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Theorem pythagtriplem4 14007
Description: Lemma for pythagtrip 14022. Show that  C  -  B and  C  +  B are relatively prime. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
pythagtriplem4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( C  -  B
)  gcd  ( C  +  B ) )  =  1 )

Proof of Theorem pythagtriplem4
StepHypRef Expression
1 simp3r 1017 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  -.  2  ||  A )
2 nnz 10782 . . . . . . . . . . . . 13  |-  ( C  e.  NN  ->  C  e.  ZZ )
3 nnz 10782 . . . . . . . . . . . . 13  |-  ( B  e.  NN  ->  B  e.  ZZ )
4 zsubcl 10801 . . . . . . . . . . . . 13  |-  ( ( C  e.  ZZ  /\  B  e.  ZZ )  ->  ( C  -  B
)  e.  ZZ )
52, 3, 4syl2anr 478 . . . . . . . . . . . 12  |-  ( ( B  e.  NN  /\  C  e.  NN )  ->  ( C  -  B
)  e.  ZZ )
653adant1 1006 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( C  -  B )  e.  ZZ )
763ad2ant1 1009 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  ( C  -  B )  e.  ZZ )
8 simp13 1020 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  C  e.  NN )
9 simp12 1019 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  B  e.  NN )
108, 9nnaddcld 10482 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  ( C  +  B )  e.  NN )
1110nnzd 10860 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  ( C  +  B )  e.  ZZ )
12 gcddvds 13820 . . . . . . . . . 10  |-  ( ( ( C  -  B
)  e.  ZZ  /\  ( C  +  B
)  e.  ZZ )  ->  ( ( ( C  -  B )  gcd  ( C  +  B ) )  ||  ( C  -  B
)  /\  ( ( C  -  B )  gcd  ( C  +  B
) )  ||  ( C  +  B )
) )
137, 11, 12syl2anc 661 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( ( C  -  B )  gcd  ( C  +  B )
)  ||  ( C  -  B )  /\  (
( C  -  B
)  gcd  ( C  +  B ) )  ||  ( C  +  B
) ) )
1413simprd 463 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( C  -  B
)  gcd  ( C  +  B ) )  ||  ( C  +  B
) )
15 breq1 4406 . . . . . . . . 9  |-  ( ( ( C  -  B
)  gcd  ( C  +  B ) )  =  2  ->  ( (
( C  -  B
)  gcd  ( C  +  B ) )  ||  ( C  +  B
)  <->  2  ||  ( C  +  B )
) )
1615biimpd 207 . . . . . . . 8  |-  ( ( ( C  -  B
)  gcd  ( C  +  B ) )  =  2  ->  ( (
( C  -  B
)  gcd  ( C  +  B ) )  ||  ( C  +  B
)  ->  2  ||  ( C  +  B
) ) )
1714, 16mpan9 469 . . . . . . 7  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  (
( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 )  /\  (
( A  gcd  B
)  =  1  /\ 
-.  2  ||  A
) )  /\  (
( C  -  B
)  gcd  ( C  +  B ) )  =  2 )  ->  2  ||  ( C  +  B
) )
18 simpl13 1065 . . . . . . . . . 10  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  (
( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 )  /\  (
( A  gcd  B
)  =  1  /\ 
-.  2  ||  A
) )  /\  (
( C  -  B
)  gcd  ( C  +  B ) )  =  2 )  ->  C  e.  NN )
1918nnzd 10860 . . . . . . . . 9  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  (
( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 )  /\  (
( A  gcd  B
)  =  1  /\ 
-.  2  ||  A
) )  /\  (
( C  -  B
)  gcd  ( C  +  B ) )  =  2 )  ->  C  e.  ZZ )
20 simpl12 1064 . . . . . . . . . 10  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  (
( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 )  /\  (
( A  gcd  B
)  =  1  /\ 
-.  2  ||  A
) )  /\  (
( C  -  B
)  gcd  ( C  +  B ) )  =  2 )  ->  B  e.  NN )
2120nnzd 10860 . . . . . . . . 9  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  (
( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 )  /\  (
( A  gcd  B
)  =  1  /\ 
-.  2  ||  A
) )  /\  (
( C  -  B
)  gcd  ( C  +  B ) )  =  2 )  ->  B  e.  ZZ )
2219, 21zaddcld 10865 . . . . . . . 8  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  (
( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 )  /\  (
( A  gcd  B
)  =  1  /\ 
-.  2  ||  A
) )  /\  (
( C  -  B
)  gcd  ( C  +  B ) )  =  2 )  ->  ( C  +  B )  e.  ZZ )
2319, 21zsubcld 10866 . . . . . . . 8  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  (
( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 )  /\  (
( A  gcd  B
)  =  1  /\ 
-.  2  ||  A
) )  /\  (
( C  -  B
)  gcd  ( C  +  B ) )  =  2 )  ->  ( C  -  B )  e.  ZZ )
24 2z 10792 . . . . . . . . 9  |-  2  e.  ZZ
25 dvdsmultr1 13688 . . . . . . . . 9  |-  ( ( 2  e.  ZZ  /\  ( C  +  B
)  e.  ZZ  /\  ( C  -  B
)  e.  ZZ )  ->  ( 2  ||  ( C  +  B
)  ->  2  ||  ( ( C  +  B )  x.  ( C  -  B )
) ) )
2624, 25mp3an1 1302 . . . . . . . 8  |-  ( ( ( C  +  B
)  e.  ZZ  /\  ( C  -  B
)  e.  ZZ )  ->  ( 2  ||  ( C  +  B
)  ->  2  ||  ( ( C  +  B )  x.  ( C  -  B )
) ) )
2722, 23, 26syl2anc 661 . . . . . . 7  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  (
( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 )  /\  (
( A  gcd  B
)  =  1  /\ 
-.  2  ||  A
) )  /\  (
( C  -  B
)  gcd  ( C  +  B ) )  =  2 )  ->  (
2  ||  ( C  +  B )  ->  2  ||  ( ( C  +  B )  x.  ( C  -  B )
) ) )
2817, 27mpd 15 . . . . . 6  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  (
( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 )  /\  (
( A  gcd  B
)  =  1  /\ 
-.  2  ||  A
) )  /\  (
( C  -  B
)  gcd  ( C  +  B ) )  =  2 )  ->  2  ||  ( ( C  +  B )  x.  ( C  -  B )
) )
2918nncnd 10452 . . . . . . 7  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  (
( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 )  /\  (
( A  gcd  B
)  =  1  /\ 
-.  2  ||  A
) )  /\  (
( C  -  B
)  gcd  ( C  +  B ) )  =  2 )  ->  C  e.  CC )
3020nncnd 10452 . . . . . . 7  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  (
( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 )  /\  (
( A  gcd  B
)  =  1  /\ 
-.  2  ||  A
) )  /\  (
( C  -  B
)  gcd  ( C  +  B ) )  =  2 )  ->  B  e.  CC )
31 subsq 12093 . . . . . . 7  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C ^
2 )  -  ( B ^ 2 ) )  =  ( ( C  +  B )  x.  ( C  -  B
) ) )
3229, 30, 31syl2anc 661 . . . . . 6  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  (
( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 )  /\  (
( A  gcd  B
)  =  1  /\ 
-.  2  ||  A
) )  /\  (
( C  -  B
)  gcd  ( C  +  B ) )  =  2 )  ->  (
( C ^ 2 )  -  ( B ^ 2 ) )  =  ( ( C  +  B )  x.  ( C  -  B
) ) )
3328, 32breqtrrd 4429 . . . . 5  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  (
( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 )  /\  (
( A  gcd  B
)  =  1  /\ 
-.  2  ||  A
) )  /\  (
( C  -  B
)  gcd  ( C  +  B ) )  =  2 )  ->  2  ||  ( ( C ^
2 )  -  ( B ^ 2 ) ) )
34 simpl2 992 . . . . . . 7  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  (
( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 )  /\  (
( A  gcd  B
)  =  1  /\ 
-.  2  ||  A
) )  /\  (
( C  -  B
)  gcd  ( C  +  B ) )  =  2 )  ->  (
( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 ) )
3534oveq1d 6218 . . . . . 6  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  (
( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 )  /\  (
( A  gcd  B
)  =  1  /\ 
-.  2  ||  A
) )  /\  (
( C  -  B
)  gcd  ( C  +  B ) )  =  2 )  ->  (
( ( A ^
2 )  +  ( B ^ 2 ) )  -  ( B ^ 2 ) )  =  ( ( C ^ 2 )  -  ( B ^ 2 ) ) )
36 simpl11 1063 . . . . . . . . 9  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  (
( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 )  /\  (
( A  gcd  B
)  =  1  /\ 
-.  2  ||  A
) )  /\  (
( C  -  B
)  gcd  ( C  +  B ) )  =  2 )  ->  A  e.  NN )
3736nnsqcld 12148 . . . . . . . 8  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  (
( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 )  /\  (
( A  gcd  B
)  =  1  /\ 
-.  2  ||  A
) )  /\  (
( C  -  B
)  gcd  ( C  +  B ) )  =  2 )  ->  ( A ^ 2 )  e.  NN )
3837nncnd 10452 . . . . . . 7  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  (
( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 )  /\  (
( A  gcd  B
)  =  1  /\ 
-.  2  ||  A
) )  /\  (
( C  -  B
)  gcd  ( C  +  B ) )  =  2 )  ->  ( A ^ 2 )  e.  CC )
3920nnsqcld 12148 . . . . . . . 8  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  (
( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 )  /\  (
( A  gcd  B
)  =  1  /\ 
-.  2  ||  A
) )  /\  (
( C  -  B
)  gcd  ( C  +  B ) )  =  2 )  ->  ( B ^ 2 )  e.  NN )
4039nncnd 10452 . . . . . . 7  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  (
( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 )  /\  (
( A  gcd  B
)  =  1  /\ 
-.  2  ||  A
) )  /\  (
( C  -  B
)  gcd  ( C  +  B ) )  =  2 )  ->  ( B ^ 2 )  e.  CC )
4138, 40pncand 9834 . . . . . 6  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  (
( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 )  /\  (
( A  gcd  B
)  =  1  /\ 
-.  2  ||  A
) )  /\  (
( C  -  B
)  gcd  ( C  +  B ) )  =  2 )  ->  (
( ( A ^
2 )  +  ( B ^ 2 ) )  -  ( B ^ 2 ) )  =  ( A ^
2 ) )
4235, 41eqtr3d 2497 . . . . 5  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  (
( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 )  /\  (
( A  gcd  B
)  =  1  /\ 
-.  2  ||  A
) )  /\  (
( C  -  B
)  gcd  ( C  +  B ) )  =  2 )  ->  (
( C ^ 2 )  -  ( B ^ 2 ) )  =  ( A ^
2 ) )
4333, 42breqtrd 4427 . . . 4  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  (
( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 )  /\  (
( A  gcd  B
)  =  1  /\ 
-.  2  ||  A
) )  /\  (
( C  -  B
)  gcd  ( C  +  B ) )  =  2 )  ->  2  ||  ( A ^ 2 ) )
44 nnz 10782 . . . . . . . 8  |-  ( A  e.  NN  ->  A  e.  ZZ )
45443ad2ant1 1009 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  A  e.  ZZ )
46453ad2ant1 1009 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  A  e.  ZZ )
4746adantr 465 . . . . 5  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  (
( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 )  /\  (
( A  gcd  B
)  =  1  /\ 
-.  2  ||  A
) )  /\  (
( C  -  B
)  gcd  ( C  +  B ) )  =  2 )  ->  A  e.  ZZ )
48 2prm 13900 . . . . . 6  |-  2  e.  Prime
49 2nn 10593 . . . . . 6  |-  2  e.  NN
50 prmdvdsexp 13921 . . . . . 6  |-  ( ( 2  e.  Prime  /\  A  e.  ZZ  /\  2  e.  NN )  ->  (
2  ||  ( A ^ 2 )  <->  2  ||  A ) )
5148, 49, 50mp3an13 1306 . . . . 5  |-  ( A  e.  ZZ  ->  (
2  ||  ( A ^ 2 )  <->  2  ||  A ) )
5247, 51syl 16 . . . 4  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  (
( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 )  /\  (
( A  gcd  B
)  =  1  /\ 
-.  2  ||  A
) )  /\  (
( C  -  B
)  gcd  ( C  +  B ) )  =  2 )  ->  (
2  ||  ( A ^ 2 )  <->  2  ||  A ) )
5343, 52mpbid 210 . . 3  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  (
( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 )  /\  (
( A  gcd  B
)  =  1  /\ 
-.  2  ||  A
) )  /\  (
( C  -  B
)  gcd  ( C  +  B ) )  =  2 )  ->  2  ||  A )
541, 53mtand 659 . 2  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  -.  ( ( C  -  B )  gcd  ( C  +  B )
)  =  2 )
55 neg1z 10795 . . . . . . . . 9  |-  -u 1  e.  ZZ
56 gcdaddm 13834 . . . . . . . . 9  |-  ( (
-u 1  e.  ZZ  /\  ( C  -  B
)  e.  ZZ  /\  ( C  +  B
)  e.  ZZ )  ->  ( ( C  -  B )  gcd  ( C  +  B
) )  =  ( ( C  -  B
)  gcd  ( ( C  +  B )  +  ( -u 1  x.  ( C  -  B
) ) ) ) )
5755, 56mp3an1 1302 . . . . . . . 8  |-  ( ( ( C  -  B
)  e.  ZZ  /\  ( C  +  B
)  e.  ZZ )  ->  ( ( C  -  B )  gcd  ( C  +  B
) )  =  ( ( C  -  B
)  gcd  ( ( C  +  B )  +  ( -u 1  x.  ( C  -  B
) ) ) ) )
587, 11, 57syl2anc 661 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( C  -  B
)  gcd  ( C  +  B ) )  =  ( ( C  -  B )  gcd  (
( C  +  B
)  +  ( -u
1  x.  ( C  -  B ) ) ) ) )
598nncnd 10452 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  C  e.  CC )
609nncnd 10452 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  B  e.  CC )
61 pnncan 9764 . . . . . . . . . . 11  |-  ( ( C  e.  CC  /\  B  e.  CC  /\  B  e.  CC )  ->  (
( C  +  B
)  -  ( C  -  B ) )  =  ( B  +  B ) )
62613anidm23 1278 . . . . . . . . . 10  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  -  ( C  -  B )
)  =  ( B  +  B ) )
63 subcl 9723 . . . . . . . . . . . . 13  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C  -  B
)  e.  CC )
6463mulm1d 9910 . . . . . . . . . . . 12  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( -u 1  x.  ( C  -  B
) )  =  -u ( C  -  B
) )
6564oveq2d 6219 . . . . . . . . . . 11  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  +  (
-u 1  x.  ( C  -  B )
) )  =  ( ( C  +  B
)  +  -u ( C  -  B )
) )
66 addcl 9478 . . . . . . . . . . . 12  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C  +  B
)  e.  CC )
6766, 63negsubd 9839 . . . . . . . . . . 11  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  +  -u ( C  -  B
) )  =  ( ( C  +  B
)  -  ( C  -  B ) ) )
6865, 67eqtrd 2495 . . . . . . . . . 10  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  +  (
-u 1  x.  ( C  -  B )
) )  =  ( ( C  +  B
)  -  ( C  -  B ) ) )
69 2times 10554 . . . . . . . . . . 11  |-  ( B  e.  CC  ->  (
2  x.  B )  =  ( B  +  B ) )
7069adantl 466 . . . . . . . . . 10  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  B
)  =  ( B  +  B ) )
7162, 68, 703eqtr4d 2505 . . . . . . . . 9  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  +  (
-u 1  x.  ( C  -  B )
) )  =  ( 2  x.  B ) )
7271oveq2d 6219 . . . . . . . 8  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  -  B )  gcd  (
( C  +  B
)  +  ( -u
1  x.  ( C  -  B ) ) ) )  =  ( ( C  -  B
)  gcd  ( 2  x.  B ) ) )
7359, 60, 72syl2anc 661 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( C  -  B
)  gcd  ( ( C  +  B )  +  ( -u 1  x.  ( C  -  B
) ) ) )  =  ( ( C  -  B )  gcd  ( 2  x.  B
) ) )
7458, 73eqtrd 2495 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( C  -  B
)  gcd  ( C  +  B ) )  =  ( ( C  -  B )  gcd  (
2  x.  B ) ) )
759nnzd 10860 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  B  e.  ZZ )
76 zmulcl 10807 . . . . . . . . 9  |-  ( ( 2  e.  ZZ  /\  B  e.  ZZ )  ->  ( 2  x.  B
)  e.  ZZ )
7724, 75, 76sylancr 663 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
2  x.  B )  e.  ZZ )
78 gcddvds 13820 . . . . . . . 8  |-  ( ( ( C  -  B
)  e.  ZZ  /\  ( 2  x.  B
)  e.  ZZ )  ->  ( ( ( C  -  B )  gcd  ( 2  x.  B ) )  ||  ( C  -  B
)  /\  ( ( C  -  B )  gcd  ( 2  x.  B
) )  ||  (
2  x.  B ) ) )
797, 77, 78syl2anc 661 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( ( C  -  B )  gcd  (
2  x.  B ) )  ||  ( C  -  B )  /\  ( ( C  -  B )  gcd  (
2  x.  B ) )  ||  ( 2  x.  B ) ) )
8079simprd 463 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( C  -  B
)  gcd  ( 2  x.  B ) ) 
||  ( 2  x.  B ) )
8174, 80eqbrtrd 4423 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( C  -  B
)  gcd  ( C  +  B ) )  ||  ( 2  x.  B
) )
82 1z 10790 . . . . . . . . 9  |-  1  e.  ZZ
83 gcdaddm 13834 . . . . . . . . 9  |-  ( ( 1  e.  ZZ  /\  ( C  -  B
)  e.  ZZ  /\  ( C  +  B
)  e.  ZZ )  ->  ( ( C  -  B )  gcd  ( C  +  B
) )  =  ( ( C  -  B
)  gcd  ( ( C  +  B )  +  ( 1  x.  ( C  -  B
) ) ) ) )
8482, 83mp3an1 1302 . . . . . . . 8  |-  ( ( ( C  -  B
)  e.  ZZ  /\  ( C  +  B
)  e.  ZZ )  ->  ( ( C  -  B )  gcd  ( C  +  B
) )  =  ( ( C  -  B
)  gcd  ( ( C  +  B )  +  ( 1  x.  ( C  -  B
) ) ) ) )
857, 11, 84syl2anc 661 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( C  -  B
)  gcd  ( C  +  B ) )  =  ( ( C  -  B )  gcd  (
( C  +  B
)  +  ( 1  x.  ( C  -  B ) ) ) ) )
86 ppncan 9765 . . . . . . . . . . 11  |-  ( ( C  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( C  +  B
)  +  ( C  -  B ) )  =  ( C  +  C ) )
87863anidm13 1277 . . . . . . . . . 10  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  +  ( C  -  B ) )  =  ( C  +  C ) )
8863mulid2d 9518 . . . . . . . . . . 11  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( 1  x.  ( C  -  B )
)  =  ( C  -  B ) )
8988oveq2d 6219 . . . . . . . . . 10  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  +  ( 1  x.  ( C  -  B ) ) )  =  ( ( C  +  B )  +  ( C  -  B ) ) )
90 2times 10554 . . . . . . . . . . 11  |-  ( C  e.  CC  ->  (
2  x.  C )  =  ( C  +  C ) )
9190adantr 465 . . . . . . . . . 10  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  C
)  =  ( C  +  C ) )
9287, 89, 913eqtr4d 2505 . . . . . . . . 9  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  +  ( 1  x.  ( C  -  B ) ) )  =  ( 2  x.  C ) )
9359, 60, 92syl2anc 661 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( C  +  B
)  +  ( 1  x.  ( C  -  B ) ) )  =  ( 2  x.  C ) )
9493oveq2d 6219 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( C  -  B
)  gcd  ( ( C  +  B )  +  ( 1  x.  ( C  -  B
) ) ) )  =  ( ( C  -  B )  gcd  ( 2  x.  C
) ) )
9585, 94eqtrd 2495 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( C  -  B
)  gcd  ( C  +  B ) )  =  ( ( C  -  B )  gcd  (
2  x.  C ) ) )
968nnzd 10860 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  C  e.  ZZ )
97 zmulcl 10807 . . . . . . . . 9  |-  ( ( 2  e.  ZZ  /\  C  e.  ZZ )  ->  ( 2  x.  C
)  e.  ZZ )
9824, 96, 97sylancr 663 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
2  x.  C )  e.  ZZ )
99 gcddvds 13820 . . . . . . . 8  |-  ( ( ( C  -  B
)  e.  ZZ  /\  ( 2  x.  C
)  e.  ZZ )  ->  ( ( ( C  -  B )  gcd  ( 2  x.  C ) )  ||  ( C  -  B
)  /\  ( ( C  -  B )  gcd  ( 2  x.  C
) )  ||  (
2  x.  C ) ) )
1007, 98, 99syl2anc 661 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( ( C  -  B )  gcd  (
2  x.  C ) )  ||  ( C  -  B )  /\  ( ( C  -  B )  gcd  (
2  x.  C ) )  ||  ( 2  x.  C ) ) )
101100simprd 463 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( C  -  B
)  gcd  ( 2  x.  C ) ) 
||  ( 2  x.  C ) )
10295, 101eqbrtrd 4423 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( C  -  B
)  gcd  ( C  +  B ) )  ||  ( 2  x.  C
) )
103 nnaddcl 10458 . . . . . . . . . . . . . 14  |-  ( ( C  e.  NN  /\  B  e.  NN )  ->  ( C  +  B
)  e.  NN )
104103nnne0d 10480 . . . . . . . . . . . . 13  |-  ( ( C  e.  NN  /\  B  e.  NN )  ->  ( C  +  B
)  =/=  0 )
105104ancoms 453 . . . . . . . . . . . 12  |-  ( ( B  e.  NN  /\  C  e.  NN )  ->  ( C  +  B
)  =/=  0 )
1061053adant1 1006 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( C  +  B )  =/=  0 )
1071063ad2ant1 1009 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  ( C  +  B )  =/=  0 )
108107neneqd 2655 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  -.  ( C  +  B
)  =  0 )
109108intnand 907 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  -.  ( ( C  -  B )  =  0  /\  ( C  +  B )  =  0 ) )
110 gcdn0cl 13819 . . . . . . . 8  |-  ( ( ( ( C  -  B )  e.  ZZ  /\  ( C  +  B
)  e.  ZZ )  /\  -.  ( ( C  -  B )  =  0  /\  ( C  +  B )  =  0 ) )  ->  ( ( C  -  B )  gcd  ( C  +  B
) )  e.  NN )
1117, 11, 109, 110syl21anc 1218 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( C  -  B
)  gcd  ( C  +  B ) )  e.  NN )
112111nnzd 10860 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( C  -  B
)  gcd  ( C  +  B ) )  e.  ZZ )
113 dvdsgcd 13848 . . . . . 6  |-  ( ( ( ( C  -  B )  gcd  ( C  +  B )
)  e.  ZZ  /\  ( 2  x.  B
)  e.  ZZ  /\  ( 2  x.  C
)  e.  ZZ )  ->  ( ( ( ( C  -  B
)  gcd  ( C  +  B ) )  ||  ( 2  x.  B
)  /\  ( ( C  -  B )  gcd  ( C  +  B
) )  ||  (
2  x.  C ) )  ->  ( ( C  -  B )  gcd  ( C  +  B
) )  ||  (
( 2  x.  B
)  gcd  ( 2  x.  C ) ) ) )
114112, 77, 98, 113syl3anc 1219 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( ( ( C  -  B )  gcd  ( C  +  B
) )  ||  (
2  x.  B )  /\  ( ( C  -  B )  gcd  ( C  +  B
) )  ||  (
2  x.  C ) )  ->  ( ( C  -  B )  gcd  ( C  +  B
) )  ||  (
( 2  x.  B
)  gcd  ( 2  x.  C ) ) ) )
11581, 102, 114mp2and 679 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( C  -  B
)  gcd  ( C  +  B ) )  ||  ( ( 2  x.  B )  gcd  (
2  x.  C ) ) )
116 2nn0 10710 . . . . . . 7  |-  2  e.  NN0
117 mulgcd 13851 . . . . . . 7  |-  ( ( 2  e.  NN0  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  (
( 2  x.  B
)  gcd  ( 2  x.  C ) )  =  ( 2  x.  ( B  gcd  C
) ) )
118116, 117mp3an1 1302 . . . . . 6  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  ( ( 2  x.  B )  gcd  (
2  x.  C ) )  =  ( 2  x.  ( B  gcd  C ) ) )
11975, 96, 118syl2anc 661 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( 2  x.  B
)  gcd  ( 2  x.  C ) )  =  ( 2  x.  ( B  gcd  C
) ) )
120 pythagtriplem3 14006 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  ( B  gcd  C )  =  1 )
121120oveq2d 6219 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
2  x.  ( B  gcd  C ) )  =  ( 2  x.  1 ) )
122 2t1e2 10584 . . . . . 6  |-  ( 2  x.  1 )  =  2
123121, 122syl6eq 2511 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
2  x.  ( B  gcd  C ) )  =  2 )
124119, 123eqtrd 2495 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( 2  x.  B
)  gcd  ( 2  x.  C ) )  =  2 )
125115, 124breqtrd 4427 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( C  -  B
)  gcd  ( C  +  B ) )  ||  2 )
126 dvdsprime 13897 . . . 4  |-  ( ( 2  e.  Prime  /\  (
( C  -  B
)  gcd  ( C  +  B ) )  e.  NN )  ->  (
( ( C  -  B )  gcd  ( C  +  B )
)  ||  2  <->  ( (
( C  -  B
)  gcd  ( C  +  B ) )  =  2  \/  ( ( C  -  B )  gcd  ( C  +  B ) )  =  1 ) ) )
12748, 111, 126sylancr 663 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( ( C  -  B )  gcd  ( C  +  B )
)  ||  2  <->  ( (
( C  -  B
)  gcd  ( C  +  B ) )  =  2  \/  ( ( C  -  B )  gcd  ( C  +  B ) )  =  1 ) ) )
128125, 127mpbid 210 . 2  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( ( C  -  B )  gcd  ( C  +  B )
)  =  2  \/  ( ( C  -  B )  gcd  ( C  +  B )
)  =  1 ) )
129 orel1 382 . 2  |-  ( -.  ( ( C  -  B )  gcd  ( C  +  B )
)  =  2  -> 
( ( ( ( C  -  B )  gcd  ( C  +  B ) )  =  2  \/  ( ( C  -  B )  gcd  ( C  +  B ) )  =  1 )  ->  (
( C  -  B
)  gcd  ( C  +  B ) )  =  1 ) )
13054, 128, 129sylc 60 1  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( C  -  B
)  gcd  ( C  +  B ) )  =  1 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   class class class wbr 4403  (class class class)co 6203   CCcc 9394   0cc0 9396   1c1 9397    + caddc 9399    x. cmul 9401    - cmin 9709   -ucneg 9710   NNcn 10436   2c2 10485   NN0cn0 10693   ZZcz 10760   ^cexp 11985    || cdivides 13656    gcd cgcd 13811   Primecprime 13884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-2o 7034  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-sup 7805  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-3 10495  df-n0 10694  df-z 10761  df-uz 10976  df-rp 11106  df-fz 11558  df-fl 11762  df-mod 11829  df-seq 11927  df-exp 11986  df-cj 12709  df-re 12710  df-im 12711  df-sqr 12845  df-abs 12846  df-dvds 13657  df-gcd 13812  df-prm 13885
This theorem is referenced by:  pythagtriplem6  14009  pythagtriplem7  14010
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