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Theorem pythagtriplem4 13080
Description: Lemma for pythagtrip 13095. 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 985 . . 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 10196 . . . . . . . . . . . . 13  |-  ( C  e.  NN  ->  C  e.  ZZ )
3 nnz 10196 . . . . . . . . . . . . 13  |-  ( B  e.  NN  ->  B  e.  ZZ )
4 zsubcl 10212 . . . . . . . . . . . . 13  |-  ( ( C  e.  ZZ  /\  B  e.  ZZ )  ->  ( C  -  B
)  e.  ZZ )
52, 3, 4syl2anr 464 . . . . . . . . . . . 12  |-  ( ( B  e.  NN  /\  C  e.  NN )  ->  ( C  -  B
)  e.  ZZ )
653adant1 974 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( C  -  B )  e.  ZZ )
763ad2ant1 977 . . . . . . . . . 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 988 . . . . . . . . . . . 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 987 . . . . . . . . . . . 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 )
10 nnaddcl 9915 . . . . . . . . . . . 12  |-  ( ( C  e.  NN  /\  B  e.  NN )  ->  ( C  +  B
)  e.  NN )
118, 9, 10syl2anc 642 . . . . . . . . . . 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 )
1211nnzd 10267 . . . . . . . . . 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 )
13 gcddvds 12902 . . . . . . . . . 10  |-  ( ( ( C  -  B
)  e.  ZZ  /\  ( C  +  B
)  e.  ZZ )  ->  ( ( ( C  -  B )  gcd  ( C  +  B ) )  ||  ( C  -  B
)  /\  ( ( C  -  B )  gcd  ( C  +  B
) )  ||  ( C  +  B )
) )
147, 12, 13syl2anc 642 . . . . . . . . 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
) ) )
1514simprd 449 . . . . . . . 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
) )
16 breq1 4128 . . . . . . . . 9  |-  ( ( ( C  -  B
)  gcd  ( C  +  B ) )  =  2  ->  ( (
( C  -  B
)  gcd  ( C  +  B ) )  ||  ( C  +  B
)  <->  2  ||  ( C  +  B )
) )
1716biimpd 198 . . . . . . . 8  |-  ( ( ( C  -  B
)  gcd  ( C  +  B ) )  =  2  ->  ( (
( C  -  B
)  gcd  ( C  +  B ) )  ||  ( C  +  B
)  ->  2  ||  ( C  +  B
) ) )
1815, 17mpan9 455 . . . . . . 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
) )
19 simpl13 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 )  ->  C  e.  NN )
2019nnzd 10267 . . . . . . . . 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 )
21 simpl12 1032 . . . . . . . . . 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 )
2221nnzd 10267 . . . . . . . . 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 )
2320, 22zaddcld 10272 . . . . . . . 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 )
2420, 22zsubcld 10273 . . . . . . . 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 )
25 2z 10205 . . . . . . . . 9  |-  2  e.  ZZ
26 dvdsmultr1 12771 . . . . . . . . 9  |-  ( ( 2  e.  ZZ  /\  ( C  +  B
)  e.  ZZ  /\  ( C  -  B
)  e.  ZZ )  ->  ( 2  ||  ( C  +  B
)  ->  2  ||  ( ( C  +  B )  x.  ( C  -  B )
) ) )
2725, 26mp3an1 1265 . . . . . . . 8  |-  ( ( ( C  +  B
)  e.  ZZ  /\  ( C  -  B
)  e.  ZZ )  ->  ( 2  ||  ( C  +  B
)  ->  2  ||  ( ( C  +  B )  x.  ( C  -  B )
) ) )
2823, 24, 27syl2anc 642 . . . . . . 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 )
) ) )
2918, 28mpd 14 . . . . . 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 )
) )
3019nncnd 9909 . . . . . . 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 )
3121nncnd 9909 . . . . . . 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 )
32 subsq 11375 . . . . . . 7  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C ^
2 )  -  ( B ^ 2 ) )  =  ( ( C  +  B )  x.  ( C  -  B
) ) )
3330, 31, 32syl2anc 642 . . . . . 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
) ) )
3429, 33breqtrrd 4151 . . . . 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 ) ) )
35 simpl2 960 . . . . . . 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 ) )
3635oveq1d 5996 . . . . . 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 ) ) )
37 simpl11 1031 . . . . . . . . 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 )
3837nnsqcld 11430 . . . . . . . 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 )
3938nncnd 9909 . . . . . . 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 )
4021nnsqcld 11430 . . . . . . . 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 )
4140nncnd 9909 . . . . . . 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 )
4239, 41pncand 9305 . . . . . 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 ) )
4336, 42eqtr3d 2400 . . . . 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 ) )
4434, 43breqtrd 4149 . . . 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 ) )
45 nnz 10196 . . . . . . . 8  |-  ( A  e.  NN  ->  A  e.  ZZ )
46453ad2ant1 977 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  A  e.  ZZ )
47463ad2ant1 977 . . . . . 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 )
4847adantr 451 . . . . 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 )
49 2prm 12982 . . . . . 6  |-  2  e.  Prime
50 2nn 10026 . . . . . 6  |-  2  e.  NN
51 prmdvdsexp 13001 . . . . . 6  |-  ( ( 2  e.  Prime  /\  A  e.  ZZ  /\  2  e.  NN )  ->  (
2  ||  ( A ^ 2 )  <->  2  ||  A ) )
5249, 50, 51mp3an13 1269 . . . . 5  |-  ( A  e.  ZZ  ->  (
2  ||  ( A ^ 2 )  <->  2  ||  A ) )
5348, 52syl 15 . . . 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 ) )
5444, 53mpbid 201 . . 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 )
551, 54mtand 640 . 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 )
56 1nn0 10130 . . . . . . . . . 10  |-  1  e.  NN0
5756nn0negzi 10209 . . . . . . . . 9  |-  -u 1  e.  ZZ
58 gcdaddm 12916 . . . . . . . . 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
) ) ) ) )
5957, 58mp3an1 1265 . . . . . . . 8  |-  ( ( ( C  -  B
)  e.  ZZ  /\  ( C  +  B
)  e.  ZZ )  ->  ( ( C  -  B )  gcd  ( C  +  B
) )  =  ( ( C  -  B
)  gcd  ( ( C  +  B )  +  ( -u 1  x.  ( C  -  B
) ) ) ) )
607, 12, 59syl2anc 642 . . . . . . 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 ) ) ) ) )
618nncnd 9909 . . . . . . . 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 )
629nncnd 9909 . . . . . . . 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 )
63 pnncan 9235 . . . . . . . . . . 11  |-  ( ( C  e.  CC  /\  B  e.  CC  /\  B  e.  CC )  ->  (
( C  +  B
)  -  ( C  -  B ) )  =  ( B  +  B ) )
64633anidm23 1242 . . . . . . . . . 10  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  -  ( C  -  B )
)  =  ( B  +  B ) )
65 subcl 9198 . . . . . . . . . . . . 13  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C  -  B
)  e.  CC )
6665mulm1d 9378 . . . . . . . . . . . 12  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( -u 1  x.  ( C  -  B
) )  =  -u ( C  -  B
) )
6766oveq2d 5997 . . . . . . . . . . 11  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  +  (
-u 1  x.  ( C  -  B )
) )  =  ( ( C  +  B
)  +  -u ( C  -  B )
) )
68 addcl 8966 . . . . . . . . . . . 12  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C  +  B
)  e.  CC )
6968, 65negsubd 9310 . . . . . . . . . . 11  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  +  -u ( C  -  B
) )  =  ( ( C  +  B
)  -  ( C  -  B ) ) )
7067, 69eqtrd 2398 . . . . . . . . . 10  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  +  (
-u 1  x.  ( C  -  B )
) )  =  ( ( C  +  B
)  -  ( C  -  B ) ) )
71 2times 9992 . . . . . . . . . . 11  |-  ( B  e.  CC  ->  (
2  x.  B )  =  ( B  +  B ) )
7271adantl 452 . . . . . . . . . 10  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  B
)  =  ( B  +  B ) )
7364, 70, 723eqtr4d 2408 . . . . . . . . 9  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  +  (
-u 1  x.  ( C  -  B )
) )  =  ( 2  x.  B ) )
7473oveq2d 5997 . . . . . . . 8  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  -  B )  gcd  (
( C  +  B
)  +  ( -u
1  x.  ( C  -  B ) ) ) )  =  ( ( C  -  B
)  gcd  ( 2  x.  B ) ) )
7561, 62, 74syl2anc 642 . . . . . . 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
) ) )
7660, 75eqtrd 2398 . . . . . 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 ) ) )
779nnzd 10267 . . . . . . . . 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 )
78 zmulcl 10217 . . . . . . . . 9  |-  ( ( 2  e.  ZZ  /\  B  e.  ZZ )  ->  ( 2  x.  B
)  e.  ZZ )
7925, 77, 78sylancr 644 . . . . . . . 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 )
80 gcddvds 12902 . . . . . . . 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 ) ) )
817, 79, 80syl2anc 642 . . . . . . 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 ) ) )
8281simprd 449 . . . . . 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 ) )
8376, 82eqbrtrd 4145 . . . . 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
) )
84 1z 10204 . . . . . . . . 9  |-  1  e.  ZZ
85 gcdaddm 12916 . . . . . . . . 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
) ) ) ) )
8684, 85mp3an1 1265 . . . . . . . 8  |-  ( ( ( C  -  B
)  e.  ZZ  /\  ( C  +  B
)  e.  ZZ )  ->  ( ( C  -  B )  gcd  ( C  +  B
) )  =  ( ( C  -  B
)  gcd  ( ( C  +  B )  +  ( 1  x.  ( C  -  B
) ) ) ) )
877, 12, 86syl2anc 642 . . . . . . 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 ) ) ) ) )
88 ppncan 9236 . . . . . . . . . . 11  |-  ( ( C  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( C  +  B
)  +  ( C  -  B ) )  =  ( C  +  C ) )
89883anidm13 1241 . . . . . . . . . 10  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  +  ( C  -  B ) )  =  ( C  +  C ) )
9065mulid2d 9000 . . . . . . . . . . 11  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( 1  x.  ( C  -  B )
)  =  ( C  -  B ) )
9190oveq2d 5997 . . . . . . . . . 10  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  +  ( 1  x.  ( C  -  B ) ) )  =  ( ( C  +  B )  +  ( C  -  B ) ) )
92 2times 9992 . . . . . . . . . . 11  |-  ( C  e.  CC  ->  (
2  x.  C )  =  ( C  +  C ) )
9392adantr 451 . . . . . . . . . 10  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  C
)  =  ( C  +  C ) )
9489, 91, 933eqtr4d 2408 . . . . . . . . 9  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  +  B )  +  ( 1  x.  ( C  -  B ) ) )  =  ( 2  x.  C ) )
9561, 62, 94syl2anc 642 . . . . . . . 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 ) )
9695oveq2d 5997 . . . . . . 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
) ) )
9787, 96eqtrd 2398 . . . . . 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 ) ) )
988nnzd 10267 . . . . . . . . 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 )
99 zmulcl 10217 . . . . . . . . 9  |-  ( ( 2  e.  ZZ  /\  C  e.  ZZ )  ->  ( 2  x.  C
)  e.  ZZ )
10025, 98, 99sylancr 644 . . . . . . . 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 )
101 gcddvds 12902 . . . . . . . 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 ) ) )
1027, 100, 101syl2anc 642 . . . . . . 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 ) ) )
103102simprd 449 . . . . . 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 ) )
10497, 103eqbrtrd 4145 . . . . 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
) )
10510nnne0d 9937 . . . . . . . . . . . . 13  |-  ( ( C  e.  NN  /\  B  e.  NN )  ->  ( C  +  B
)  =/=  0 )
106105ancoms 439 . . . . . . . . . . . 12  |-  ( ( B  e.  NN  /\  C  e.  NN )  ->  ( C  +  B
)  =/=  0 )
1071063adant1 974 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( C  +  B )  =/=  0 )
1081073ad2ant1 977 . . . . . . . . . 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 2545 . . . . . . . . 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 882 . . . . . . . 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 12901 . . . . . . . 8  |-  ( ( ( ( C  -  B )  e.  ZZ  /\  ( C  +  B
)  e.  ZZ )  /\  -.  ( ( C  -  B )  =  0  /\  ( C  +  B )  =  0 ) )  ->  ( ( C  -  B )  gcd  ( C  +  B
) )  e.  NN )
1127, 12, 110, 111syl21anc 1182 . . . . . . 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 10267 . . . . . 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 12930 . . . . . 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, 79, 100, 114syl3anc 1183 . . . . 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 ) ) ) )
11683, 104, 115mp2and 660 . . . 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 10131 . . . . . . 7  |-  2  e.  NN0
118 mulgcd 12933 . . . . . . 7  |-  ( ( 2  e.  NN0  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  (
( 2  x.  B
)  gcd  ( 2  x.  C ) )  =  ( 2  x.  ( B  gcd  C
) ) )
119117, 118mp3an1 1265 . . . . . 6  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  ( ( 2  x.  B )  gcd  (
2  x.  C ) )  =  ( 2  x.  ( B  gcd  C ) ) )
12077, 98, 119syl2anc 642 . . . . 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 13079 . . . . . . 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 5997 . . . . . 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 9963 . . . . . . 7  |-  2  e.  CC
124123mulid1i 8986 . . . . . 6  |-  ( 2  x.  1 )  =  2
125122, 124syl6eq 2414 . . . . 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 2398 . . . 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 4149 . . 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 12979 . . . 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 ) ) )
12949, 112, 128sylancr 644 . . 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 201 . 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 371 . 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 ) )
13255, 130, 131sylc 56 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 176    \/ wo 357    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715    =/= wne 2529   class class class wbr 4125  (class class class)co 5981   CCcc 8882   0cc0 8884   1c1 8885    + caddc 8887    x. cmul 8889    - cmin 9184   -ucneg 9185   NNcn 9893   2c2 9942   NN0cn0 10114   ZZcz 10175   ^cexp 11269    || cdivides 12739    gcd cgcd 12893   Primecprime 12966
This theorem is referenced by:  pythagtriplem6  13082  pythagtriplem7  13083
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-2o 6622  df-oadd 6625  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-sup 7341  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-nn 9894  df-2 9951  df-3 9952  df-n0 10115  df-z 10176  df-uz 10382  df-rp 10506  df-fz 10936  df-fl 11089  df-mod 11138  df-seq 11211  df-exp 11270  df-cj 11791  df-re 11792  df-im 11793  df-sqr 11927  df-abs 11928  df-dvds 12740  df-gcd 12894  df-prm 12967
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