MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pythagtriplem4 Structured version   Unicode version

Theorem pythagtriplem4 14205
Description: Lemma for pythagtrip 14220. 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 1025 . . 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 10887 . . . . . . . . . . . . 13  |-  ( C  e.  NN  ->  C  e.  ZZ )
3 nnz 10887 . . . . . . . . . . . . 13  |-  ( B  e.  NN  ->  B  e.  ZZ )
4 zsubcl 10906 . . . . . . . . . . . . 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 1014 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( C  -  B )  e.  ZZ )
763ad2ant1 1017 . . . . . . . . . 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 1028 . . . . . . . . . . . 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 1027 . . . . . . . . . . . 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 10583 . . . . . . . . . . 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 10966 . . . . . . . . . 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 14015 . . . . . . . . . 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 4450 . . . . . . . . 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 1073 . . . . . . . . . 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 10966 . . . . . . . . 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 1072 . . . . . . . . . 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 10966 . . . . . . . . 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 10971 . . . . . . . 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 10972 . . . . . . . 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 10897 . . . . . . . . 9  |-  2  e.  ZZ
25 dvdsmultr1 13882 . . . . . . . . 9  |-  ( ( 2  e.  ZZ  /\  ( C  +  B
)  e.  ZZ  /\  ( C  -  B
)  e.  ZZ )  ->  ( 2  ||  ( C  +  B
)  ->  2  ||  ( ( C  +  B )  x.  ( C  -  B )
) ) )
2624, 25mp3an1 1311 . . . . . . . 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 10553 . . . . . . 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 10553 . . . . . . 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 12244 . . . . . . 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 4473 . . . . 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 1000 . . . . . . 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 6300 . . . . . 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 1071 . . . . . . . . 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 12299 . . . . . . . 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 10553 . . . . . . 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 12299 . . . . . . . 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 10553 . . . . . . 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 9932 . . . . . 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 2510 . . . . 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 4471 . . . 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 10887 . . . . . . . 8  |-  ( A  e.  NN  ->  A  e.  ZZ )
45443ad2ant1 1017 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  A  e.  ZZ )
46453ad2ant1 1017 . . . . . 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 14095 . . . . . 6  |-  2  e.  Prime
49 2nn 10694 . . . . . 6  |-  2  e.  NN
50 prmdvdsexp 14117 . . . . . 6  |-  ( ( 2  e.  Prime  /\  A  e.  ZZ  /\  2  e.  NN )  ->  (
2  ||  ( A ^ 2 )  <->  2  ||  A ) )
5148, 49, 50mp3an13 1315 . . . . 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 10900 . . . . . . . . 9  |-  -u 1  e.  ZZ
56 gcdaddm 14029 . . . . . . . . 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 1311 . . . . . . . 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 10553 . . . . . . . 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 10553 . . . . . . . 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 9861 . . . . . . . . . . 11  |-  ( ( C  e.  CC  /\  B  e.  CC  /\  B  e.  CC )  ->  (
( C  +  B
)  -  ( C  -  B ) )  =  ( B  +  B ) )
62613anidm23 1287 . . . . . . . . . 10  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  -  ( C  -  B )
)  =  ( B  +  B ) )
63 subcl 9820 . . . . . . . . . . . . 13  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C  -  B
)  e.  CC )
6463mulm1d 10009 . . . . . . . . . . . 12  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( -u 1  x.  ( C  -  B
) )  =  -u ( C  -  B
) )
6564oveq2d 6301 . . . . . . . . . . 11  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  +  (
-u 1  x.  ( C  -  B )
) )  =  ( ( C  +  B
)  +  -u ( C  -  B )
) )
66 addcl 9575 . . . . . . . . . . . 12  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C  +  B
)  e.  CC )
6766, 63negsubd 9937 . . . . . . . . . . 11  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  +  -u ( C  -  B
) )  =  ( ( C  +  B
)  -  ( C  -  B ) ) )
6865, 67eqtrd 2508 . . . . . . . . . 10  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  +  (
-u 1  x.  ( C  -  B )
) )  =  ( ( C  +  B
)  -  ( C  -  B ) ) )
69 2times 10655 . . . . . . . . . . 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 2518 . . . . . . . . 9  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  +  (
-u 1  x.  ( C  -  B )
) )  =  ( 2  x.  B ) )
7271oveq2d 6301 . . . . . . . 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 2508 . . . . . 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 10966 . . . . . . . . 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 10912 . . . . . . . . 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 14015 . . . . . . . 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 4467 . . . . 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 10895 . . . . . . . . 9  |-  1  e.  ZZ
83 gcdaddm 14029 . . . . . . . . 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 1311 . . . . . . . 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 9862 . . . . . . . . . . 11  |-  ( ( C  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( C  +  B
)  +  ( C  -  B ) )  =  ( C  +  C ) )
87863anidm13 1286 . . . . . . . . . 10  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  +  ( C  -  B ) )  =  ( C  +  C ) )
8863mulid2d 9615 . . . . . . . . . . 11  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( 1  x.  ( C  -  B )
)  =  ( C  -  B ) )
8988oveq2d 6301 . . . . . . . . . 10  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  +  ( 1  x.  ( C  -  B ) ) )  =  ( ( C  +  B )  +  ( C  -  B ) ) )
90 2times 10655 . . . . . . . . . . 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 2518 . . . . . . . . 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 6301 . . . . . . 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 2508 . . . . . 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 10966 . . . . . . . . 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 10912 . . . . . . . . 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 14015 . . . . . . . 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 4467 . . . . 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 10559 . . . . . . . . . . . . . 14  |-  ( ( C  e.  NN  /\  B  e.  NN )  ->  ( C  +  B
)  e.  NN )
104103nnne0d 10581 . . . . . . . . . . . . 13  |-  ( ( C  e.  NN  /\  B  e.  NN )  ->  ( C  +  B
)  =/=  0 )
105104ancoms 453 . . . . . . . . . . . 12  |-  ( ( B  e.  NN  /\  C  e.  NN )  ->  ( C  +  B
)  =/=  0 )
1061053adant1 1014 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( C  +  B )  =/=  0 )
1071063ad2ant1 1017 . . . . . . . . . 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 2669 . . . . . . . . 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 914 . . . . . . . 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 14014 . . . . . . . 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 1227 . . . . . . 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 10966 . . . . . 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 14043 . . . . . 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 1228 . . . . 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 10813 . . . . . . 7  |-  2  e.  NN0
117 mulgcd 14046 . . . . . . 7  |-  ( ( 2  e.  NN0  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  (
( 2  x.  B
)  gcd  ( 2  x.  C ) )  =  ( 2  x.  ( B  gcd  C
) ) )
118116, 117mp3an1 1311 . . . . . 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 14204 . . . . . . 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 6301 . . . . . 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 10685 . . . . . 6  |-  ( 2  x.  1 )  =  2
123121, 122syl6eq 2524 . . . . 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 2508 . . . 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 4471 . . 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 14092 . . . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447  (class class class)co 6285   CCcc 9491   0cc0 9493   1c1 9494    + caddc 9496    x. cmul 9498    - cmin 9806   -ucneg 9807   NNcn 10537   2c2 10586   NN0cn0 10796   ZZcz 10865   ^cexp 12135    || cdivides 13850    gcd cgcd 14006   Primecprime 14079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-2o 7132  df-oadd 7135  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-sup 7902  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11084  df-rp 11222  df-fz 11674  df-fl 11898  df-mod 11966  df-seq 12077  df-exp 12136  df-cj 12898  df-re 12899  df-im 12900  df-sqrt 13034  df-abs 13035  df-dvds 13851  df-gcd 14007  df-prm 14080
This theorem is referenced by:  pythagtriplem6  14207  pythagtriplem7  14208
  Copyright terms: Public domain W3C validator