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Theorem qredeq 11813
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 9083 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  CC )
21adantr 274 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  M  e.  CC )
3 nncn 8752 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  CC )
43adantl 275 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N  e.  CC )
5 nnap0 8773 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N #  0 )
65adantl 275 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N #  0 )
72, 4, 6divclapd 8574 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  /  N
)  e.  CC )
873adant3 1002 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  ( M  /  N )  e.  CC )
98adantr 274 . . . . . 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 9083 . . . . . . . . . 10  |-  ( P  e.  ZZ  ->  P  e.  CC )
1110adantr 274 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  Q  e.  NN )  ->  P  e.  CC )
12 nncn 8752 . . . . . . . . . 10  |-  ( Q  e.  NN  ->  Q  e.  CC )
1312adantl 275 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  Q  e.  NN )  ->  Q  e.  CC )
14 nnap0 8773 . . . . . . . . . 10  |-  ( Q  e.  NN  ->  Q #  0 )
1514adantl 275 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  Q  e.  NN )  ->  Q #  0 )
1611, 13, 15divclapd 8574 . . . . . . . 8  |-  ( ( P  e.  ZZ  /\  Q  e.  NN )  ->  ( P  /  Q
)  e.  CC )
17163adant3 1002 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  ( P  /  Q )  e.  CC )
1817adantl 275 . . . . . 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 1004 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  N  e.  CC )
2019adantr 274 . . . . . 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 1004 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  N #  0 )
2221adantr 274 . . . . . 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 8446 . . . . 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 8576 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  x.  ( M  /  N ) )  =  M )
25243adant3 1002 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  ( N  x.  ( M  /  N ) )  =  M )
2625adantr 274 . . . . . 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 2149 . . . . 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 189 . . . 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 1003 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  M  e.  CC )
3029adantr 274 . . . . . 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 7771 . . . . . . 7  |-  ( ( N  e.  CC  /\  ( P  /  Q
)  e.  CC )  ->  ( N  x.  ( P  /  Q
) )  e.  CC )
3219, 17, 31syl2an 287 . . . . . 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 1004 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  Q  e.  CC )
3433adantl 275 . . . . . 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 1004 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  Q #  0 )
3635adantl 275 . . . . . 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 8447 . . . . 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 7813 . . . . . . 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 8575 . . . . . . . . . 10  |-  ( ( P  e.  ZZ  /\  Q  e.  NN )  ->  ( ( P  /  Q )  x.  Q
)  =  P )
40393adant3 1002 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  (
( P  /  Q
)  x.  Q )  =  P )
4140adantl 275 . . . . . . . 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 5798 . . . . . . 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 2173 . . . . . 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 2152 . . . . 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 189 . . . 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 187 . . 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 9097 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  ZZ )
48473ad2ant2 1004 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  N  e.  ZZ )
49 simp2 983 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  Q  e.  NN )
5048, 49anim12i 336 . . . . . . . 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 274 . . . . . . 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 274 . . . . . . . . . 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 985 . . . . . . . . . 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 9097 . . . . . . . . . . . 12  |-  ( Q  e.  NN  ->  Q  e.  ZZ )
55543ad2ant2 1004 . . . . . . . . . . 11  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  Q  e.  ZZ )
5655adantl 275 . . . . . . . . . 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 1162 . . . . . . . . 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 274 . . . . . . . 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 982 . . . . . . . . . . . 12  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  P  e.  ZZ )
60 dvdsmul1 11551 . . . . . . . . . . . 12  |-  ( ( N  e.  ZZ  /\  P  e.  ZZ )  ->  N  ||  ( N  x.  P ) )
6148, 59, 60syl2an 287 . . . . . . . . . . 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 274 . . . . . . . . . 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 109 . . . . . . . . . 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 3964 . . . . . . . . 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 11698 . . . . . . . . . . . . . 14  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( N  gcd  M
)  =  ( M  gcd  N ) )
6647, 65sylan 281 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  M  e.  ZZ )  ->  ( N  gcd  M
)  =  ( M  gcd  N ) )
6766ancoms 266 . . . . . . . . . . . 12  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  gcd  M
)  =  ( M  gcd  N ) )
68673adant3 1002 . . . . . . . . . . 11  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  ( N  gcd  M )  =  ( M  gcd  N
) )
69 simp3 984 . . . . . . . . . . 11  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  ( M  gcd  N )  =  1 )
7068, 69eqtrd 2173 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  ( N  gcd  M )  =  1 )
7170ad2antrr 480 . . . . . . . . 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 304 . . . . . . . 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 11809 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  Q  e.  ZZ )  ->  (
( N  ||  ( M  x.  Q )  /\  ( N  gcd  M
)  =  1 )  ->  N  ||  Q
) )
7458, 72, 73sylc 62 . . . . . . 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 11578 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  Q  e.  NN )  ->  ( N  ||  Q  ->  N  <_  Q )
)
7651, 74, 75sylc 62 . . . . . 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 983 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  N  e.  NN )
7855, 77anim12i 336 . . . . . . . . 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 266 . . . . . . . 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 274 . . . . . . 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 988 . . . . . . . . . 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 1162 . . . . . . . . 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 274 . . . . . . . 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 982 . . . . . . . . . . . 12  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  M  e.  ZZ )
85 dvdsmul2 11552 . . . . . . . . . . . 12  |-  ( ( M  e.  ZZ  /\  Q  e.  ZZ )  ->  Q  ||  ( M  x.  Q ) )
8684, 55, 85syl2an 287 . . . . . . . . . . 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 274 . . . . . . . . . 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 1003 . . . . . . . . . . . . 13  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  P  e.  CC )
89 mulcom 7773 . . . . . . . . . . . . 13  |-  ( ( N  e.  CC  /\  P  e.  CC )  ->  ( N  x.  P
)  =  ( P  x.  N ) )
9019, 88, 89syl2an 287 . . . . . . . . . . . 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 274 . . . . . . . . . . 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 2173 . . . . . . . . . 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 3962 . . . . . . . . 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 11698 . . . . . . . . . . . . . 14  |-  ( ( Q  e.  ZZ  /\  P  e.  ZZ )  ->  ( Q  gcd  P
)  =  ( P  gcd  Q ) )
9554, 94sylan 281 . . . . . . . . . . . . 13  |-  ( ( Q  e.  NN  /\  P  e.  ZZ )  ->  ( Q  gcd  P
)  =  ( P  gcd  Q ) )
9695ancoms 266 . . . . . . . . . . . 12  |-  ( ( P  e.  ZZ  /\  Q  e.  NN )  ->  ( Q  gcd  P
)  =  ( P  gcd  Q ) )
97963adant3 1002 . . . . . . . . . . 11  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  ( Q  gcd  P )  =  ( P  gcd  Q
) )
98 simp3 984 . . . . . . . . . . 11  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  ( P  gcd  Q )  =  1 )
9997, 98eqtrd 2173 . . . . . . . . . 10  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  ( Q  gcd  P )  =  1 )
10099ad2antlr 481 . . . . . . . . 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 304 . . . . . . . 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 11809 . . . . . . . 8  |-  ( ( Q  e.  ZZ  /\  P  e.  ZZ  /\  N  e.  ZZ )  ->  (
( Q  ||  ( P  x.  N )  /\  ( Q  gcd  P
)  =  1 )  ->  Q  ||  N
) )
10383, 101, 102sylc 62 . . . . . . 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 11578 . . . . . . 7  |-  ( ( Q  e.  ZZ  /\  N  e.  NN )  ->  ( Q  ||  N  ->  Q  <_  N )
)
10580, 103, 104sylc 62 . . . . . 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 8751 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  RR )
1071063ad2ant2 1004 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  N  e.  RR )
108107ad2antrr 480 . . . . . . 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 8751 . . . . . . . . 9  |-  ( Q  e.  NN  ->  Q  e.  RR )
1101093ad2ant2 1004 . . . . . . . 8  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  Q  e.  RR )
111110ad2antlr 481 . . . . . . 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 7903 . . . . . 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 929 . . . . 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 5790 . . . . . . . . . 10  |-  ( N  =  Q  ->  ( M  x.  N )  =  ( M  x.  Q ) )
115114eqeq1d 2149 . . . . . . . . 9  |-  ( N  =  Q  ->  (
( M  x.  N
)  =  ( N  x.  P )  <->  ( M  x.  Q )  =  ( N  x.  P ) ) )
116115anbi2d 460 . . . . . . . 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 7773 . . . . . . . . . . . . . 14  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( M  x.  N
)  =  ( N  x.  M ) )
1181, 3, 117syl2an 287 . . . . . . . . . . . . 13  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  x.  N
)  =  ( N  x.  M ) )
1191183adant3 1002 . . . . . . . . . . . 12  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  ( M  x.  N )  =  ( N  x.  M ) )
120119adantr 274 . . . . . . . . . . 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 2149 . . . . . . . . . 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 275 . . . . . . . . . . 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 8446 . . . . . . . . . 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 187 . . . . . . . . 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 294 . . . . . . . 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 163 . . . . . . 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 324 . . . . 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 114 . . 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 149 . 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 1179 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 103    /\ w3a 963    = wceq 1332    e. wcel 1481   class class class wbr 3937  (class class class)co 5782   CCcc 7642   RRcr 7643   0cc0 7644   1c1 7645    x. cmul 7649    <_ cle 7825   # cap 8367    / cdiv 8456   NNcn 8744   ZZcz 9078    || cdvds 11529    gcd cgcd 11671
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762  ax-arch 7763  ax-caucvg 7764
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-frec 6296  df-sup 6879  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-2 8803  df-3 8804  df-4 8805  df-n0 9002  df-z 9079  df-uz 9351  df-q 9439  df-rp 9471  df-fz 9822  df-fzo 9951  df-fl 10074  df-mod 10127  df-seqfrec 10250  df-exp 10324  df-cj 10646  df-re 10647  df-im 10648  df-rsqrt 10802  df-abs 10803  df-dvds 11530  df-gcd 11672
This theorem is referenced by:  qredeu  11814
  Copyright terms: Public domain W3C validator