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Theorem pythagtriplem2 13191
Description: Lemma for pythagtrip 13208. Prove the full version of one direction of the theorem. (Contributed by Scott Fenton, 28-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
pythagtriplem2  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( E. n  e.  NN  E. m  e.  NN  E. k  e.  NN  ( { A ,  B }  =  {
( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) ) ,  ( k  x.  ( 2  x.  ( m  x.  n
) ) ) }  /\  C  =  ( k  x.  ( ( m ^ 2 )  +  ( n ^
2 ) ) ) )  ->  ( ( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) ) )
Distinct variable groups:    A, n, m, k    B, n, m, k    C, n, m, k

Proof of Theorem pythagtriplem2
StepHypRef Expression
1 ovex 6106 . . . . . . . 8  |-  ( k  x.  ( ( m ^ 2 )  -  ( n ^ 2 ) ) )  e. 
_V
2 ovex 6106 . . . . . . . 8  |-  ( k  x.  ( 2  x.  ( m  x.  n
) ) )  e. 
_V
3 preq12bg 3977 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( ( k  x.  ( ( m ^ 2 )  -  ( n ^ 2 ) ) )  e. 
_V  /\  ( k  x.  ( 2  x.  (
m  x.  n ) ) )  e.  _V ) )  ->  ( { A ,  B }  =  { ( k  x.  ( ( m ^
2 )  -  (
n ^ 2 ) ) ) ,  ( k  x.  ( 2  x.  ( m  x.  n ) ) ) }  <->  ( ( A  =  ( k  x.  ( ( m ^
2 )  -  (
n ^ 2 ) ) )  /\  B  =  ( k  x.  ( 2  x.  (
m  x.  n ) ) ) )  \/  ( A  =  ( k  x.  ( 2  x.  ( m  x.  n ) ) )  /\  B  =  ( k  x.  ( ( m ^ 2 )  -  ( n ^
2 ) ) ) ) ) ) )
41, 2, 3mpanr12 667 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( { A ,  B }  =  {
( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) ) ,  ( k  x.  ( 2  x.  ( m  x.  n
) ) ) }  <-> 
( ( A  =  ( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) )  /\  B  =  ( k  x.  (
2  x.  ( m  x.  n ) ) ) )  \/  ( A  =  ( k  x.  ( 2  x.  (
m  x.  n ) ) )  /\  B  =  ( k  x.  ( ( m ^
2 )  -  (
n ^ 2 ) ) ) ) ) ) )
54anbi1d 686 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( { A ,  B }  =  {
( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) ) ,  ( k  x.  ( 2  x.  ( m  x.  n
) ) ) }  /\  C  =  ( k  x.  ( ( m ^ 2 )  +  ( n ^
2 ) ) ) )  <->  ( ( ( A  =  ( k  x.  ( ( m ^ 2 )  -  ( n ^ 2 ) ) )  /\  B  =  ( k  x.  ( 2  x.  (
m  x.  n ) ) ) )  \/  ( A  =  ( k  x.  ( 2  x.  ( m  x.  n ) ) )  /\  B  =  ( k  x.  ( ( m ^ 2 )  -  ( n ^
2 ) ) ) ) )  /\  C  =  ( k  x.  ( ( m ^
2 )  +  ( n ^ 2 ) ) ) ) ) )
6 andir 839 . . . . . . 7  |-  ( ( ( ( A  =  ( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) )  /\  B  =  ( k  x.  (
2  x.  ( m  x.  n ) ) ) )  \/  ( A  =  ( k  x.  ( 2  x.  (
m  x.  n ) ) )  /\  B  =  ( k  x.  ( ( m ^
2 )  -  (
n ^ 2 ) ) ) ) )  /\  C  =  ( k  x.  ( ( m ^ 2 )  +  ( n ^
2 ) ) ) )  <->  ( ( ( A  =  ( k  x.  ( ( m ^ 2 )  -  ( n ^ 2 ) ) )  /\  B  =  ( k  x.  ( 2  x.  (
m  x.  n ) ) ) )  /\  C  =  ( k  x.  ( ( m ^
2 )  +  ( n ^ 2 ) ) ) )  \/  ( ( A  =  ( k  x.  (
2  x.  ( m  x.  n ) ) )  /\  B  =  ( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) ) )  /\  C  =  ( k  x.  ( ( m ^
2 )  +  ( n ^ 2 ) ) ) ) ) )
7 df-3an 938 . . . . . . . 8  |-  ( ( A  =  ( k  x.  ( ( m ^ 2 )  -  ( n ^ 2 ) ) )  /\  B  =  ( k  x.  ( 2  x.  (
m  x.  n ) ) )  /\  C  =  ( k  x.  ( ( m ^
2 )  +  ( n ^ 2 ) ) ) )  <->  ( ( A  =  ( k  x.  ( ( m ^
2 )  -  (
n ^ 2 ) ) )  /\  B  =  ( k  x.  ( 2  x.  (
m  x.  n ) ) ) )  /\  C  =  ( k  x.  ( ( m ^
2 )  +  ( n ^ 2 ) ) ) ) )
8 df-3an 938 . . . . . . . 8  |-  ( ( A  =  ( k  x.  ( 2  x.  ( m  x.  n
) ) )  /\  B  =  ( k  x.  ( ( m ^
2 )  -  (
n ^ 2 ) ) )  /\  C  =  ( k  x.  ( ( m ^
2 )  +  ( n ^ 2 ) ) ) )  <->  ( ( A  =  ( k  x.  ( 2  x.  (
m  x.  n ) ) )  /\  B  =  ( k  x.  ( ( m ^
2 )  -  (
n ^ 2 ) ) ) )  /\  C  =  ( k  x.  ( ( m ^
2 )  +  ( n ^ 2 ) ) ) ) )
97, 8orbi12i 508 . . . . . . 7  |-  ( ( ( A  =  ( k  x.  ( ( m ^ 2 )  -  ( n ^
2 ) ) )  /\  B  =  ( k  x.  ( 2  x.  ( m  x.  n ) ) )  /\  C  =  ( k  x.  ( ( m ^ 2 )  +  ( n ^
2 ) ) ) )  \/  ( A  =  ( k  x.  ( 2  x.  (
m  x.  n ) ) )  /\  B  =  ( k  x.  ( ( m ^
2 )  -  (
n ^ 2 ) ) )  /\  C  =  ( k  x.  ( ( m ^
2 )  +  ( n ^ 2 ) ) ) ) )  <-> 
( ( ( A  =  ( k  x.  ( ( m ^
2 )  -  (
n ^ 2 ) ) )  /\  B  =  ( k  x.  ( 2  x.  (
m  x.  n ) ) ) )  /\  C  =  ( k  x.  ( ( m ^
2 )  +  ( n ^ 2 ) ) ) )  \/  ( ( A  =  ( k  x.  (
2  x.  ( m  x.  n ) ) )  /\  B  =  ( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) ) )  /\  C  =  ( k  x.  ( ( m ^
2 )  +  ( n ^ 2 ) ) ) ) ) )
106, 9bitr4i 244 . . . . . 6  |-  ( ( ( ( A  =  ( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) )  /\  B  =  ( k  x.  (
2  x.  ( m  x.  n ) ) ) )  \/  ( A  =  ( k  x.  ( 2  x.  (
m  x.  n ) ) )  /\  B  =  ( k  x.  ( ( m ^
2 )  -  (
n ^ 2 ) ) ) ) )  /\  C  =  ( k  x.  ( ( m ^ 2 )  +  ( n ^
2 ) ) ) )  <->  ( ( A  =  ( k  x.  ( ( m ^
2 )  -  (
n ^ 2 ) ) )  /\  B  =  ( k  x.  ( 2  x.  (
m  x.  n ) ) )  /\  C  =  ( k  x.  ( ( m ^
2 )  +  ( n ^ 2 ) ) ) )  \/  ( A  =  ( k  x.  ( 2  x.  ( m  x.  n ) ) )  /\  B  =  ( k  x.  ( ( m ^ 2 )  -  ( n ^
2 ) ) )  /\  C  =  ( k  x.  ( ( m ^ 2 )  +  ( n ^
2 ) ) ) ) ) )
115, 10syl6bb 253 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( { A ,  B }  =  {
( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) ) ,  ( k  x.  ( 2  x.  ( m  x.  n
) ) ) }  /\  C  =  ( k  x.  ( ( m ^ 2 )  +  ( n ^
2 ) ) ) )  <->  ( ( A  =  ( k  x.  ( ( m ^
2 )  -  (
n ^ 2 ) ) )  /\  B  =  ( k  x.  ( 2  x.  (
m  x.  n ) ) )  /\  C  =  ( k  x.  ( ( m ^
2 )  +  ( n ^ 2 ) ) ) )  \/  ( A  =  ( k  x.  ( 2  x.  ( m  x.  n ) ) )  /\  B  =  ( k  x.  ( ( m ^ 2 )  -  ( n ^
2 ) ) )  /\  C  =  ( k  x.  ( ( m ^ 2 )  +  ( n ^
2 ) ) ) ) ) ) )
1211rexbidv 2726 . . . 4  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( E. k  e.  NN  ( { A ,  B }  =  {
( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) ) ,  ( k  x.  ( 2  x.  ( m  x.  n
) ) ) }  /\  C  =  ( k  x.  ( ( m ^ 2 )  +  ( n ^
2 ) ) ) )  <->  E. k  e.  NN  ( ( A  =  ( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) )  /\  B  =  ( k  x.  (
2  x.  ( m  x.  n ) ) )  /\  C  =  ( k  x.  (
( m ^ 2 )  +  ( n ^ 2 ) ) ) )  \/  ( A  =  ( k  x.  ( 2  x.  (
m  x.  n ) ) )  /\  B  =  ( k  x.  ( ( m ^
2 )  -  (
n ^ 2 ) ) )  /\  C  =  ( k  x.  ( ( m ^
2 )  +  ( n ^ 2 ) ) ) ) ) ) )
13122rexbidv 2748 . . 3  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( E. n  e.  NN  E. m  e.  NN  E. k  e.  NN  ( { A ,  B }  =  {
( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) ) ,  ( k  x.  ( 2  x.  ( m  x.  n
) ) ) }  /\  C  =  ( k  x.  ( ( m ^ 2 )  +  ( n ^
2 ) ) ) )  <->  E. n  e.  NN  E. m  e.  NN  E. k  e.  NN  (
( A  =  ( k  x.  ( ( m ^ 2 )  -  ( n ^
2 ) ) )  /\  B  =  ( k  x.  ( 2  x.  ( m  x.  n ) ) )  /\  C  =  ( k  x.  ( ( m ^ 2 )  +  ( n ^
2 ) ) ) )  \/  ( A  =  ( k  x.  ( 2  x.  (
m  x.  n ) ) )  /\  B  =  ( k  x.  ( ( m ^
2 )  -  (
n ^ 2 ) ) )  /\  C  =  ( k  x.  ( ( m ^
2 )  +  ( n ^ 2 ) ) ) ) ) ) )
14 r19.43 2863 . . . . 5  |-  ( E. k  e.  NN  (
( A  =  ( k  x.  ( ( m ^ 2 )  -  ( n ^
2 ) ) )  /\  B  =  ( k  x.  ( 2  x.  ( m  x.  n ) ) )  /\  C  =  ( k  x.  ( ( m ^ 2 )  +  ( n ^
2 ) ) ) )  \/  ( A  =  ( k  x.  ( 2  x.  (
m  x.  n ) ) )  /\  B  =  ( k  x.  ( ( m ^
2 )  -  (
n ^ 2 ) ) )  /\  C  =  ( k  x.  ( ( m ^
2 )  +  ( n ^ 2 ) ) ) ) )  <-> 
( E. k  e.  NN  ( A  =  ( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) )  /\  B  =  ( k  x.  (
2  x.  ( m  x.  n ) ) )  /\  C  =  ( k  x.  (
( m ^ 2 )  +  ( n ^ 2 ) ) ) )  \/  E. k  e.  NN  ( A  =  ( k  x.  ( 2  x.  (
m  x.  n ) ) )  /\  B  =  ( k  x.  ( ( m ^
2 )  -  (
n ^ 2 ) ) )  /\  C  =  ( k  x.  ( ( m ^
2 )  +  ( n ^ 2 ) ) ) ) ) )
15142rexbii 2732 . . . 4  |-  ( E. n  e.  NN  E. m  e.  NN  E. k  e.  NN  ( ( A  =  ( k  x.  ( ( m ^
2 )  -  (
n ^ 2 ) ) )  /\  B  =  ( k  x.  ( 2  x.  (
m  x.  n ) ) )  /\  C  =  ( k  x.  ( ( m ^
2 )  +  ( n ^ 2 ) ) ) )  \/  ( A  =  ( k  x.  ( 2  x.  ( m  x.  n ) ) )  /\  B  =  ( k  x.  ( ( m ^ 2 )  -  ( n ^
2 ) ) )  /\  C  =  ( k  x.  ( ( m ^ 2 )  +  ( n ^
2 ) ) ) ) )  <->  E. n  e.  NN  E. m  e.  NN  ( E. k  e.  NN  ( A  =  ( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) )  /\  B  =  ( k  x.  (
2  x.  ( m  x.  n ) ) )  /\  C  =  ( k  x.  (
( m ^ 2 )  +  ( n ^ 2 ) ) ) )  \/  E. k  e.  NN  ( A  =  ( k  x.  ( 2  x.  (
m  x.  n ) ) )  /\  B  =  ( k  x.  ( ( m ^
2 )  -  (
n ^ 2 ) ) )  /\  C  =  ( k  x.  ( ( m ^
2 )  +  ( n ^ 2 ) ) ) ) ) )
16 r19.43 2863 . . . . 5  |-  ( E. m  e.  NN  ( E. k  e.  NN  ( A  =  (
k  x.  ( ( m ^ 2 )  -  ( n ^
2 ) ) )  /\  B  =  ( k  x.  ( 2  x.  ( m  x.  n ) ) )  /\  C  =  ( k  x.  ( ( m ^ 2 )  +  ( n ^
2 ) ) ) )  \/  E. k  e.  NN  ( A  =  ( k  x.  (
2  x.  ( m  x.  n ) ) )  /\  B  =  ( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) )  /\  C  =  ( k  x.  (
( m ^ 2 )  +  ( n ^ 2 ) ) ) ) )  <->  ( E. m  e.  NN  E. k  e.  NN  ( A  =  ( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) )  /\  B  =  ( k  x.  (
2  x.  ( m  x.  n ) ) )  /\  C  =  ( k  x.  (
( m ^ 2 )  +  ( n ^ 2 ) ) ) )  \/  E. m  e.  NN  E. k  e.  NN  ( A  =  ( k  x.  (
2  x.  ( m  x.  n ) ) )  /\  B  =  ( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) )  /\  C  =  ( k  x.  (
( m ^ 2 )  +  ( n ^ 2 ) ) ) ) ) )
1716rexbii 2730 . . . 4  |-  ( E. n  e.  NN  E. m  e.  NN  ( E. k  e.  NN  ( A  =  (
k  x.  ( ( m ^ 2 )  -  ( n ^
2 ) ) )  /\  B  =  ( k  x.  ( 2  x.  ( m  x.  n ) ) )  /\  C  =  ( k  x.  ( ( m ^ 2 )  +  ( n ^
2 ) ) ) )  \/  E. k  e.  NN  ( A  =  ( k  x.  (
2  x.  ( m  x.  n ) ) )  /\  B  =  ( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) )  /\  C  =  ( k  x.  (
( m ^ 2 )  +  ( n ^ 2 ) ) ) ) )  <->  E. n  e.  NN  ( E. m  e.  NN  E. k  e.  NN  ( A  =  ( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) )  /\  B  =  ( k  x.  (
2  x.  ( m  x.  n ) ) )  /\  C  =  ( k  x.  (
( m ^ 2 )  +  ( n ^ 2 ) ) ) )  \/  E. m  e.  NN  E. k  e.  NN  ( A  =  ( k  x.  (
2  x.  ( m  x.  n ) ) )  /\  B  =  ( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) )  /\  C  =  ( k  x.  (
( m ^ 2 )  +  ( n ^ 2 ) ) ) ) ) )
18 r19.43 2863 . . . 4  |-  ( E. n  e.  NN  ( E. m  e.  NN  E. k  e.  NN  ( A  =  ( k  x.  ( ( m ^
2 )  -  (
n ^ 2 ) ) )  /\  B  =  ( k  x.  ( 2  x.  (
m  x.  n ) ) )  /\  C  =  ( k  x.  ( ( m ^
2 )  +  ( n ^ 2 ) ) ) )  \/ 
E. m  e.  NN  E. k  e.  NN  ( A  =  ( k  x.  ( 2  x.  (
m  x.  n ) ) )  /\  B  =  ( k  x.  ( ( m ^
2 )  -  (
n ^ 2 ) ) )  /\  C  =  ( k  x.  ( ( m ^
2 )  +  ( n ^ 2 ) ) ) ) )  <-> 
( E. n  e.  NN  E. m  e.  NN  E. k  e.  NN  ( A  =  ( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) )  /\  B  =  ( k  x.  (
2  x.  ( m  x.  n ) ) )  /\  C  =  ( k  x.  (
( m ^ 2 )  +  ( n ^ 2 ) ) ) )  \/  E. n  e.  NN  E. m  e.  NN  E. k  e.  NN  ( A  =  ( k  x.  (
2  x.  ( m  x.  n ) ) )  /\  B  =  ( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) )  /\  C  =  ( k  x.  (
( m ^ 2 )  +  ( n ^ 2 ) ) ) ) ) )
1915, 17, 183bitri 263 . . 3  |-  ( E. n  e.  NN  E. m  e.  NN  E. k  e.  NN  ( ( A  =  ( k  x.  ( ( m ^
2 )  -  (
n ^ 2 ) ) )  /\  B  =  ( k  x.  ( 2  x.  (
m  x.  n ) ) )  /\  C  =  ( k  x.  ( ( m ^
2 )  +  ( n ^ 2 ) ) ) )  \/  ( A  =  ( k  x.  ( 2  x.  ( m  x.  n ) ) )  /\  B  =  ( k  x.  ( ( m ^ 2 )  -  ( n ^
2 ) ) )  /\  C  =  ( k  x.  ( ( m ^ 2 )  +  ( n ^
2 ) ) ) ) )  <->  ( E. n  e.  NN  E. m  e.  NN  E. k  e.  NN  ( A  =  ( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) )  /\  B  =  ( k  x.  (
2  x.  ( m  x.  n ) ) )  /\  C  =  ( k  x.  (
( m ^ 2 )  +  ( n ^ 2 ) ) ) )  \/  E. n  e.  NN  E. m  e.  NN  E. k  e.  NN  ( A  =  ( k  x.  (
2  x.  ( m  x.  n ) ) )  /\  B  =  ( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) )  /\  C  =  ( k  x.  (
( m ^ 2 )  +  ( n ^ 2 ) ) ) ) ) )
2013, 19syl6bb 253 . 2  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( E. n  e.  NN  E. m  e.  NN  E. k  e.  NN  ( { A ,  B }  =  {
( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) ) ,  ( k  x.  ( 2  x.  ( m  x.  n
) ) ) }  /\  C  =  ( k  x.  ( ( m ^ 2 )  +  ( n ^
2 ) ) ) )  <->  ( E. n  e.  NN  E. m  e.  NN  E. k  e.  NN  ( A  =  ( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) )  /\  B  =  ( k  x.  (
2  x.  ( m  x.  n ) ) )  /\  C  =  ( k  x.  (
( m ^ 2 )  +  ( n ^ 2 ) ) ) )  \/  E. n  e.  NN  E. m  e.  NN  E. k  e.  NN  ( A  =  ( k  x.  (
2  x.  ( m  x.  n ) ) )  /\  B  =  ( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) )  /\  C  =  ( k  x.  (
( m ^ 2 )  +  ( n ^ 2 ) ) ) ) ) ) )
21 pythagtriplem1 13190 . . . 4  |-  ( E. n  e.  NN  E. m  e.  NN  E. k  e.  NN  ( A  =  ( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) )  /\  B  =  ( k  x.  (
2  x.  ( m  x.  n ) ) )  /\  C  =  ( k  x.  (
( m ^ 2 )  +  ( n ^ 2 ) ) ) )  ->  (
( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 ) )
2221a1i 11 . . 3  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( E. n  e.  NN  E. m  e.  NN  E. k  e.  NN  ( A  =  ( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) )  /\  B  =  ( k  x.  (
2  x.  ( m  x.  n ) ) )  /\  C  =  ( k  x.  (
( m ^ 2 )  +  ( n ^ 2 ) ) ) )  ->  (
( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 ) ) )
23 3ancoma 943 . . . . . . 7  |-  ( ( A  =  ( k  x.  ( 2  x.  ( m  x.  n
) ) )  /\  B  =  ( k  x.  ( ( m ^
2 )  -  (
n ^ 2 ) ) )  /\  C  =  ( k  x.  ( ( m ^
2 )  +  ( n ^ 2 ) ) ) )  <->  ( B  =  ( k  x.  ( ( m ^
2 )  -  (
n ^ 2 ) ) )  /\  A  =  ( k  x.  ( 2  x.  (
m  x.  n ) ) )  /\  C  =  ( k  x.  ( ( m ^
2 )  +  ( n ^ 2 ) ) ) ) )
2423rexbii 2730 . . . . . 6  |-  ( E. k  e.  NN  ( A  =  ( k  x.  ( 2  x.  (
m  x.  n ) ) )  /\  B  =  ( k  x.  ( ( m ^
2 )  -  (
n ^ 2 ) ) )  /\  C  =  ( k  x.  ( ( m ^
2 )  +  ( n ^ 2 ) ) ) )  <->  E. k  e.  NN  ( B  =  ( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) )  /\  A  =  ( k  x.  (
2  x.  ( m  x.  n ) ) )  /\  C  =  ( k  x.  (
( m ^ 2 )  +  ( n ^ 2 ) ) ) ) )
25242rexbii 2732 . . . . 5  |-  ( E. n  e.  NN  E. m  e.  NN  E. k  e.  NN  ( A  =  ( k  x.  (
2  x.  ( m  x.  n ) ) )  /\  B  =  ( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) )  /\  C  =  ( k  x.  (
( m ^ 2 )  +  ( n ^ 2 ) ) ) )  <->  E. n  e.  NN  E. m  e.  NN  E. k  e.  NN  ( B  =  ( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) )  /\  A  =  ( k  x.  (
2  x.  ( m  x.  n ) ) )  /\  C  =  ( k  x.  (
( m ^ 2 )  +  ( n ^ 2 ) ) ) ) )
26 pythagtriplem1 13190 . . . . 5  |-  ( E. n  e.  NN  E. m  e.  NN  E. k  e.  NN  ( B  =  ( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) )  /\  A  =  ( k  x.  (
2  x.  ( m  x.  n ) ) )  /\  C  =  ( k  x.  (
( m ^ 2 )  +  ( n ^ 2 ) ) ) )  ->  (
( B ^ 2 )  +  ( A ^ 2 ) )  =  ( C ^
2 ) )
2725, 26sylbi 188 . . . 4  |-  ( E. n  e.  NN  E. m  e.  NN  E. k  e.  NN  ( A  =  ( k  x.  (
2  x.  ( m  x.  n ) ) )  /\  B  =  ( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) )  /\  C  =  ( k  x.  (
( m ^ 2 )  +  ( n ^ 2 ) ) ) )  ->  (
( B ^ 2 )  +  ( A ^ 2 ) )  =  ( C ^
2 ) )
28 nncn 10008 . . . . . . 7  |-  ( A  e.  NN  ->  A  e.  CC )
2928sqcld 11521 . . . . . 6  |-  ( A  e.  NN  ->  ( A ^ 2 )  e.  CC )
30 nncn 10008 . . . . . . 7  |-  ( B  e.  NN  ->  B  e.  CC )
3130sqcld 11521 . . . . . 6  |-  ( B  e.  NN  ->  ( B ^ 2 )  e.  CC )
32 addcom 9252 . . . . . 6  |-  ( ( ( A ^ 2 )  e.  CC  /\  ( B ^ 2 )  e.  CC )  -> 
( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( ( B ^ 2 )  +  ( A ^
2 ) ) )
3329, 31, 32syl2an 464 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( ( B ^ 2 )  +  ( A ^
2 ) ) )
3433eqeq1d 2444 . . . 4  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( ( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  <-> 
( ( B ^
2 )  +  ( A ^ 2 ) )  =  ( C ^ 2 ) ) )
3527, 34syl5ibr 213 . . 3  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( E. n  e.  NN  E. m  e.  NN  E. k  e.  NN  ( A  =  ( k  x.  (
2  x.  ( m  x.  n ) ) )  /\  B  =  ( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) )  /\  C  =  ( k  x.  (
( m ^ 2 )  +  ( n ^ 2 ) ) ) )  ->  (
( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 ) ) )
3622, 35jaod 370 . 2  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( E. n  e.  NN  E. m  e.  NN  E. k  e.  NN  ( A  =  ( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) )  /\  B  =  ( k  x.  (
2  x.  ( m  x.  n ) ) )  /\  C  =  ( k  x.  (
( m ^ 2 )  +  ( n ^ 2 ) ) ) )  \/  E. n  e.  NN  E. m  e.  NN  E. k  e.  NN  ( A  =  ( k  x.  (
2  x.  ( m  x.  n ) ) )  /\  B  =  ( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) )  /\  C  =  ( k  x.  (
( m ^ 2 )  +  ( n ^ 2 ) ) ) ) )  -> 
( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) ) )
3720, 36sylbid 207 1  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( E. n  e.  NN  E. m  e.  NN  E. k  e.  NN  ( { A ,  B }  =  {
( k  x.  (
( m ^ 2 )  -  ( n ^ 2 ) ) ) ,  ( k  x.  ( 2  x.  ( m  x.  n
) ) ) }  /\  C  =  ( k  x.  ( ( m ^ 2 )  +  ( n ^
2 ) ) ) )  ->  ( ( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   E.wrex 2706   _Vcvv 2956   {cpr 3815  (class class class)co 6081   CCcc 8988    + caddc 8993    x. cmul 8995    - cmin 9291   NNcn 10000   2c2 10049   ^cexp 11382
This theorem is referenced by:  pythagtrip  13208
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
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-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-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-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-n0 10222  df-z 10283  df-uz 10489  df-seq 11324  df-exp 11383
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