HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  polid Unicode version

Theorem polid 21698
Description: Polarization identity. Recovers inner product from norm. Exercise 4(a) of [ReedSimon] p. 63. The outermost operation is + instead of - due to our mathematicians' (rather than physicists') version of axiom ax-his3 21623. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
Assertion
Ref Expression
polid  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  B
)  =  ( ( ( ( ( normh `  ( A  +h  B
) ) ^ 2 )  -  ( (
normh `  ( A  -h  B ) ) ^
2 ) )  +  ( _i  x.  (
( ( normh `  ( A  +h  ( _i  .h  B ) ) ) ^ 2 )  -  ( ( normh `  ( A  -h  ( _i  .h  B ) ) ) ^ 2 ) ) ) )  /  4
) )

Proof of Theorem polid
StepHypRef Expression
1 oveq1 5799 . . 3  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  .ih  B )  =  ( if ( A  e.  ~H ,  A ,  0h )  .ih  B
) )
2 oveq1 5799 . . . . . . . 8  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  +h  B )  =  ( if ( A  e.  ~H ,  A ,  0h )  +h  B
) )
32fveq2d 5462 . . . . . . 7  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( normh `  ( A  +h  B ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  B ) ) )
43oveq1d 5807 . . . . . 6  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( normh `  ( A  +h  B ) ) ^
2 )  =  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  B ) ) ^
2 ) )
5 oveq1 5799 . . . . . . . 8  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  -h  B )  =  ( if ( A  e.  ~H ,  A ,  0h )  -h  B
) )
65fveq2d 5462 . . . . . . 7  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( normh `  ( A  -h  B ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) ) )
76oveq1d 5807 . . . . . 6  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( normh `  ( A  -h  B ) ) ^
2 )  =  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) ) ^
2 ) )
84, 7oveq12d 5810 . . . . 5  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( ( normh `  ( A  +h  B ) ) ^ 2 )  -  ( ( normh `  ( A  -h  B ) ) ^ 2 ) )  =  ( ( (
normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  B ) ) ^
2 )  -  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) ) ^
2 ) ) )
9 oveq1 5799 . . . . . . . . 9  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  +h  ( _i  .h  B ) )  =  ( if ( A  e.  ~H ,  A ,  0h )  +h  (
_i  .h  B )
) )
109fveq2d 5462 . . . . . . . 8  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( normh `  ( A  +h  ( _i  .h  B
) ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  ( _i  .h  B
) ) ) )
1110oveq1d 5807 . . . . . . 7  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( normh `  ( A  +h  ( _i  .h  B
) ) ) ^
2 )  =  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  ( _i  .h  B
) ) ) ^
2 ) )
12 oveq1 5799 . . . . . . . . 9  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  -h  ( _i  .h  B ) )  =  ( if ( A  e.  ~H ,  A ,  0h )  -h  (
_i  .h  B )
) )
1312fveq2d 5462 . . . . . . . 8  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( normh `  ( A  -h  ( _i  .h  B
) ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  ( _i  .h  B
) ) ) )
1413oveq1d 5807 . . . . . . 7  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( normh `  ( A  -h  ( _i  .h  B
) ) ) ^
2 )  =  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  ( _i  .h  B
) ) ) ^
2 ) )
1511, 14oveq12d 5810 . . . . . 6  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( ( normh `  ( A  +h  ( _i  .h  B ) ) ) ^ 2 )  -  ( ( normh `  ( A  -h  ( _i  .h  B ) ) ) ^ 2 ) )  =  ( ( (
normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  ( _i  .h  B
) ) ) ^
2 )  -  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  ( _i  .h  B
) ) ) ^
2 ) ) )
1615oveq2d 5808 . . . . 5  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
_i  x.  ( (
( normh `  ( A  +h  ( _i  .h  B
) ) ) ^
2 )  -  (
( normh `  ( A  -h  ( _i  .h  B
) ) ) ^
2 ) ) )  =  ( _i  x.  ( ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  (
_i  .h  B )
) ) ^ 2 )  -  ( (
normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  ( _i  .h  B
) ) ) ^
2 ) ) ) )
178, 16oveq12d 5810 . . . 4  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( ( ( normh `  ( A  +h  B
) ) ^ 2 )  -  ( (
normh `  ( A  -h  B ) ) ^
2 ) )  +  ( _i  x.  (
( ( normh `  ( A  +h  ( _i  .h  B ) ) ) ^ 2 )  -  ( ( normh `  ( A  -h  ( _i  .h  B ) ) ) ^ 2 ) ) ) )  =  ( ( ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  B
) ) ^ 2 )  -  ( (
normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) ) ^
2 ) )  +  ( _i  x.  (
( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  ( _i  .h  B
) ) ) ^
2 )  -  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  ( _i  .h  B
) ) ) ^
2 ) ) ) ) )
1817oveq1d 5807 . . 3  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( ( ( (
normh `  ( A  +h  B ) ) ^
2 )  -  (
( normh `  ( A  -h  B ) ) ^
2 ) )  +  ( _i  x.  (
( ( normh `  ( A  +h  ( _i  .h  B ) ) ) ^ 2 )  -  ( ( normh `  ( A  -h  ( _i  .h  B ) ) ) ^ 2 ) ) ) )  /  4
)  =  ( ( ( ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  B
) ) ^ 2 )  -  ( (
normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) ) ^
2 ) )  +  ( _i  x.  (
( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  ( _i  .h  B
) ) ) ^
2 )  -  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  ( _i  .h  B
) ) ) ^
2 ) ) ) )  /  4 ) )
191, 18eqeq12d 2272 . 2  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( A  .ih  B
)  =  ( ( ( ( ( normh `  ( A  +h  B
) ) ^ 2 )  -  ( (
normh `  ( A  -h  B ) ) ^
2 ) )  +  ( _i  x.  (
( ( normh `  ( A  +h  ( _i  .h  B ) ) ) ^ 2 )  -  ( ( normh `  ( A  -h  ( _i  .h  B ) ) ) ^ 2 ) ) ) )  /  4
)  <->  ( if ( A  e.  ~H ,  A ,  0h )  .ih  B )  =  ( ( ( ( (
normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  B ) ) ^
2 )  -  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) ) ^
2 ) )  +  ( _i  x.  (
( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  ( _i  .h  B
) ) ) ^
2 )  -  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  ( _i  .h  B
) ) ) ^
2 ) ) ) )  /  4 ) ) )
20 oveq2 5800 . . 3  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( if ( A  e.  ~H ,  A ,  0h )  .ih  B )  =  ( if ( A  e. 
~H ,  A ,  0h )  .ih  if ( B  e.  ~H ,  B ,  0h )
) )
21 oveq2 5800 . . . . . . . 8  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( if ( A  e.  ~H ,  A ,  0h )  +h  B )  =  ( if ( A  e. 
~H ,  A ,  0h )  +h  if ( B  e.  ~H ,  B ,  0h )
) )
2221fveq2d 5462 . . . . . . 7  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  B ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  if ( B  e. 
~H ,  B ,  0h ) ) ) )
2322oveq1d 5807 . . . . . 6  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  B ) ) ^
2 )  =  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  if ( B  e. 
~H ,  B ,  0h ) ) ) ^
2 ) )
24 oveq2 5800 . . . . . . . 8  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( if ( A  e.  ~H ,  A ,  0h )  -h  B )  =  ( if ( A  e. 
~H ,  A ,  0h )  -h  if ( B  e.  ~H ,  B ,  0h )
) )
2524fveq2d 5462 . . . . . . 7  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) ) )
2625oveq1d 5807 . . . . . 6  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) ) ^
2 )  =  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) ) ^
2 ) )
2723, 26oveq12d 5810 . . . . 5  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  B ) ) ^
2 )  -  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) ) ^
2 ) )  =  ( ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  if ( B  e.  ~H ,  B ,  0h )
) ) ^ 2 )  -  ( (
normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) ) ^
2 ) ) )
28 oveq2 5800 . . . . . . . . . 10  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
_i  .h  B )  =  ( _i  .h  if ( B  e.  ~H ,  B ,  0h )
) )
2928oveq2d 5808 . . . . . . . . 9  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( if ( A  e.  ~H ,  A ,  0h )  +h  ( _i  .h  B
) )  =  ( if ( A  e. 
~H ,  A ,  0h )  +h  (
_i  .h  if ( B  e.  ~H ,  B ,  0h ) ) ) )
3029fveq2d 5462 . . . . . . . 8  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  ( _i  .h  B
) ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  ( _i  .h  if ( B  e.  ~H ,  B ,  0h )
) ) ) )
3130oveq1d 5807 . . . . . . 7  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  ( _i  .h  B
) ) ) ^
2 )  =  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  ( _i  .h  if ( B  e.  ~H ,  B ,  0h )
) ) ) ^
2 ) )
3228oveq2d 5808 . . . . . . . . 9  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( if ( A  e.  ~H ,  A ,  0h )  -h  ( _i  .h  B
) )  =  ( if ( A  e. 
~H ,  A ,  0h )  -h  (
_i  .h  if ( B  e.  ~H ,  B ,  0h ) ) ) )
3332fveq2d 5462 . . . . . . . 8  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  ( _i  .h  B
) ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  ( _i  .h  if ( B  e.  ~H ,  B ,  0h )
) ) ) )
3433oveq1d 5807 . . . . . . 7  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  ( _i  .h  B
) ) ) ^
2 )  =  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  ( _i  .h  if ( B  e.  ~H ,  B ,  0h )
) ) ) ^
2 ) )
3531, 34oveq12d 5810 . . . . . 6  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  ( _i  .h  B
) ) ) ^
2 )  -  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  ( _i  .h  B
) ) ) ^
2 ) )  =  ( ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  (
_i  .h  if ( B  e.  ~H ,  B ,  0h ) ) ) ) ^ 2 )  -  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  (
_i  .h  if ( B  e.  ~H ,  B ,  0h ) ) ) ) ^ 2 ) ) )
3635oveq2d 5808 . . . . 5  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
_i  x.  ( (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  ( _i  .h  B
) ) ) ^
2 )  -  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  ( _i  .h  B
) ) ) ^
2 ) ) )  =  ( _i  x.  ( ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  (
_i  .h  if ( B  e.  ~H ,  B ,  0h ) ) ) ) ^ 2 )  -  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  (
_i  .h  if ( B  e.  ~H ,  B ,  0h ) ) ) ) ^ 2 ) ) ) )
3727, 36oveq12d 5810 . . . 4  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  B
) ) ^ 2 )  -  ( (
normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) ) ^
2 ) )  +  ( _i  x.  (
( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  ( _i  .h  B
) ) ) ^
2 )  -  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  ( _i  .h  B
) ) ) ^
2 ) ) ) )  =  ( ( ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  if ( B  e. 
~H ,  B ,  0h ) ) ) ^
2 )  -  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) ) ^
2 ) )  +  ( _i  x.  (
( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  ( _i  .h  if ( B  e.  ~H ,  B ,  0h )
) ) ) ^
2 )  -  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  ( _i  .h  if ( B  e.  ~H ,  B ,  0h )
) ) ) ^
2 ) ) ) ) )
3837oveq1d 5807 . . 3  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( ( ( (
normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  B ) ) ^
2 )  -  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) ) ^
2 ) )  +  ( _i  x.  (
( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  ( _i  .h  B
) ) ) ^
2 )  -  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  ( _i  .h  B
) ) ) ^
2 ) ) ) )  /  4 )  =  ( ( ( ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  if ( B  e. 
~H ,  B ,  0h ) ) ) ^
2 )  -  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) ) ^
2 ) )  +  ( _i  x.  (
( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  ( _i  .h  if ( B  e.  ~H ,  B ,  0h )
) ) ) ^
2 )  -  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  ( _i  .h  if ( B  e.  ~H ,  B ,  0h )
) ) ) ^
2 ) ) ) )  /  4 ) )
3920, 38eqeq12d 2272 . 2  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( if ( A  e.  ~H ,  A ,  0h )  .ih  B
)  =  ( ( ( ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  B
) ) ^ 2 )  -  ( (
normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) ) ^
2 ) )  +  ( _i  x.  (
( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  ( _i  .h  B
) ) ) ^
2 )  -  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  ( _i  .h  B
) ) ) ^
2 ) ) ) )  /  4 )  <-> 
( if ( A  e.  ~H ,  A ,  0h )  .ih  if ( B  e.  ~H ,  B ,  0h )
)  =  ( ( ( ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  if ( B  e.  ~H ,  B ,  0h )
) ) ^ 2 )  -  ( (
normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) ) ^
2 ) )  +  ( _i  x.  (
( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  ( _i  .h  if ( B  e.  ~H ,  B ,  0h )
) ) ) ^
2 )  -  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  ( _i  .h  if ( B  e.  ~H ,  B ,  0h )
) ) ) ^
2 ) ) ) )  /  4 ) ) )
40 ax-hv0cl 21543 . . . 4  |-  0h  e.  ~H
4140elimel 3591 . . 3  |-  if ( A  e.  ~H ,  A ,  0h )  e.  ~H
4240elimel 3591 . . 3  |-  if ( B  e.  ~H ,  B ,  0h )  e.  ~H
4341, 42polidi 21697 . 2  |-  ( if ( A  e.  ~H ,  A ,  0h )  .ih  if ( B  e. 
~H ,  B ,  0h ) )  =  ( ( ( ( (
normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  if ( B  e. 
~H ,  B ,  0h ) ) ) ^
2 )  -  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) ) ) ^
2 ) )  +  ( _i  x.  (
( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  ( _i  .h  if ( B  e.  ~H ,  B ,  0h )
) ) ) ^
2 )  -  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  ( _i  .h  if ( B  e.  ~H ,  B ,  0h )
) ) ) ^
2 ) ) ) )  /  4 )
4419, 39, 43dedth2h 3581 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  B
)  =  ( ( ( ( ( normh `  ( A  +h  B
) ) ^ 2 )  -  ( (
normh `  ( A  -h  B ) ) ^
2 ) )  +  ( _i  x.  (
( ( normh `  ( A  +h  ( _i  .h  B ) ) ) ^ 2 )  -  ( ( normh `  ( A  -h  ( _i  .h  B ) ) ) ^ 2 ) ) ) )  /  4
) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   ifcif 3539   ` cfv 4673  (class class class)co 5792   _ici 8707    + caddc 8708    x. cmul 8710    - cmin 9005    / cdiv 9391   2c2 9763   4c4 9765   ^cexp 11070   ~Hchil 21459    +h cva 21460    .h csm 21461    .ih csp 21462   normhcno 21463   0hc0v 21464    -h cmv 21465
This theorem is referenced by:  hhip  21716
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783  ax-hfvadd 21540  ax-hv0cl 21543  ax-hfvmul 21545  ax-hvmul0 21550  ax-hfi 21618  ax-his1 21621  ax-his2 21622  ax-his3 21623  ax-his4 21624
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-sup 7162  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-n0 9933  df-z 9992  df-uz 10198  df-rp 10322  df-seq 11013  df-exp 11071  df-cj 11549  df-re 11550  df-im 11551  df-sqr 11685  df-hnorm 21508  df-hvsub 21511
  Copyright terms: Public domain W3C validator