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Theorem lcmgcdeq 11775
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 11761 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  ( M lcm  N )  /\  N  ||  ( M lcm  N ) ) )
21simpld 111 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  M  ||  ( M lcm 
N ) )
32adantr 274 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  M  ||  ( M lcm  N ) )
4 gcddvds 11663 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  gcd  N )  ||  M  /\  ( M  gcd  N ) 
||  N ) )
54simprd 113 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  ||  N )
6 breq1 3932 . . . . . . 7  |-  ( ( M lcm  N )  =  ( M  gcd  N
)  ->  ( ( M lcm  N )  ||  N  <->  ( M  gcd  N ) 
||  N ) )
75, 6syl5ibrcom 156 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M lcm  N
)  =  ( M  gcd  N )  -> 
( M lcm  N ) 
||  N ) )
87imp 123 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  ( M lcm  N
)  ||  N )
9 lcmcl 11764 . . . . . . . . . . 11  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  N )  e.  NN0 )
109nn0zd 9183 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  N )  e.  ZZ )
11 dvdstr 11541 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  ( M lcm  N )  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  ||  ( M lcm  N )  /\  ( M lcm  N )  ||  N
)  ->  M  ||  N
) )
1210, 11syl3an2 1250 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  e.  ZZ )  ->  ( ( M 
||  ( M lcm  N
)  /\  ( M lcm  N )  ||  N )  ->  M  ||  N
) )
13123com12 1185 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M 
||  ( M lcm  N
)  /\  ( M lcm  N )  ||  N )  ->  M  ||  N
) )
14133expb 1182 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( ( M  ||  ( M lcm  N )  /\  ( M lcm  N ) 
||  N )  ->  M  ||  N ) )
1514anidms 394 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  ||  ( M lcm  N )  /\  ( M lcm  N ) 
||  N )  ->  M  ||  N ) )
1615adantr 274 . . . . 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 429 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  M  ||  N
)
18 absdvdsb 11522 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  ( abs `  M ) 
||  N ) )
19 zabscl 10870 . . . . . . 7  |-  ( M  e.  ZZ  ->  ( abs `  M )  e.  ZZ )
20 dvdsabsb 11523 . . . . . . 7  |-  ( ( ( abs `  M
)  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  M
)  ||  N  <->  ( abs `  M )  ||  ( abs `  N ) ) )
2119, 20sylan 281 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  M
)  ||  N  <->  ( abs `  M )  ||  ( abs `  N ) ) )
2218, 21bitrd 187 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  ( abs `  M ) 
||  ( abs `  N
) ) )
2322adantr 274 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  ( M  ||  N 
<->  ( abs `  M
)  ||  ( abs `  N ) ) )
2417, 23mpbid 146 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  ( abs `  M
)  ||  ( abs `  N ) )
251simprd 113 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  N  ||  ( M lcm 
N ) )
2625adantr 274 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  N  ||  ( M lcm  N ) )
274simpld 111 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  ||  M )
28 breq1 3932 . . . . . . 7  |-  ( ( M lcm  N )  =  ( M  gcd  N
)  ->  ( ( M lcm  N )  ||  M  <->  ( M  gcd  N ) 
||  M ) )
2927, 28syl5ibrcom 156 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M lcm  N
)  =  ( M  gcd  N )  -> 
( M lcm  N ) 
||  M ) )
3029imp 123 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  ( M lcm  N
)  ||  M )
31 dvdstr 11541 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  ( M lcm  N )  e.  ZZ  /\  M  e.  ZZ )  ->  (
( N  ||  ( M lcm  N )  /\  ( M lcm  N )  ||  M
)  ->  N  ||  M
) )
3210, 31syl3an2 1250 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  e.  ZZ )  ->  ( ( N 
||  ( M lcm  N
)  /\  ( M lcm  N )  ||  M )  ->  N  ||  M
) )
33323coml 1188 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( N 
||  ( M lcm  N
)  /\  ( M lcm  N )  ||  M )  ->  N  ||  M
) )
34333expb 1182 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( ( N  ||  ( M lcm  N )  /\  ( M lcm  N ) 
||  M )  ->  N  ||  M ) )
3534anidms 394 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( N  ||  ( M lcm  N )  /\  ( M lcm  N ) 
||  M )  ->  N  ||  M ) )
3635adantr 274 . . . . 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 429 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  N  ||  M
)
38 absdvdsb 11522 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( N  ||  M  <->  ( abs `  N ) 
||  M ) )
39 zabscl 10870 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( abs `  N )  e.  ZZ )
40 dvdsabsb 11523 . . . . . . . 8  |-  ( ( ( abs `  N
)  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( abs `  N
)  ||  M  <->  ( abs `  N )  ||  ( abs `  M ) ) )
4139, 40sylan 281 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( abs `  N
)  ||  M  <->  ( abs `  N )  ||  ( abs `  M ) ) )
4238, 41bitrd 187 . . . . . 6  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( N  ||  M  <->  ( abs `  N ) 
||  ( abs `  M
) ) )
4342ancoms 266 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  ||  M  <->  ( abs `  N ) 
||  ( abs `  M
) ) )
4443adantr 274 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  ( N  ||  M 
<->  ( abs `  N
)  ||  ( abs `  M ) ) )
4537, 44mpbid 146 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  ( abs `  N
)  ||  ( abs `  M ) )
46 nn0abscl 10869 . . . . . . 7  |-  ( M  e.  ZZ  ->  ( abs `  M )  e. 
NN0 )
47 nn0abscl 10869 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( abs `  N )  e. 
NN0 )
4846, 47anim12i 336 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  M
)  e.  NN0  /\  ( abs `  N )  e.  NN0 ) )
49 dvdseq 11557 . . . . . 6  |-  ( ( ( ( abs `  M
)  e.  NN0  /\  ( abs `  N )  e.  NN0 )  /\  ( ( abs `  M
)  ||  ( abs `  N )  /\  ( abs `  N )  ||  ( abs `  M ) ) )  ->  ( abs `  M )  =  ( abs `  N
) )
5048, 49sylan 281 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( abs `  M )  ||  ( abs `  N )  /\  ( abs `  N ) 
||  ( abs `  M
) ) )  -> 
( abs `  M
)  =  ( abs `  N ) )
5150ex 114 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( abs `  M )  ||  ( abs `  N )  /\  ( abs `  N ) 
||  ( abs `  M
) )  ->  ( abs `  M )  =  ( abs `  N
) ) )
5251adantr 274 . . 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 429 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  ( abs `  M
)  =  ( abs `  N ) )
54 lcmid 11772 . . . . . . . 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 11685 . . . . . . . 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 2175 . . . . . 6  |-  ( M  e.  ZZ  ->  (
( abs `  M
) lcm  ( abs `  M
) )  =  ( ( abs `  M
)  gcd  ( abs `  M ) ) )
59 oveq2 5782 . . . . . . 7  |-  ( ( abs `  M )  =  ( abs `  N
)  ->  ( ( abs `  M ) lcm  ( abs `  M ) )  =  ( ( abs `  M ) lcm  ( abs `  N ) ) )
60 oveq2 5782 . . . . . . 7  |-  ( ( abs `  M )  =  ( abs `  N
)  ->  ( ( abs `  M )  gcd  ( abs `  M
) )  =  ( ( abs `  M
)  gcd  ( abs `  N ) ) )
6159, 60eqeq12d 2154 . . . . . 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 154 . . . . 5  |-  ( M  e.  ZZ  ->  (
( abs `  M
)  =  ( abs `  N )  ->  (
( abs `  M
) lcm  ( abs `  N
) )  =  ( ( abs `  M
)  gcd  ( abs `  N ) ) ) )
6362imp 123 . . . 4  |-  ( ( M  e.  ZZ  /\  ( abs `  M )  =  ( abs `  N
) )  ->  (
( abs `  M
) lcm  ( abs `  N
) )  =  ( ( abs `  M
)  gcd  ( abs `  N ) ) )
6463adantlr 468 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( abs `  M
)  =  ( abs `  N ) )  -> 
( ( abs `  M
) lcm  ( abs `  N
) )  =  ( ( abs `  M
)  gcd  ( abs `  N ) ) )
65 lcmabs 11768 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  M
) lcm  ( abs `  N
) )  =  ( M lcm  N ) )
66 gcdabs 11687 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  M
)  gcd  ( abs `  N ) )  =  ( M  gcd  N
) )
6765, 66eqeq12d 2154 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( abs `  M ) lcm  ( abs `  N ) )  =  ( ( abs `  M
)  gcd  ( abs `  N ) )  <->  ( M lcm  N )  =  ( M  gcd  N ) ) )
6867adantr 274 . . 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 146 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( abs `  M
)  =  ( abs `  N ) )  -> 
( M lcm  N )  =  ( M  gcd  N ) )
7053, 69impbida 585 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 103    <-> wb 104    = wceq 1331    e. wcel 1480   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   NN0cn0 8989   ZZcz 9066   abscabs 10781    || cdvds 11504    gcd cgcd 11646   lcm clcm 11752
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulrcl 7731  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-precex 7742  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748  ax-pre-mulgt0 7749  ax-pre-mulext 7750  ax-arch 7751  ax-caucvg 7752
This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-sup 6871  df-inf 6872  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-reap 8349  df-ap 8356  df-div 8445  df-inn 8733  df-2 8791  df-3 8792  df-4 8793  df-n0 8990  df-z 9067  df-uz 9339  df-q 9424  df-rp 9454  df-fz 9803  df-fzo 9932  df-fl 10055  df-mod 10108  df-seqfrec 10231  df-exp 10305  df-cj 10626  df-re 10627  df-im 10628  df-rsqrt 10782  df-abs 10783  df-dvds 11505  df-gcd 11647  df-lcm 11753
This theorem is referenced by: (None)
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