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Theorem pythagtriplem2 12870
Description: Lemma for pythagtrip 12887. 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 5883 . . . . . . . 8  |-  ( k  x.  ( ( m ^ 2 )  -  ( n ^ 2 ) ) )  e. 
_V
2 ovex 5883 . . . . . . . 8  |-  ( k  x.  ( 2  x.  ( m  x.  n
) ) )  e. 
_V
3 preq12bg 3791 . . . . . . . 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 666 . . . . . . 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 685 . . . . . 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 838 . . . . . . 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 936 . . . . . . . 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 936 . . . . . . . 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 507 . . . . . . 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 243 . . . . . 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 252 . . . . 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 2564 . . . 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 2586 . . 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 2695 . . . . 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 2570 . . . 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 2695 . . . . 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 2568 . . . 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 2695 . . . 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 262 . . 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 252 . 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 12869 . . . 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 10 . . 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 941 . . . . . . 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 2568 . . . . . 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 2570 . . . . 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 12869 . . . . 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 187 . . . 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 9754 . . . . . . 7  |-  ( A  e.  NN  ->  A  e.  CC )
2928sqcld 11243 . . . . . 6  |-  ( A  e.  NN  ->  ( A ^ 2 )  e.  CC )
30 nncn 9754 . . . . . . 7  |-  ( B  e.  NN  ->  B  e.  CC )
3130sqcld 11243 . . . . . 6  |-  ( B  e.  NN  ->  ( B ^ 2 )  e.  CC )
32 addcom 8998 . . . . . 6  |-  ( ( ( A ^ 2 )  e.  CC  /\  ( B ^ 2 )  e.  CC )  -> 
( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( ( B ^ 2 )  +  ( A ^
2 ) ) )
3329, 31, 32syl2an 463 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( ( B ^ 2 )  +  ( A ^
2 ) ) )
3433eqeq1d 2291 . . . 4  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( ( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  <-> 
( ( B ^
2 )  +  ( A ^ 2 ) )  =  ( C ^ 2 ) ) )
3527, 34syl5ibr 212 . . 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 369 . 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 206 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788   {cpr 3641  (class class class)co 5858   CCcc 8735    + caddc 8740    x. cmul 8742    - cmin 9037   NNcn 9746   2c2 9795   ^cexp 11104
This theorem is referenced by:  pythagtrip  12887
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-n0 9966  df-z 10025  df-uz 10231  df-seq 11047  df-exp 11105
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