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Theorem pythagtriplem4 14712
Description: Lemma for pythagtrip 14727. 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 1034 . . 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 10910 . . . . . . . . . . . . 13  |-  ( C  e.  NN  ->  C  e.  ZZ )
3 nnz 10910 . . . . . . . . . . . . 13  |-  ( B  e.  NN  ->  B  e.  ZZ )
4 zsubcl 10930 . . . . . . . . . . . . 13  |-  ( ( C  e.  ZZ  /\  B  e.  ZZ )  ->  ( C  -  B
)  e.  ZZ )
52, 3, 4syl2anr 480 . . . . . . . . . . . 12  |-  ( ( B  e.  NN  /\  C  e.  NN )  ->  ( C  -  B
)  e.  ZZ )
653adant1 1023 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( C  -  B )  e.  ZZ )
763ad2ant1 1026 . . . . . . . . . 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 1037 . . . . . . . . . . . 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 1036 . . . . . . . . . . . 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 10607 . . . . . . . . . . 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 10990 . . . . . . . . . 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 14420 . . . . . . . . . 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 665 . . . . . . . . 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 464 . . . . . . . 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 4369 . . . . . . . . 9  |-  ( ( ( C  -  B
)  gcd  ( C  +  B ) )  =  2  ->  ( (
( C  -  B
)  gcd  ( C  +  B ) )  ||  ( C  +  B
)  <->  2  ||  ( C  +  B )
) )
1615biimpd 210 . . . . . . . 8  |-  ( ( ( C  -  B
)  gcd  ( C  +  B ) )  =  2  ->  ( (
( C  -  B
)  gcd  ( C  +  B ) )  ||  ( C  +  B
)  ->  2  ||  ( C  +  B
) ) )
1714, 16mpan9 471 . . . . . . 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 1082 . . . . . . . . . 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 10990 . . . . . . . . 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 1081 . . . . . . . . . 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 10990 . . . . . . . . 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 10995 . . . . . . . 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 10996 . . . . . . . 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 10920 . . . . . . . . 9  |-  2  e.  ZZ
25 dvdsmultr1 14281 . . . . . . . . 9  |-  ( ( 2  e.  ZZ  /\  ( C  +  B
)  e.  ZZ  /\  ( C  -  B
)  e.  ZZ )  ->  ( 2  ||  ( C  +  B
)  ->  2  ||  ( ( C  +  B )  x.  ( C  -  B )
) ) )
2624, 25mp3an1 1347 . . . . . . . 8  |-  ( ( ( C  +  B
)  e.  ZZ  /\  ( C  -  B
)  e.  ZZ )  ->  ( 2  ||  ( C  +  B
)  ->  2  ||  ( ( C  +  B )  x.  ( C  -  B )
) ) )
2722, 23, 26syl2anc 665 . . . . . . 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 10576 . . . . . . 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 10576 . . . . . . 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 12332 . . . . . . 7  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C ^
2 )  -  ( B ^ 2 ) )  =  ( ( C  +  B )  x.  ( C  -  B
) ) )
3229, 30, 31syl2anc 665 . . . . . 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 4393 . . . . 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 1009 . . . . . . 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 6264 . . . . . 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 1080 . . . . . . . . 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 12386 . . . . . . . 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 10576 . . . . . . 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 12386 . . . . . . . 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 10576 . . . . . . 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 9938 . . . . . 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 2464 . . . . 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 4391 . . . 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 10910 . . . . . . . 8  |-  ( A  e.  NN  ->  A  e.  ZZ )
45443ad2ant1 1026 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  A  e.  ZZ )
46453ad2ant1 1026 . . . . . 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 466 . . . . 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 14583 . . . . . 6  |-  2  e.  Prime
49 2nn 10718 . . . . . 6  |-  2  e.  NN
50 prmdvdsexp 14610 . . . . . 6  |-  ( ( 2  e.  Prime  /\  A  e.  ZZ  /\  2  e.  NN )  ->  (
2  ||  ( A ^ 2 )  <->  2  ||  A ) )
5148, 49, 50mp3an13 1351 . . . . 5  |-  ( A  e.  ZZ  ->  (
2  ||  ( A ^ 2 )  <->  2  ||  A ) )
5247, 51syl 17 . . . 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 213 . . 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 663 . 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 10924 . . . . . . . . 9  |-  -u 1  e.  ZZ
56 gcdaddm 14436 . . . . . . . . 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 1347 . . . . . . . 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 665 . . . . . . 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 10576 . . . . . . . 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 10576 . . . . . . . 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 9866 . . . . . . . . . . 11  |-  ( ( C  e.  CC  /\  B  e.  CC  /\  B  e.  CC )  ->  (
( C  +  B
)  -  ( C  -  B ) )  =  ( B  +  B ) )
62613anidm23 1323 . . . . . . . . . 10  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  -  ( C  -  B )
)  =  ( B  +  B ) )
63 subcl 9825 . . . . . . . . . . . . 13  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C  -  B
)  e.  CC )
6463mulm1d 10021 . . . . . . . . . . . 12  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( -u 1  x.  ( C  -  B
) )  =  -u ( C  -  B
) )
6564oveq2d 6265 . . . . . . . . . . 11  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  +  (
-u 1  x.  ( C  -  B )
) )  =  ( ( C  +  B
)  +  -u ( C  -  B )
) )
66 addcl 9572 . . . . . . . . . . . 12  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C  +  B
)  e.  CC )
6766, 63negsubd 9943 . . . . . . . . . . 11  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  +  -u ( C  -  B
) )  =  ( ( C  +  B
)  -  ( C  -  B ) ) )
6865, 67eqtrd 2462 . . . . . . . . . 10  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  +  (
-u 1  x.  ( C  -  B )
) )  =  ( ( C  +  B
)  -  ( C  -  B ) ) )
69 2times 10679 . . . . . . . . . . 11  |-  ( B  e.  CC  ->  (
2  x.  B )  =  ( B  +  B ) )
7069adantl 467 . . . . . . . . . 10  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  B
)  =  ( B  +  B ) )
7162, 68, 703eqtr4d 2472 . . . . . . . . 9  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  +  (
-u 1  x.  ( C  -  B )
) )  =  ( 2  x.  B ) )
7271oveq2d 6265 . . . . . . . 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 665 . . . . . . 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 2462 . . . . . 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 10990 . . . . . . . . 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 10936 . . . . . . . . 9  |-  ( ( 2  e.  ZZ  /\  B  e.  ZZ )  ->  ( 2  x.  B
)  e.  ZZ )
7724, 75, 76sylancr 667 . . . . . . . 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 14420 . . . . . . . 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 665 . . . . . . 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 464 . . . . . 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 4387 . . . . 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 10918 . . . . . . . . 9  |-  1  e.  ZZ
83 gcdaddm 14436 . . . . . . . . 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 1347 . . . . . . . 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 665 . . . . . . 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 9867 . . . . . . . . . . 11  |-  ( ( C  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( C  +  B
)  +  ( C  -  B ) )  =  ( C  +  C ) )
87863anidm13 1322 . . . . . . . . . 10  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  +  ( C  -  B ) )  =  ( C  +  C ) )
8863mulid2d 9612 . . . . . . . . . . 11  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( 1  x.  ( C  -  B )
)  =  ( C  -  B ) )
8988oveq2d 6265 . . . . . . . . . 10  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  +  ( 1  x.  ( C  -  B ) ) )  =  ( ( C  +  B )  +  ( C  -  B ) ) )
90 2times 10679 . . . . . . . . . . 11  |-  ( C  e.  CC  ->  (
2  x.  C )  =  ( C  +  C ) )
9190adantr 466 . . . . . . . . . 10  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  C
)  =  ( C  +  C ) )
9287, 89, 913eqtr4d 2472 . . . . . . . . 9  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  +  ( 1  x.  ( C  -  B ) ) )  =  ( 2  x.  C ) )
9359, 60, 92syl2anc 665 . . . . . . . 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 6265 . . . . . . 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 2462 . . . . . 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 10990 . . . . . . . . 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 10936 . . . . . . . . 9  |-  ( ( 2  e.  ZZ  /\  C  e.  ZZ )  ->  ( 2  x.  C
)  e.  ZZ )
9824, 96, 97sylancr 667 . . . . . . . 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 14420 . . . . . . . 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 665 . . . . . . 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 464 . . . . . 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 4387 . . . . 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 10582 . . . . . . . . . . . . . 14  |-  ( ( C  e.  NN  /\  B  e.  NN )  ->  ( C  +  B
)  e.  NN )
104103nnne0d 10605 . . . . . . . . . . . . 13  |-  ( ( C  e.  NN  /\  B  e.  NN )  ->  ( C  +  B
)  =/=  0 )
105104ancoms 454 . . . . . . . . . . . 12  |-  ( ( B  e.  NN  /\  C  e.  NN )  ->  ( C  +  B
)  =/=  0 )
1061053adant1 1023 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( C  +  B )  =/=  0 )
1071063ad2ant1 1026 . . . . . . . . . 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 2606 . . . . . . . . 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 924 . . . . . . . 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 14419 . . . . . . . 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 1263 . . . . . . 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 10990 . . . . . 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 14454 . . . . . 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 1264 . . . . 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 683 . . . 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 10837 . . . . . . 7  |-  2  e.  NN0
117 mulgcd 14457 . . . . . . 7  |-  ( ( 2  e.  NN0  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  (
( 2  x.  B
)  gcd  ( 2  x.  C ) )  =  ( 2  x.  ( B  gcd  C
) ) )
118116, 117mp3an1 1347 . . . . . 6  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  ( ( 2  x.  B )  gcd  (
2  x.  C ) )  =  ( 2  x.  ( B  gcd  C ) ) )
11975, 96, 118syl2anc 665 . . . . 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 14711 . . . . . . 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 6265 . . . . . 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 10709 . . . . . 6  |-  ( 2  x.  1 )  =  2
123121, 122syl6eq 2478 . . . . 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 2462 . . . 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 4391 . . 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 14580 . . . 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 667 . . 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 213 . 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 383 . 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 62 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 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2599   class class class wbr 4366  (class class class)co 6249   CCcc 9488   0cc0 9490   1c1 9491    + caddc 9493    x. cmul 9495    - cmin 9811   -ucneg 9812   NNcn 10560   2c2 10610   NN0cn0 10820   ZZcz 10888   ^cexp 12222    || cdvds 14248    gcd cgcd 14411   Primecprime 14565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-1st 6751  df-2nd 6752  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-1o 7137  df-2o 7138  df-oadd 7141  df-er 7318  df-en 7525  df-dom 7526  df-sdom 7527  df-fin 7528  df-sup 7909  df-inf 7910  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-div 10221  df-nn 10561  df-2 10619  df-3 10620  df-n0 10821  df-z 10889  df-uz 11111  df-rp 11254  df-fz 11736  df-fl 11978  df-mod 12047  df-seq 12164  df-exp 12223  df-cj 13106  df-re 13107  df-im 13108  df-sqrt 13242  df-abs 13243  df-dvds 14249  df-gcd 14412  df-prm 14566
This theorem is referenced by:  pythagtriplem6  14714  pythagtriplem7  14715
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