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Theorem qredeq 11171
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 8725 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  CC )
21adantr 270 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  M  e.  CC )
3 nncn 8402 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  CC )
43adantl 271 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N  e.  CC )
5 nnap0 8423 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N #  0 )
65adantl 271 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N #  0 )
72, 4, 6divclapd 8230 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  /  N
)  e.  CC )
873adant3 963 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  ( M  /  N )  e.  CC )
98adantr 270 . . . . . 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 8725 . . . . . . . . . 10  |-  ( P  e.  ZZ  ->  P  e.  CC )
1110adantr 270 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  Q  e.  NN )  ->  P  e.  CC )
12 nncn 8402 . . . . . . . . . 10  |-  ( Q  e.  NN  ->  Q  e.  CC )
1312adantl 271 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  Q  e.  NN )  ->  Q  e.  CC )
14 nnap0 8423 . . . . . . . . . 10  |-  ( Q  e.  NN  ->  Q #  0 )
1514adantl 271 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  Q  e.  NN )  ->  Q #  0 )
1611, 13, 15divclapd 8230 . . . . . . . 8  |-  ( ( P  e.  ZZ  /\  Q  e.  NN )  ->  ( P  /  Q
)  e.  CC )
17163adant3 963 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  ( P  /  Q )  e.  CC )
1817adantl 271 . . . . . 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 965 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  N  e.  CC )
2019adantr 270 . . . . . 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 965 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  N #  0 )
2221adantr 270 . . . . . 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, 22mulcanapd 8104 . . . . 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, 6divcanap2d 8232 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  x.  ( M  /  N ) )  =  M )
25243adant3 963 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  ( N  x.  ( M  /  N ) )  =  M )
2625adantr 270 . . . . . 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 2096 . . . . 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 188 . . . 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 964 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  M  e.  CC )
3029adantr 270 . . . . . 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 7448 . . . . . . 7  |-  ( ( N  e.  CC  /\  ( P  /  Q
)  e.  CC )  ->  ( N  x.  ( P  /  Q
) )  e.  CC )
3219, 17, 31syl2an 283 . . . . . 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 965 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  Q  e.  CC )
3433adantl 271 . . . . . 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 965 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  Q #  0 )
3635adantl 271 . . . . . 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, 36mulcanap2d 8105 . . . . 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 7490 . . . . . . 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, 15divcanap1d 8231 . . . . . . . . . 10  |-  ( ( P  e.  ZZ  /\  Q  e.  NN )  ->  ( ( P  /  Q )  x.  Q
)  =  P )
40393adant3 963 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  (
( P  /  Q
)  x.  Q )  =  P )
4140adantl 271 . . . . . . . 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 5650 . . . . . . 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 2120 . . . . . 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 2099 . . . . 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 188 . . . 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 186 . . 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 8739 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  ZZ )
48473ad2ant2 965 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  N  e.  ZZ )
49 simp2 944 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  Q  e.  NN )
5048, 49anim12i 331 . . . . . . . 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 270 . . . . . . 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 270 . . . . . . . . . 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 946 . . . . . . . . . 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 8739 . . . . . . . . . . . 12  |-  ( Q  e.  NN  ->  Q  e.  ZZ )
55543ad2ant2 965 . . . . . . . . . . 11  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  Q  e.  ZZ )
5655adantl 271 . . . . . . . . . 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 1123 . . . . . . . . 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 270 . . . . . . . 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 943 . . . . . . . . . . . 12  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  P  e.  ZZ )
60 dvdsmul1 10911 . . . . . . . . . . . 12  |-  ( ( N  e.  ZZ  /\  P  e.  ZZ )  ->  N  ||  ( N  x.  P ) )
6148, 59, 60syl2an 283 . . . . . . . . . . 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 270 . . . . . . . . . 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 108 . . . . . . . . . 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 3863 . . . . . . . . 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 11058 . . . . . . . . . . . . . 14  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( N  gcd  M
)  =  ( M  gcd  N ) )
6647, 65sylan 277 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  M  e.  ZZ )  ->  ( N  gcd  M
)  =  ( M  gcd  N ) )
6766ancoms 264 . . . . . . . . . . . 12  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  gcd  M
)  =  ( M  gcd  N ) )
68673adant3 963 . . . . . . . . . . 11  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  ( N  gcd  M )  =  ( M  gcd  N
) )
69 simp3 945 . . . . . . . . . . 11  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  ( M  gcd  N )  =  1 )
7068, 69eqtrd 2120 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  ( N  gcd  M )  =  1 )
7170ad2antrr 472 . . . . . . . . 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 300 . . . . . . . 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 11167 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  Q  e.  ZZ )  ->  (
( N  ||  ( M  x.  Q )  /\  ( N  gcd  M
)  =  1 )  ->  N  ||  Q
) )
7458, 72, 73sylc 61 . . . . . . 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 10938 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  Q  e.  NN )  ->  ( N  ||  Q  ->  N  <_  Q )
)
7651, 74, 75sylc 61 . . . . . 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 944 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  N  e.  NN )
7855, 77anim12i 331 . . . . . . . . 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 264 . . . . . . . 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 270 . . . . . . 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 949 . . . . . . . . . 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 1123 . . . . . . . . 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 270 . . . . . . . 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 943 . . . . . . . . . . . 12  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  M  e.  ZZ )
85 dvdsmul2 10912 . . . . . . . . . . . 12  |-  ( ( M  e.  ZZ  /\  Q  e.  ZZ )  ->  Q  ||  ( M  x.  Q ) )
8684, 55, 85syl2an 283 . . . . . . . . . . 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 270 . . . . . . . . . 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 964 . . . . . . . . . . . . 13  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  P  e.  CC )
89 mulcom 7450 . . . . . . . . . . . . 13  |-  ( ( N  e.  CC  /\  P  e.  CC )  ->  ( N  x.  P
)  =  ( P  x.  N ) )
9019, 88, 89syl2an 283 . . . . . . . . . . . 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 270 . . . . . . . . . . 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 2120 . . . . . . . . . 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 3861 . . . . . . . . 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 11058 . . . . . . . . . . . . . 14  |-  ( ( Q  e.  ZZ  /\  P  e.  ZZ )  ->  ( Q  gcd  P
)  =  ( P  gcd  Q ) )
9554, 94sylan 277 . . . . . . . . . . . . 13  |-  ( ( Q  e.  NN  /\  P  e.  ZZ )  ->  ( Q  gcd  P
)  =  ( P  gcd  Q ) )
9695ancoms 264 . . . . . . . . . . . 12  |-  ( ( P  e.  ZZ  /\  Q  e.  NN )  ->  ( Q  gcd  P
)  =  ( P  gcd  Q ) )
97963adant3 963 . . . . . . . . . . 11  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  ( Q  gcd  P )  =  ( P  gcd  Q
) )
98 simp3 945 . . . . . . . . . . 11  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  ( P  gcd  Q )  =  1 )
9997, 98eqtrd 2120 . . . . . . . . . 10  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  ( Q  gcd  P )  =  1 )
10099ad2antlr 473 . . . . . . . . 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 300 . . . . . . . 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 11167 . . . . . . . 8  |-  ( ( Q  e.  ZZ  /\  P  e.  ZZ  /\  N  e.  ZZ )  ->  (
( Q  ||  ( P  x.  N )  /\  ( Q  gcd  P
)  =  1 )  ->  Q  ||  N
) )
10383, 101, 102sylc 61 . . . . . . 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 10938 . . . . . . 7  |-  ( ( Q  e.  ZZ  /\  N  e.  NN )  ->  ( Q  ||  N  ->  Q  <_  N )
)
10580, 103, 104sylc 61 . . . . . 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 8401 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  RR )
1071063ad2ant2 965 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  N  e.  RR )
108107ad2antrr 472 . . . . . . 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 8401 . . . . . . . . 9  |-  ( Q  e.  NN  ->  Q  e.  RR )
1101093ad2ant2 965 . . . . . . . 8  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  Q  e.  RR )
111110ad2antlr 473 . . . . . . 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 7579 . . . . . 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 890 . . . . 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 5642 . . . . . . . . . 10  |-  ( N  =  Q  ->  ( M  x.  N )  =  ( M  x.  Q ) )
115114eqeq1d 2096 . . . . . . . . 9  |-  ( N  =  Q  ->  (
( M  x.  N
)  =  ( N  x.  P )  <->  ( M  x.  Q )  =  ( N  x.  P ) ) )
116115anbi2d 452 . . . . . . . 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 7450 . . . . . . . . . . . . . 14  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( M  x.  N
)  =  ( N  x.  M ) )
1181, 3, 117syl2an 283 . . . . . . . . . . . . 13  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  x.  N
)  =  ( N  x.  M ) )
1191183adant3 963 . . . . . . . . . . . 12  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  ( M  x.  N )  =  ( N  x.  M ) )
120119adantr 270 . . . . . . . . . . 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 2096 . . . . . . . . . 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 271 . . . . . . . . . . 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, 22mulcanapd 8104 . . . . . . . . . 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 186 . . . . . . . . 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 290 . . . . . . . 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 162 . . . . . . 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 30 . . . . . 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 319 . . . . 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 13 . . . 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 113 . . 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 148 . 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 1140 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 102    /\ w3a 924    = wceq 1289    e. wcel 1438   class class class wbr 3837  (class class class)co 5634   CCcc 7327   RRcr 7328   0cc0 7329   1c1 7330    x. cmul 7334    <_ cle 7502   # cap 8034    / cdiv 8113   NNcn 8394   ZZcz 8720    || cdvds 10889    gcd cgcd 11031
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442  ax-arch 7443  ax-caucvg 7444
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-frec 6138  df-sup 6658  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-2 8452  df-3 8453  df-4 8454  df-n0 8644  df-z 8721  df-uz 8989  df-q 9074  df-rp 9104  df-fz 9394  df-fzo 9519  df-fl 9642  df-mod 9695  df-iseq 9818  df-seq3 9819  df-exp 9920  df-cj 10241  df-re 10242  df-im 10243  df-rsqrt 10396  df-abs 10397  df-dvds 10890  df-gcd 11032
This theorem is referenced by:  qredeu  11172
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