  /\  ( M lcm  N )  ||  M )  ->  N  ||  M
) )
34333expb 1140 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( ( N  ||  ( M lcm  N )  /\  ( M lcm  N ) 
||  M )  ->  N  ||  M ) )
3534anidms 389 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( N  ||  ( M lcm  N )  /\  ( M lcm  N ) 
||  M )  ->  N  ||  M ) )
3635adantr 270 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  ( ( N 
||  ( M lcm  N
)  /\  ( M lcm  N )  ||  M )  ->  N  ||  M
) )
3726, 30, 36mp2and 424 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  N  ||  M
)
38 absdvdsb 10421 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( N  ||  M  <->  ( abs `  N ) 
||  M ) )
39 zabscl 10173 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( abs `  N )  e.  ZZ )
40 dvdsabsb 10422 . . . . . . . 8  |-  ( ( ( abs `  N
)  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( abs `  N
)  ||  M  <->  ( abs `  N )  ||  ( abs `  M ) ) )
4139, 40sylan 277 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( abs `  N
)  ||  M  <->  ( abs `  N )  ||  ( abs `  M ) ) )
4238, 41bitrd 186 . . . . . 6  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( N  ||  M  <->  ( abs `  N ) 
||  ( abs `  M
) ) )
4342ancoms 264 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  ||  M  <->  ( abs `  N ) 
||  ( abs `  M
) ) )
4443adantr 270 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  ( N  ||  M 
<->  ( abs `  N
)  ||  ( abs `  M ) ) )
4537, 44mpbid 145 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  ( abs `  N
)  ||  ( abs `  M ) )
46 nn0abscl 10172 . . . . . . 7  |-  ( M  e.  ZZ  ->  ( abs `  M )  e. 
NN0 )
47 nn0abscl 10172 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( abs `  N )  e. 
NN0 )
4846, 47anim12i 331 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  M
)  e.  NN0  /\  ( abs `  N )  e.  NN0 ) )
49 dvdseq 10456 . . . . . 6  |-  ( ( ( ( abs `  M
)  e.  NN0  /\  ( abs `  N )  e.  NN0 )  /\  ( ( abs `  M
)  ||  ( abs `  N )  /\  ( abs `  N )  ||  ( abs `  M ) ) )  ->  ( abs `  M )  =  ( abs `  N
) )
5048, 49sylan 277 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( abs `  M )  ||  ( abs `  N )  /\  ( abs `  N ) 
||  ( abs `  M
) ) )  -> 
( abs `  M
)  =  ( abs `  N ) )
5150ex 113 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( abs `  M )  ||  ( abs `  N )  /\  ( abs `  N ) 
||  ( abs `  M
) )  ->  ( abs `  M )  =  ( abs `  N
) ) )
5251adantr 270 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  ( ( ( abs `  M ) 
||  ( abs `  N
)  /\  ( abs `  N )  ||  ( abs `  M ) )  ->  ( abs `  M
)  =  ( abs `  N ) ) )
5324, 45, 52mp2and 424 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  ( abs `  M
)  =  ( abs `  N ) )
54 lcmid 10669 . . . . . . . 8  |-  ( ( abs `  M )  e.  ZZ  ->  (
( abs `  M
) lcm  ( abs `  M
) )  =  ( abs `  ( abs `  M ) ) )
5519, 54syl 14 . . . . . . 7  |-  ( M  e.  ZZ  ->  (
( abs `  M
) lcm  ( abs `  M
) )  =  ( abs `  ( abs `  M ) ) )
56 gcdid 10584 . . . . . . . 8  |-  ( ( abs `  M )  e.  ZZ  ->  (
( abs `  M
)  gcd  ( abs `  M ) )  =  ( abs `  ( abs `  M ) ) )
5719, 56syl 14 . . . . . . 7  |-  ( M  e.  ZZ  ->  (
( abs `  M
)  gcd  ( abs `  M ) )  =  ( abs `  ( abs `  M ) ) )
5855, 57eqtr4d 2118 . . . . . 6  |-  ( M  e.  ZZ  ->  (
( abs `  M
) lcm  ( abs `  M
) )  =  ( ( abs `  M
)  gcd  ( abs `  M ) ) )
59 oveq2 5571 . . . . . . 7  |-  ( ( abs `  M )  =  ( abs `  N
)  ->  ( ( abs `  M ) lcm  ( abs `  M ) )  =  ( ( abs `  M ) lcm  ( abs `  N ) ) )
60 oveq2 5571 . . . . . . 7  |-  ( ( abs `  M )  =  ( abs `  N
)  ->  ( ( abs `  M )  gcd  ( abs `  M
) )  =  ( ( abs `  M
)  gcd  ( abs `  N ) ) )
6159, 60eqeq12d 2097 . . . . . 6  |-  ( ( abs `  M )  =  ( abs `  N
)  ->  ( (
( abs `  M
) lcm  ( abs `  M
) )  =  ( ( abs `  M
)  gcd  ( abs `  M ) )  <->  ( ( abs `  M ) lcm  ( abs `  N ) )  =  ( ( abs `  M )  gcd  ( abs `  N ) ) ) )
6258, 61syl5ibcom 153 . . . . 5  |-  ( M  e.  ZZ  ->  (
( abs `  M
)  =  ( abs `  N )  ->  (
( abs `  M
) lcm  ( abs `  N
) )  =  ( ( abs `  M
)  gcd  ( abs `  N ) ) ) )
6362imp 122 . . . 4  |-  ( ( M  e.  ZZ  /\  ( abs `  M )  =  ( abs `  N
) )  ->  (
( abs `  M
) lcm  ( abs `  N
) )  =  ( ( abs `  M
)  gcd  ( abs `  N ) ) )
6463adantlr 461 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( abs `  M
)  =  ( abs `  N ) )  -> 
( ( abs `  M
) lcm  ( abs `  N
) )  =  ( ( abs `  M
)  gcd  ( abs `  N ) ) )
65 lcmabs 10665 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  M
) lcm  ( abs `  N
) )  =  ( M lcm  N ) )
66 gcdabs 10586 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  M
)  gcd  ( abs `  N ) )  =  ( M  gcd  N
) )
6765, 66eqeq12d 2097 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( abs `  M ) lcm  ( abs `  N ) )  =  ( ( abs `  M
)  gcd  ( abs `  N ) )  <->  ( M lcm  N )  =  ( M  gcd  N ) ) )
6867adantr 270 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( abs `  M
)  =  ( abs `  N ) )  -> 
( ( ( abs `  M ) lcm  ( abs `  N ) )  =  ( ( abs `  M
)  gcd  ( abs `  N ) )  <->  ( M lcm  N )  =  ( M  gcd  N ) ) )
6964, 68mpbid 145 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( abs `  M
)  =  ( abs `  N ) )  -> 
( M lcm  N )  =  ( M  gcd  N ) )
7053, 69impbida 561 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M lcm  N
)  =  ( M  gcd  N )  <->  ( abs `  M )  =  ( abs `  N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   class class class wbr 3805   ` cfv 4952  (class class class)co 5563   NN0cn0 8407   ZZcz 8484   abscabs 10084    || cdvds 10403    gcd cgcd 10545   lcm clcm 10649
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 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-mulrcl 7189  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-precex 7200  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-apti 7205  ax-pre-ltadd 7206  ax-pre-mulgt0 7207  ax-pre-mulext 7208  ax-arch 7209  ax-caucvg 7210
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-isom 4961  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-frec 6060  df-sup 6491  df-inf 6492  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-reap 7794  df-ap 7801  df-div 7880  df-inn 8159  df-2 8217  df-3 8218  df-4 8219  df-n0 8408  df-z 8485  df-uz 8753  df-q 8838  df-rp 8868  df-fz 9158  df-fzo 9282  df-fl 9404  df-mod 9457  df-iseq 9574  df-iexp 9625  df-cj 9930  df-re 9931  df-im 9932  df-rsqrt 10085  df-abs 10086  df-dvds 10404  df-gcd 10546  df-lcm 10650
This theorem is referenced by: (None)
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