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Theorem pythagtriplem4 13193
Description: Lemma for pythagtrip 13208. 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 986 . . 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 10303 . . . . . . . . . . . . 13  |-  ( C  e.  NN  ->  C  e.  ZZ )
3 nnz 10303 . . . . . . . . . . . . 13  |-  ( B  e.  NN  ->  B  e.  ZZ )
4 zsubcl 10319 . . . . . . . . . . . . 13  |-  ( ( C  e.  ZZ  /\  B  e.  ZZ )  ->  ( C  -  B
)  e.  ZZ )
52, 3, 4syl2anr 465 . . . . . . . . . . . 12  |-  ( ( B  e.  NN  /\  C  e.  NN )  ->  ( C  -  B
)  e.  ZZ )
653adant1 975 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( C  -  B )  e.  ZZ )
763ad2ant1 978 . . . . . . . . . 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 989 . . . . . . . . . . . 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 988 . . . . . . . . . . . 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 10046 . . . . . . . . . . 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 10374 . . . . . . . . . 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 13015 . . . . . . . . . 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 643 . . . . . . . . 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 450 . . . . . . . 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 4215 . . . . . . . . 9  |-  ( ( ( C  -  B
)  gcd  ( C  +  B ) )  =  2  ->  ( (
( C  -  B
)  gcd  ( C  +  B ) )  ||  ( C  +  B
)  <->  2  ||  ( C  +  B )
) )
1615biimpd 199 . . . . . . . 8  |-  ( ( ( C  -  B
)  gcd  ( C  +  B ) )  =  2  ->  ( (
( C  -  B
)  gcd  ( C  +  B ) )  ||  ( C  +  B
)  ->  2  ||  ( C  +  B
) ) )
1714, 16mpan9 456 . . . . . . 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 1034 . . . . . . . . . 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 10374 . . . . . . . . 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 1033 . . . . . . . . . 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 10374 . . . . . . . . 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 10379 . . . . . . . 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 10380 . . . . . . . 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 10312 . . . . . . . . 9  |-  2  e.  ZZ
25 dvdsmultr1 12884 . . . . . . . . 9  |-  ( ( 2  e.  ZZ  /\  ( C  +  B
)  e.  ZZ  /\  ( C  -  B
)  e.  ZZ )  ->  ( 2  ||  ( C  +  B
)  ->  2  ||  ( ( C  +  B )  x.  ( C  -  B )
) ) )
2624, 25mp3an1 1266 . . . . . . . 8  |-  ( ( ( C  +  B
)  e.  ZZ  /\  ( C  -  B
)  e.  ZZ )  ->  ( 2  ||  ( C  +  B
)  ->  2  ||  ( ( C  +  B )  x.  ( C  -  B )
) ) )
2722, 23, 26syl2anc 643 . . . . . . 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 10016 . . . . . . 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 10016 . . . . . . 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 11488 . . . . . . 7  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C ^
2 )  -  ( B ^ 2 ) )  =  ( ( C  +  B )  x.  ( C  -  B
) ) )
3229, 30, 31syl2anc 643 . . . . . 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 4238 . . . . 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 961 . . . . . . 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 6096 . . . . . 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 1032 . . . . . . . . 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 11543 . . . . . . . 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 10016 . . . . . . 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 11543 . . . . . . . 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 10016 . . . . . . 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 9412 . . . . . 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 2470 . . . . 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 4236 . . . 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 10303 . . . . . . . 8  |-  ( A  e.  NN  ->  A  e.  ZZ )
45443ad2ant1 978 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  A  e.  ZZ )
46453ad2ant1 978 . . . . . 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 452 . . . . 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 13095 . . . . . 6  |-  2  e.  Prime
49 2nn 10133 . . . . . 6  |-  2  e.  NN
50 prmdvdsexp 13114 . . . . . 6  |-  ( ( 2  e.  Prime  /\  A  e.  ZZ  /\  2  e.  NN )  ->  (
2  ||  ( A ^ 2 )  <->  2  ||  A ) )
5148, 49, 50mp3an13 1270 . . . . 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 202 . . 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 641 . 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 1nn0 10237 . . . . . . . . . 10  |-  1  e.  NN0
5655nn0negzi 10316 . . . . . . . . 9  |-  -u 1  e.  ZZ
57 gcdaddm 13029 . . . . . . . . 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
) ) ) ) )
5856, 57mp3an1 1266 . . . . . . . 8  |-  ( ( ( C  -  B
)  e.  ZZ  /\  ( C  +  B
)  e.  ZZ )  ->  ( ( C  -  B )  gcd  ( C  +  B
) )  =  ( ( C  -  B
)  gcd  ( ( C  +  B )  +  ( -u 1  x.  ( C  -  B
) ) ) ) )
597, 11, 58syl2anc 643 . . . . . . 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 ) ) ) ) )
608nncnd 10016 . . . . . . . 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 )
619nncnd 10016 . . . . . . . 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 )
62 pnncan 9342 . . . . . . . . . . 11  |-  ( ( C  e.  CC  /\  B  e.  CC  /\  B  e.  CC )  ->  (
( C  +  B
)  -  ( C  -  B ) )  =  ( B  +  B ) )
63623anidm23 1243 . . . . . . . . . 10  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  -  ( C  -  B )
)  =  ( B  +  B ) )
64 subcl 9305 . . . . . . . . . . . . 13  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C  -  B
)  e.  CC )
6564mulm1d 9485 . . . . . . . . . . . 12  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( -u 1  x.  ( C  -  B
) )  =  -u ( C  -  B
) )
6665oveq2d 6097 . . . . . . . . . . 11  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  +  (
-u 1  x.  ( C  -  B )
) )  =  ( ( C  +  B
)  +  -u ( C  -  B )
) )
67 addcl 9072 . . . . . . . . . . . 12  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C  +  B
)  e.  CC )
6867, 64negsubd 9417 . . . . . . . . . . 11  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  +  -u ( C  -  B
) )  =  ( ( C  +  B
)  -  ( C  -  B ) ) )
6966, 68eqtrd 2468 . . . . . . . . . 10  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  +  (
-u 1  x.  ( C  -  B )
) )  =  ( ( C  +  B
)  -  ( C  -  B ) ) )
70 2times 10099 . . . . . . . . . . 11  |-  ( B  e.  CC  ->  (
2  x.  B )  =  ( B  +  B ) )
7170adantl 453 . . . . . . . . . 10  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  B
)  =  ( B  +  B ) )
7263, 69, 713eqtr4d 2478 . . . . . . . . 9  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  +  (
-u 1  x.  ( C  -  B )
) )  =  ( 2  x.  B ) )
7372oveq2d 6097 . . . . . . . 8  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  -  B )  gcd  (
( C  +  B
)  +  ( -u
1  x.  ( C  -  B ) ) ) )  =  ( ( C  -  B
)  gcd  ( 2  x.  B ) ) )
7460, 61, 73syl2anc 643 . . . . . . 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
) ) )
7559, 74eqtrd 2468 . . . . . 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 ) ) )
769nnzd 10374 . . . . . . . . 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 )
77 zmulcl 10324 . . . . . . . . 9  |-  ( ( 2  e.  ZZ  /\  B  e.  ZZ )  ->  ( 2  x.  B
)  e.  ZZ )
7824, 76, 77sylancr 645 . . . . . . . 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 )
79 gcddvds 13015 . . . . . . . 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 ) ) )
807, 78, 79syl2anc 643 . . . . . . 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 ) ) )
8180simprd 450 . . . . . 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 ) )
8275, 81eqbrtrd 4232 . . . . 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
) )
83 1z 10311 . . . . . . . . 9  |-  1  e.  ZZ
84 gcdaddm 13029 . . . . . . . . 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
) ) ) ) )
8583, 84mp3an1 1266 . . . . . . . 8  |-  ( ( ( C  -  B
)  e.  ZZ  /\  ( C  +  B
)  e.  ZZ )  ->  ( ( C  -  B )  gcd  ( C  +  B
) )  =  ( ( C  -  B
)  gcd  ( ( C  +  B )  +  ( 1  x.  ( C  -  B
) ) ) ) )
867, 11, 85syl2anc 643 . . . . . . 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 ) ) ) ) )
87 ppncan 9343 . . . . . . . . . . 11  |-  ( ( C  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( C  +  B
)  +  ( C  -  B ) )  =  ( C  +  C ) )
88873anidm13 1242 . . . . . . . . . 10  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  +  ( C  -  B ) )  =  ( C  +  C ) )
8964mulid2d 9106 . . . . . . . . . . 11  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( 1  x.  ( C  -  B )
)  =  ( C  -  B ) )
9089oveq2d 6097 . . . . . . . . . 10  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  +  ( 1  x.  ( C  -  B ) ) )  =  ( ( C  +  B )  +  ( C  -  B ) ) )
91 2times 10099 . . . . . . . . . . 11  |-  ( C  e.  CC  ->  (
2  x.  C )  =  ( C  +  C ) )
9291adantr 452 . . . . . . . . . 10  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  C
)  =  ( C  +  C ) )
9388, 90, 923eqtr4d 2478 . . . . . . . . 9  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  +  ( 1  x.  ( C  -  B ) ) )  =  ( 2  x.  C ) )
9460, 61, 93syl2anc 643 . . . . . . . 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 ) )
9594oveq2d 6097 . . . . . . 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
) ) )
9686, 95eqtrd 2468 . . . . . 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 ) ) )
978nnzd 10374 . . . . . . . . 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 )
98 zmulcl 10324 . . . . . . . . 9  |-  ( ( 2  e.  ZZ  /\  C  e.  ZZ )  ->  ( 2  x.  C
)  e.  ZZ )
9924, 97, 98sylancr 645 . . . . . . . 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 )
100 gcddvds 13015 . . . . . . . 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 ) ) )
1017, 99, 100syl2anc 643 . . . . . . 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 ) ) )
102101simprd 450 . . . . . 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 ) )
10396, 102eqbrtrd 4232 . . . . 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
) )
104 nnaddcl 10022 . . . . . . . . . . . . . 14  |-  ( ( C  e.  NN  /\  B  e.  NN )  ->  ( C  +  B
)  e.  NN )
105104nnne0d 10044 . . . . . . . . . . . . 13  |-  ( ( C  e.  NN  /\  B  e.  NN )  ->  ( C  +  B
)  =/=  0 )
106105ancoms 440 . . . . . . . . . . . 12  |-  ( ( B  e.  NN  /\  C  e.  NN )  ->  ( C  +  B
)  =/=  0 )
1071063adant1 975 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( C  +  B )  =/=  0 )
1081073ad2ant1 978 . . . . . . . . . 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 )
109108neneqd 2617 . . . . . . . . 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 )
110109intnand 883 . . . . . . . 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 ) )
111 gcdn0cl 13014 . . . . . . . 8  |-  ( ( ( ( C  -  B )  e.  ZZ  /\  ( C  +  B
)  e.  ZZ )  /\  -.  ( ( C  -  B )  =  0  /\  ( C  +  B )  =  0 ) )  ->  ( ( C  -  B )  gcd  ( C  +  B
) )  e.  NN )
1127, 11, 110, 111syl21anc 1183 . . . . . . 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 )
113112nnzd 10374 . . . . . 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 )
114 dvdsgcd 13043 . . . . . 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 ) ) ) )
115113, 78, 99, 114syl3anc 1184 . . . . 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 ) ) ) )
11682, 103, 115mp2and 661 . . . 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 ) ) )
117 2nn0 10238 . . . . . . 7  |-  2  e.  NN0
118 mulgcd 13046 . . . . . . 7  |-  ( ( 2  e.  NN0  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  (
( 2  x.  B
)  gcd  ( 2  x.  C ) )  =  ( 2  x.  ( B  gcd  C
) ) )
119117, 118mp3an1 1266 . . . . . 6  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  ( ( 2  x.  B )  gcd  (
2  x.  C ) )  =  ( 2  x.  ( B  gcd  C ) ) )
12076, 97, 119syl2anc 643 . . . . 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
) ) )
121 pythagtriplem3 13192 . . . . . . 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 )
122121oveq2d 6097 . . . . . 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 ) )
123 2cn 10070 . . . . . . 7  |-  2  e.  CC
124123mulid1i 9092 . . . . . 6  |-  ( 2  x.  1 )  =  2
125122, 124syl6eq 2484 . . . . 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 )
126120, 125eqtrd 2468 . . . 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 )
127116, 126breqtrd 4236 . . 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 )
128 dvdsprime 13092 . . . 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 ) ) )
12948, 112, 128sylancr 645 . . 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 ) ) )
130127, 129mpbid 202 . 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 ) )
131 orel1 372 . 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 ) )
13254, 130, 131sylc 58 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4212  (class class class)co 6081   CCcc 8988   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995    - cmin 9291   -ucneg 9292   NNcn 10000   2c2 10049   NN0cn0 10221   ZZcz 10282   ^cexp 11382    || cdivides 12852    gcd cgcd 13006   Primecprime 13079
This theorem is referenced by:  pythagtriplem6  13195  pythagtriplem7  13196
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-dvds 12853  df-gcd 13007  df-prm 13080
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