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Theorem qredeq 10685
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 8489 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  CC )
21adantr 270 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  M  e.  CC )
3 nncn 8166 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  CC )
43adantl 271 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N  e.  CC )
5 nnap0 8187 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N #  0 )
65adantl 271 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  N #  0 )
72, 4, 6divclapd 7996 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  /  N
)  e.  CC )
873adant3 959 . . . . . . 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 8489 . . . . . . . . . 10  |-  ( P  e.  ZZ  ->  P  e.  CC )
1110adantr 270 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  Q  e.  NN )  ->  P  e.  CC )
12 nncn 8166 . . . . . . . . . 10  |-  ( Q  e.  NN  ->  Q  e.  CC )
1312adantl 271 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  Q  e.  NN )  ->  Q  e.  CC )
14 nnap0 8187 . . . . . . . . . 10  |-  ( Q  e.  NN  ->  Q #  0 )
1514adantl 271 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  Q  e.  NN )  ->  Q #  0 )
1611, 13, 15divclapd 7996 . . . . . . . 8  |-  ( ( P  e.  ZZ  /\  Q  e.  NN )  ->  ( P  /  Q
)  e.  CC )
17163adant3 959 . . . . . . 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 961 . . . . . . 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 961 . . . . . . 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 7870 . . . . 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 7998 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( N  x.  ( M  /  N ) )  =  M )
25243adant3 959 . . . . . . 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 2091 . . . . 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 960 . . . . . . 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 7214 . . . . . . 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 961 . . . . . . 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 961 . . . . . . 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 7871 . . . . 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 7256 . . . . . . 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 7997 . . . . . . . . . 10  |-  ( ( P  e.  ZZ  /\  Q  e.  NN )  ->  ( ( P  /  Q )  x.  Q
)  =  P )
40393adant3 959 . . . . . . . . 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 5579 . . . . . . 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 2115 . . . . . 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 2094 . . . . 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 8503 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  ZZ )
48473ad2ant2 961 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  N  e.  ZZ )
49 simp2 940 . . . . . . . . 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 942 . . . . . . . . . 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 8503 . . . . . . . . . . . 12  |-  ( Q  e.  NN  ->  Q  e.  ZZ )
55543ad2ant2 961 . . . . . . . . . . 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 1119 . . . . . . . . 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 939 . . . . . . . . . . . 12  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  P  e.  ZZ )
60 dvdsmul1 10425 . . . . . . . . . . . 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 3831 . . . . . . . . 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 10572 . . . . . . . . . . . . . 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 959 . . . . . . . . . . 11  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  ( N  gcd  M )  =  ( M  gcd  N
) )
69 simp3 941 . . . . . . . . . . 11  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  ( M  gcd  N )  =  1 )
7068, 69eqtrd 2115 . . . . . . . . . 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 10681 . . . . . . . 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 10452 . . . . . . 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 940 . . . . . . . . . 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 945 . . . . . . . . . 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 1119 . . . . . . . . 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 939 . . . . . . . . . . . 12  |-  ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  ->  M  e.  ZZ )
85 dvdsmul2 10426 . . . . . . . . . . . 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 960 . . . . . . . . . . . . 13  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  P  e.  CC )
89 mulcom 7216 . . . . . . . . . . . . 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 2115 . . . . . . . . . 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 3829 . . . . . . . . 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 10572 . . . . . . . . . . . . . 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 959 . . . . . . . . . . 11  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  ( Q  gcd  P )  =  ( P  gcd  Q
) )
98 simp3 941 . . . . . . . . . . 11  |-  ( ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  ->  ( P  gcd  Q )  =  1 )
9997, 98eqtrd 2115 . . . . . . . . . 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 10681 . . . . . . . 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 10452 . . . . . . 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 8165 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  RR )
1071063ad2ant2 961 . . . . . . . 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 8165 . . . . . . . . 9  |-  ( Q  e.  NN  ->  Q  e.  RR )
1101093ad2ant2 961 . . . . . . . 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 7345 . . . . . 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 886 . . . . 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 5571 . . . . . . . . . 10  |-  ( N  =  Q  ->  ( M  x.  N )  =  ( M  x.  Q ) )
115114eqeq1d 2091 . . . . . . . . 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 7216 . . . . . . . . . . . . . 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 959 . . . . . . . . . . . 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 2091 . . . . . . . . . 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 7870 . . . . . . . . . 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 1136 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 920    = wceq 1285    e. wcel 1434   class class class wbr 3805  (class class class)co 5563   CCcc 7093   RRcr 7094   0cc0 7095   1c1 7096    x. cmul 7100    <_ cle 7268   # cap 7800    / cdiv 7879   NNcn 8158   ZZcz 8484    || cdvds 10403    gcd cgcd 10545
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-mulrcl 7189  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-precex 7200  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-apti 7205  ax-pre-ltadd 7206  ax-pre-mulgt0 7207  ax-pre-mulext 7208  ax-arch 7209  ax-caucvg 7210
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-if 3369  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-frec 6060  df-sup 6491  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-reap 7794  df-ap 7801  df-div 7880  df-inn 8159  df-2 8217  df-3 8218  df-4 8219  df-n0 8408  df-z 8485  df-uz 8753  df-q 8838  df-rp 8868  df-fz 9158  df-fzo 9282  df-fl 9404  df-mod 9457  df-iseq 9574  df-iexp 9625  df-cj 9930  df-re 9931  df-im 9932  df-rsqrt 10085  df-abs 10086  df-dvds 10404  df-gcd 10546
This theorem is referenced by:  qredeu  10686
  Copyright terms: Public domain W3C validator