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Theorem lcmgcdeq 12097
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 12083 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  ( M lcm  N )  /\  N  ||  ( M lcm  N ) ) )
21simpld 112 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  M  ||  ( M lcm 
N ) )
32adantr 276 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  M  ||  ( M lcm  N ) )
4 gcddvds 11978 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  gcd  N )  ||  M  /\  ( M  gcd  N ) 
||  N ) )
54simprd 114 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  ||  N )
6 breq1 4018 . . . . . . 7  |-  ( ( M lcm  N )  =  ( M  gcd  N
)  ->  ( ( M lcm  N )  ||  N  <->  ( M  gcd  N ) 
||  N ) )
75, 6syl5ibrcom 157 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M lcm  N
)  =  ( M  gcd  N )  -> 
( M lcm  N ) 
||  N ) )
87imp 124 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  ( M lcm  N
)  ||  N )
9 lcmcl 12086 . . . . . . . . . . 11  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  N )  e.  NN0 )
109nn0zd 9387 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  N )  e.  ZZ )
11 dvdstr 11849 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  ( M lcm  N )  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  ||  ( M lcm  N )  /\  ( M lcm  N )  ||  N
)  ->  M  ||  N
) )
1210, 11syl3an2 1282 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  e.  ZZ )  ->  ( ( M 
||  ( M lcm  N
)  /\  ( M lcm  N )  ||  N )  ->  M  ||  N
) )
13123com12 1208 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M 
||  ( M lcm  N
)  /\  ( M lcm  N )  ||  N )  ->  M  ||  N
) )
14133expb 1205 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( ( M  ||  ( M lcm  N )  /\  ( M lcm  N ) 
||  N )  ->  M  ||  N ) )
1514anidms 397 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  ||  ( M lcm  N )  /\  ( M lcm  N ) 
||  N )  ->  M  ||  N ) )
1615adantr 276 . . . . 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 433 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  M  ||  N
)
18 absdvdsb 11830 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  ( abs `  M ) 
||  N ) )
19 zabscl 11109 . . . . . . 7  |-  ( M  e.  ZZ  ->  ( abs `  M )  e.  ZZ )
20 dvdsabsb 11831 . . . . . . 7  |-  ( ( ( abs `  M
)  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  M
)  ||  N  <->  ( abs `  M )  ||  ( abs `  N ) ) )
2119, 20sylan 283 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  M
)  ||  N  <->  ( abs `  M )  ||  ( abs `  N ) ) )
2218, 21bitrd 188 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  ( abs `  M ) 
||  ( abs `  N
) ) )
2322adantr 276 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  ( M  ||  N 
<->  ( abs `  M
)  ||  ( abs `  N ) ) )
2417, 23mpbid 147 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  ( abs `  M
)  ||  ( abs `  N ) )
251simprd 114 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  N  ||  ( M lcm 
N ) )
2625adantr 276 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  N  ||  ( M lcm  N ) )
274simpld 112 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  ||  M )
28 breq1 4018 . . . . . . 7  |-  ( ( M lcm  N )  =  ( M  gcd  N
)  ->  ( ( M lcm  N )  ||  M  <->  ( M  gcd  N ) 
||  M ) )
2927, 28syl5ibrcom 157 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M lcm  N
)  =  ( M  gcd  N )  -> 
( M lcm  N ) 
||  M ) )
3029imp 124 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  ( M lcm  N
)  ||  M )
31 dvdstr 11849 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  ( M lcm  N )  e.  ZZ  /\  M  e.  ZZ )  ->  (
( N  ||  ( M lcm  N )  /\  ( M lcm  N )  ||  M
)  ->  N  ||  M
) )
3210, 31syl3an2 1282 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  e.  ZZ )  ->  ( ( N 
||  ( M lcm  N
)  /\  ( M lcm  N )  ||  M )  ->  N  ||  M
) )
33323coml 1211 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( N 
||  ( M lcm  N
)  /\  ( M lcm  N )  ||  M )  ->  N  ||  M
) )
34333expb 1205 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( ( N  ||  ( M lcm  N )  /\  ( M lcm  N ) 
||  M )  ->  N  ||  M ) )
3534anidms 397 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( N  ||  ( M lcm  N )  /\  ( M lcm  N ) 
||  M )  ->  N  ||  M ) )
3635adantr 276 . . . . 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 433 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  N  ||  M
)
38 absdvdsb 11830 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( N  ||  M  <->  ( abs `  N ) 
||  M ) )
39 zabscl 11109 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( abs `  N )  e.  ZZ )
40 dvdsabsb 11831 . . . . . . . 8  |-  ( ( ( abs `  N
)  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( abs `  N
)  ||  M  <->  ( abs `  N )  ||  ( abs `  M ) ) )
4139, 40sylan 283 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( abs `  N
)  ||  M  <->  ( abs `  N )  ||  ( abs `  M ) ) )
4238, 41bitrd 188 . . . . . 6  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( N  ||  M  <->  ( abs `  N ) 
||  ( abs `  M
) ) )
4342ancoms 268 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  ||  M  <->  ( abs `  N ) 
||  ( abs `  M
) ) )
4443adantr 276 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  ( N  ||  M 
<->  ( abs `  N
)  ||  ( abs `  M ) ) )
4537, 44mpbid 147 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  ( abs `  N
)  ||  ( abs `  M ) )
46 nn0abscl 11108 . . . . . . 7  |-  ( M  e.  ZZ  ->  ( abs `  M )  e. 
NN0 )
47 nn0abscl 11108 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( abs `  N )  e. 
NN0 )
4846, 47anim12i 338 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  M
)  e.  NN0  /\  ( abs `  N )  e.  NN0 ) )
49 dvdseq 11868 . . . . . 6  |-  ( ( ( ( abs `  M
)  e.  NN0  /\  ( abs `  N )  e.  NN0 )  /\  ( ( abs `  M
)  ||  ( abs `  N )  /\  ( abs `  N )  ||  ( abs `  M ) ) )  ->  ( abs `  M )  =  ( abs `  N
) )
5048, 49sylan 283 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( abs `  M )  ||  ( abs `  N )  /\  ( abs `  N ) 
||  ( abs `  M
) ) )  -> 
( abs `  M
)  =  ( abs `  N ) )
5150ex 115 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( abs `  M )  ||  ( abs `  N )  /\  ( abs `  N ) 
||  ( abs `  M
) )  ->  ( abs `  M )  =  ( abs `  N
) ) )
5251adantr 276 . . 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 433 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M lcm  N
)  =  ( M  gcd  N ) )  ->  ( abs `  M
)  =  ( abs `  N ) )
54 lcmid 12094 . . . . . . . 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 12001 . . . . . . . 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 2223 . . . . . 6  |-  ( M  e.  ZZ  ->  (
( abs `  M
) lcm  ( abs `  M
) )  =  ( ( abs `  M
)  gcd  ( abs `  M ) ) )
59 oveq2 5896 . . . . . . 7  |-  ( ( abs `  M )  =  ( abs `  N
)  ->  ( ( abs `  M ) lcm  ( abs `  M ) )  =  ( ( abs `  M ) lcm  ( abs `  N ) ) )
60 oveq2 5896 . . . . . . 7  |-  ( ( abs `  M )  =  ( abs `  N
)  ->  ( ( abs `  M )  gcd  ( abs `  M
) )  =  ( ( abs `  M
)  gcd  ( abs `  N ) ) )
6159, 60eqeq12d 2202 . . . . . 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 155 . . . . 5  |-  ( M  e.  ZZ  ->  (
( abs `  M
)  =  ( abs `  N )  ->  (
( abs `  M
) lcm  ( abs `  N
) )  =  ( ( abs `  M
)  gcd  ( abs `  N ) ) ) )
6362imp 124 . . . 4  |-  ( ( M  e.  ZZ  /\  ( abs `  M )  =  ( abs `  N
) )  ->  (
( abs `  M
) lcm  ( abs `  N
) )  =  ( ( abs `  M
)  gcd  ( abs `  N ) ) )
6463adantlr 477 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( abs `  M
)  =  ( abs `  N ) )  -> 
( ( abs `  M
) lcm  ( abs `  N
) )  =  ( ( abs `  M
)  gcd  ( abs `  N ) ) )
65 lcmabs 12090 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  M
) lcm  ( abs `  N
) )  =  ( M lcm  N ) )
66 gcdabs 12003 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  M
)  gcd  ( abs `  N ) )  =  ( M  gcd  N
) )
6765, 66eqeq12d 2202 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( abs `  M ) lcm  ( abs `  N ) )  =  ( ( abs `  M
)  gcd  ( abs `  N ) )  <->  ( M lcm  N )  =  ( M  gcd  N ) ) )
6867adantr 276 . . 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 147 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( abs `  M
)  =  ( abs `  N ) )  -> 
( M lcm  N )  =  ( M  gcd  N ) )
7053, 69impbida 596 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 104    <-> wb 105    = wceq 1363    e. wcel 2158   class class class wbr 4015   ` cfv 5228  (class class class)co 5888   NN0cn0 9190   ZZcz 9267   abscabs 11020    || cdvds 11808    gcd cgcd 11957   lcm clcm 12074
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942  ax-pre-mulext 7943  ax-arch 7944  ax-caucvg 7945
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-isom 5237  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-frec 6406  df-sup 6997  df-inf 6998  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553  df-div 8644  df-inn 8934  df-2 8992  df-3 8993  df-4 8994  df-n0 9191  df-z 9268  df-uz 9543  df-q 9634  df-rp 9668  df-fz 10023  df-fzo 10157  df-fl 10284  df-mod 10337  df-seqfrec 10460  df-exp 10534  df-cj 10865  df-re 10866  df-im 10867  df-rsqrt 11021  df-abs 11022  df-dvds 11809  df-gcd 11958  df-lcm 12075
This theorem is referenced by: (None)
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