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Theorem qredeq 13098
Description: Two equal reduced fractions have the same numerator and denominator. (Contributed by Jeff Hankins, 29-Sep-2013.)
Assertion
Ref Expression
qredeq  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  /\  ( M  /  N
)  =  ( P  /  Q ) )  ->  ( M  =  P  /\  N  =  Q ) )

Proof of Theorem qredeq
StepHypRef Expression
1 zcn 10279 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  CC )
21adantr 452 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  M  e.  CC )
3 nncn 10000 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  CC )
43adantl 453 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N  e.  CC )
5 nnne0 10024 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  =/=  0 )
65adantl 453 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N  =/=  0 )
72, 4, 6divcld 9782 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  /  N
)  e.  CC )
873adant3 977 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  ( M  /  N )  e.  CC )
98adantr 452 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  ( M  /  N )  e.  CC )
10 zcn 10279 . . . . . . . . . 10  |-  ( P  e.  ZZ  ->  P  e.  CC )
1110adantr 452 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  Q  e.  NN )  ->  P  e.  CC )
12 nncn 10000 . . . . . . . . . 10  |-  ( Q  e.  NN  ->  Q  e.  CC )
1312adantl 453 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  Q  e.  NN )  ->  Q  e.  CC )
14 nnne0 10024 . . . . . . . . . 10  |-  ( Q  e.  NN  ->  Q  =/=  0 )
1514adantl 453 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  Q  e.  NN )  ->  Q  =/=  0 )
1611, 13, 15divcld 9782 . . . . . . . 8  |-  ( ( P  e.  ZZ  /\  Q  e.  NN )  ->  ( P  /  Q
)  e.  CC )
17163adant3 977 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  ( P  /  Q )  e.  CC )
1817adantl 453 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  ( P  /  Q )  e.  CC )
1933ad2ant2 979 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  N  e.  CC )
2019adantr 452 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  N  e.  CC )
2153ad2ant2 979 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  N  =/=  0 )
2221adantr 452 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  N  =/=  0
)
239, 18, 20, 22mulcand 9647 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  ( ( N  x.  ( M  /  N ) )  =  ( N  x.  ( P  /  Q ) )  <-> 
( M  /  N
)  =  ( P  /  Q ) ) )
242, 4, 6divcan2d 9784 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  x.  ( M  /  N ) )  =  M )
25243adant3 977 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  ( N  x.  ( M  /  N ) )  =  M )
2625adantr 452 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  ( N  x.  ( M  /  N
) )  =  M )
2726eqeq1d 2443 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  ( ( N  x.  ( M  /  N ) )  =  ( N  x.  ( P  /  Q ) )  <-> 
M  =  ( N  x.  ( P  /  Q ) ) ) )
2823, 27bitr3d 247 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  ( ( M  /  N )  =  ( P  /  Q
)  <->  M  =  ( N  x.  ( P  /  Q ) ) ) )
2913ad2ant1 978 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  M  e.  CC )
3029adantr 452 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  M  e.  CC )
31 mulcl 9066 . . . . . . 7  |-  ( ( N  e.  CC  /\  ( P  /  Q
)  e.  CC )  ->  ( N  x.  ( P  /  Q
) )  e.  CC )
3219, 17, 31syl2an 464 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  ( N  x.  ( P  /  Q
) )  e.  CC )
33123ad2ant2 979 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  Q  e.  CC )
3433adantl 453 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  Q  e.  CC )
35143ad2ant2 979 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  Q  =/=  0 )
3635adantl 453 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  Q  =/=  0
)
3730, 32, 34, 36mulcan2d 9648 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  ( ( M  x.  Q )  =  ( ( N  x.  ( P  /  Q
) )  x.  Q
)  <->  M  =  ( N  x.  ( P  /  Q ) ) ) )
3820, 18, 34mulassd 9103 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  ( ( N  x.  ( P  /  Q ) )  x.  Q )  =  ( N  x.  ( ( P  /  Q )  x.  Q ) ) )
3911, 13, 15divcan1d 9783 . . . . . . . . . 10  |-  ( ( P  e.  ZZ  /\  Q  e.  NN )  ->  ( ( P  /  Q )  x.  Q
)  =  P )
40393adant3 977 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  (
( P  /  Q
)  x.  Q )  =  P )
4140adantl 453 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  ( ( P  /  Q )  x.  Q )  =  P )
4241oveq2d 6089 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  ( N  x.  ( ( P  /  Q )  x.  Q
) )  =  ( N  x.  P ) )
4338, 42eqtrd 2467 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  ( ( N  x.  ( P  /  Q ) )  x.  Q )  =  ( N  x.  P ) )
4443eqeq2d 2446 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  ( ( M  x.  Q )  =  ( ( N  x.  ( P  /  Q
) )  x.  Q
)  <->  ( M  x.  Q )  =  ( N  x.  P ) ) )
4537, 44bitr3d 247 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  ( M  =  ( N  x.  ( P  /  Q ) )  <-> 
( M  x.  Q
)  =  ( N  x.  P ) ) )
4628, 45bitrd 245 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  ( ( M  /  N )  =  ( P  /  Q
)  <->  ( M  x.  Q )  =  ( N  x.  P ) ) )
47 nnz 10295 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  ZZ )
48473ad2ant2 979 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  N  e.  ZZ )
49 simp2 958 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  Q  e.  NN )
5048, 49anim12i 550 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  ( N  e.  ZZ  /\  Q  e.  NN ) )
5150adantr 452 . . . . . . 7  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  /\  ( M  x.  Q
)  =  ( N  x.  P ) )  ->  ( N  e.  ZZ  /\  Q  e.  NN ) )
5248adantr 452 . . . . . . . . . 10  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  N  e.  ZZ )
53 simpl1 960 . . . . . . . . . 10  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  M  e.  ZZ )
54 nnz 10295 . . . . . . . . . . . 12  |-  ( Q  e.  NN  ->  Q  e.  ZZ )
55543ad2ant2 979 . . . . . . . . . . 11  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  Q  e.  ZZ )
5655adantl 453 . . . . . . . . . 10  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  Q  e.  ZZ )
5752, 53, 563jca 1134 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  ( N  e.  ZZ  /\  M  e.  ZZ  /\  Q  e.  ZZ ) )
5857adantr 452 . . . . . . . 8  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  /\  ( M  x.  Q
)  =  ( N  x.  P ) )  ->  ( N  e.  ZZ  /\  M  e.  ZZ  /\  Q  e.  ZZ ) )
59 simp1 957 . . . . . . . . . . . 12  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  P  e.  ZZ )
60 dvdsmul1 12863 . . . . . . . . . . . 12  |-  ( ( N  e.  ZZ  /\  P  e.  ZZ )  ->  N  ||  ( N  x.  P ) )
6148, 59, 60syl2an 464 . . . . . . . . . . 11  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  N  ||  ( N  x.  P )
)
6261adantr 452 . . . . . . . . . 10  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  /\  ( M  x.  Q
)  =  ( N  x.  P ) )  ->  N  ||  ( N  x.  P )
)
63 simpr 448 . . . . . . . . . 10  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  /\  ( M  x.  Q
)  =  ( N  x.  P ) )  ->  ( M  x.  Q )  =  ( N  x.  P ) )
6462, 63breqtrrd 4230 . . . . . . . . 9  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  /\  ( M  x.  Q
)  =  ( N  x.  P ) )  ->  N  ||  ( M  x.  Q )
)
65 gcdcom 13012 . . . . . . . . . . . . . 14  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( N  gcd  M
)  =  ( M  gcd  N ) )
6647, 65sylan 458 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  M  e.  ZZ )  ->  ( N  gcd  M
)  =  ( M  gcd  N ) )
6766ancoms 440 . . . . . . . . . . . 12  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  gcd  M
)  =  ( M  gcd  N ) )
68673adant3 977 . . . . . . . . . . 11  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  ( N  gcd  M )  =  ( M  gcd  N
) )
69 simp3 959 . . . . . . . . . . 11  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  ( M  gcd  N )  =  1 )
7068, 69eqtrd 2467 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  ( N  gcd  M )  =  1 )
7170ad2antrr 707 . . . . . . . . 9  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  /\  ( M  x.  Q
)  =  ( N  x.  P ) )  ->  ( N  gcd  M )  =  1 )
7264, 71jca 519 . . . . . . . 8  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  /\  ( M  x.  Q
)  =  ( N  x.  P ) )  ->  ( N  ||  ( M  x.  Q
)  /\  ( N  gcd  M )  =  1 ) )
73 coprmdvds 13094 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  Q  e.  ZZ )  ->  (
( N  ||  ( M  x.  Q )  /\  ( N  gcd  M
)  =  1 )  ->  N  ||  Q
) )
7458, 72, 73sylc 58 . . . . . . 7  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  /\  ( M  x.  Q
)  =  ( N  x.  P ) )  ->  N  ||  Q
)
75 dvdsle 12887 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  Q  e.  NN )  ->  ( N  ||  Q  ->  N  <_  Q )
)
7651, 74, 75sylc 58 . . . . . 6  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  /\  ( M  x.  Q
)  =  ( N  x.  P ) )  ->  N  <_  Q
)
77 simp2 958 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  N  e.  NN )
7855, 77anim12i 550 . . . . . . . . 9  |-  ( ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  /\  ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 ) )  ->  ( Q  e.  ZZ  /\  N  e.  NN ) )
7978ancoms 440 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  ( Q  e.  ZZ  /\  N  e.  NN ) )
8079adantr 452 . . . . . . 7  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  /\  ( M  x.  Q
)  =  ( N  x.  P ) )  ->  ( Q  e.  ZZ  /\  N  e.  NN ) )
81 simpr1 963 . . . . . . . . . 10  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  P  e.  ZZ )
8256, 81, 523jca 1134 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  ( Q  e.  ZZ  /\  P  e.  ZZ  /\  N  e.  ZZ ) )
8382adantr 452 . . . . . . . 8  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  /\  ( M  x.  Q
)  =  ( N  x.  P ) )  ->  ( Q  e.  ZZ  /\  P  e.  ZZ  /\  N  e.  ZZ ) )
84 simp1 957 . . . . . . . . . . . 12  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  M  e.  ZZ )
85 dvdsmul2 12864 . . . . . . . . . . . 12  |-  ( ( M  e.  ZZ  /\  Q  e.  ZZ )  ->  Q  ||  ( M  x.  Q ) )
8684, 55, 85syl2an 464 . . . . . . . . . . 11  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  Q  ||  ( M  x.  Q )
)
8786adantr 452 . . . . . . . . . 10  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  /\  ( M  x.  Q
)  =  ( N  x.  P ) )  ->  Q  ||  ( M  x.  Q )
)
88103ad2ant1 978 . . . . . . . . . . . . 13  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  P  e.  CC )
89 mulcom 9068 . . . . . . . . . . . . 13  |-  ( ( N  e.  CC  /\  P  e.  CC )  ->  ( N  x.  P
)  =  ( P  x.  N ) )
9019, 88, 89syl2an 464 . . . . . . . . . . . 12  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  ( N  x.  P )  =  ( P  x.  N ) )
9190adantr 452 . . . . . . . . . . 11  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  /\  ( M  x.  Q
)  =  ( N  x.  P ) )  ->  ( N  x.  P )  =  ( P  x.  N ) )
9263, 91eqtrd 2467 . . . . . . . . . 10  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  /\  ( M  x.  Q
)  =  ( N  x.  P ) )  ->  ( M  x.  Q )  =  ( P  x.  N ) )
9387, 92breqtrd 4228 . . . . . . . . 9  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  /\  ( M  x.  Q
)  =  ( N  x.  P ) )  ->  Q  ||  ( P  x.  N )
)
94 gcdcom 13012 . . . . . . . . . . . . . 14  |-  ( ( Q  e.  ZZ  /\  P  e.  ZZ )  ->  ( Q  gcd  P
)  =  ( P  gcd  Q ) )
9554, 94sylan 458 . . . . . . . . . . . . 13  |-  ( ( Q  e.  NN  /\  P  e.  ZZ )  ->  ( Q  gcd  P
)  =  ( P  gcd  Q ) )
9695ancoms 440 . . . . . . . . . . . 12  |-  ( ( P  e.  ZZ  /\  Q  e.  NN )  ->  ( Q  gcd  P
)  =  ( P  gcd  Q ) )
97963adant3 977 . . . . . . . . . . 11  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  ( Q  gcd  P )  =  ( P  gcd  Q
) )
98 simp3 959 . . . . . . . . . . 11  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  ( P  gcd  Q )  =  1 )
9997, 98eqtrd 2467 . . . . . . . . . 10  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  ( Q  gcd  P )  =  1 )
10099ad2antlr 708 . . . . . . . . 9  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  /\  ( M  x.  Q
)  =  ( N  x.  P ) )  ->  ( Q  gcd  P )  =  1 )
10193, 100jca 519 . . . . . . . 8  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  /\  ( M  x.  Q
)  =  ( N  x.  P ) )  ->  ( Q  ||  ( P  x.  N
)  /\  ( Q  gcd  P )  =  1 ) )
102 coprmdvds 13094 . . . . . . . 8  |-  ( ( Q  e.  ZZ  /\  P  e.  ZZ  /\  N  e.  ZZ )  ->  (
( Q  ||  ( P  x.  N )  /\  ( Q  gcd  P
)  =  1 )  ->  Q  ||  N
) )
10383, 101, 102sylc 58 . . . . . . 7  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  /\  ( M  x.  Q
)  =  ( N  x.  P ) )  ->  Q  ||  N
)
104 dvdsle 12887 . . . . . . 7  |-  ( ( Q  e.  ZZ  /\  N  e.  NN )  ->  ( Q  ||  N  ->  Q  <_  N )
)
10580, 103, 104sylc 58 . . . . . 6  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  /\  ( M  x.  Q
)  =  ( N  x.  P ) )  ->  Q  <_  N
)
106 nnre 9999 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  RR )
1071063ad2ant2 979 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  N  e.  RR )
108107ad2antrr 707 . . . . . . 7  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  /\  ( M  x.  Q
)  =  ( N  x.  P ) )  ->  N  e.  RR )
109 nnre 9999 . . . . . . . . 9  |-  ( Q  e.  NN  ->  Q  e.  RR )
1101093ad2ant2 979 . . . . . . . 8  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  Q  e.  RR )
111110ad2antlr 708 . . . . . . 7  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  /\  ( M  x.  Q
)  =  ( N  x.  P ) )  ->  Q  e.  RR )
112108, 111letri3d 9207 . . . . . 6  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  /\  ( M  x.  Q
)  =  ( N  x.  P ) )  ->  ( N  =  Q  <->  ( N  <_  Q  /\  Q  <_  N
) ) )
11376, 105, 112mpbir2and 889 . . . . 5  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  /\  ( M  x.  Q
)  =  ( N  x.  P ) )  ->  N  =  Q )
114 oveq2 6081 . . . . . . . . . 10  |-  ( N  =  Q  ->  ( M  x.  N )  =  ( M  x.  Q ) )
115114eqeq1d 2443 . . . . . . . . 9  |-  ( N  =  Q  ->  (
( M  x.  N
)  =  ( N  x.  P )  <->  ( M  x.  Q )  =  ( N  x.  P ) ) )
116115anbi2d 685 . . . . . . . 8  |-  ( N  =  Q  ->  (
( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  /\  ( M  x.  N
)  =  ( N  x.  P ) )  <-> 
( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  /\  ( M  x.  Q
)  =  ( N  x.  P ) ) ) )
117 mulcom 9068 . . . . . . . . . . . . . 14  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( M  x.  N
)  =  ( N  x.  M ) )
1181, 3, 117syl2an 464 . . . . . . . . . . . . 13  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  x.  N
)  =  ( N  x.  M ) )
1191183adant3 977 . . . . . . . . . . . 12  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  ( M  x.  N )  =  ( N  x.  M ) )
120119adantr 452 . . . . . . . . . . 11  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  ( M  x.  N )  =  ( N  x.  M ) )
121120eqeq1d 2443 . . . . . . . . . 10  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  ( ( M  x.  N )  =  ( N  x.  P
)  <->  ( N  x.  M )  =  ( N  x.  P ) ) )
12288adantl 453 . . . . . . . . . . 11  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  P  e.  CC )
12330, 122, 20, 22mulcand 9647 . . . . . . . . . 10  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  ( ( N  x.  M )  =  ( N  x.  P
)  <->  M  =  P
) )
124121, 123bitrd 245 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  ( ( M  x.  N )  =  ( N  x.  P
)  <->  M  =  P
) )
125124biimpa 471 . . . . . . . 8  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  /\  ( M  x.  N
)  =  ( N  x.  P ) )  ->  M  =  P )
126116, 125syl6bir 221 . . . . . . 7  |-  ( N  =  Q  ->  (
( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  /\  ( M  x.  Q
)  =  ( N  x.  P ) )  ->  M  =  P ) )
127126com12 29 . . . . . 6  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  /\  ( M  x.  Q
)  =  ( N  x.  P ) )  ->  ( N  =  Q  ->  M  =  P ) )
128127ancrd 538 . . . . 5  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  /\  ( M  x.  Q
)  =  ( N  x.  P ) )  ->  ( N  =  Q  ->  ( M  =  P  /\  N  =  Q ) ) )
129113, 128mpd 15 . . . 4  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  /\  ( M  x.  Q
)  =  ( N  x.  P ) )  ->  ( M  =  P  /\  N  =  Q ) )
130129ex 424 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  ( ( M  x.  Q )  =  ( N  x.  P
)  ->  ( M  =  P  /\  N  =  Q ) ) )
13146, 130sylbid 207 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 ) )  ->  ( ( M  /  N )  =  ( P  /  Q
)  ->  ( M  =  P  /\  N  =  Q ) ) )
1321313impia 1150 1  |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  /\  ( M  /  N
)  =  ( P  /  Q ) )  ->  ( M  =  P  /\  N  =  Q ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    x. cmul 8987    <_ cle 9113    / cdiv 9669   NNcn 9992   ZZcz 10274    || cdivides 12844    gcd cgcd 12998
This theorem is referenced by:  qredeu  13099
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-dvds 12845  df-gcd 12999
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