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Theorem polid 21568
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 21493. (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 5717 . . 3  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  .ih  B )  =  ( if ( A  e.  ~H ,  A ,  0h )  .ih  B
) )
2 oveq1 5717 . . . . . . . 8  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  +h  B )  =  ( if ( A  e.  ~H ,  A ,  0h )  +h  B
) )
32fveq2d 5381 . . . . . . 7  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( normh `  ( A  +h  B ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  B ) ) )
43oveq1d 5725 . . . . . 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 5717 . . . . . . . 8  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  -h  B )  =  ( if ( A  e.  ~H ,  A ,  0h )  -h  B
) )
65fveq2d 5381 . . . . . . 7  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( normh `  ( A  -h  B ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  -h  B ) ) )
76oveq1d 5725 . . . . . 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 5728 . . . . 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 5717 . . . . . . . . 9  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  +h  ( _i  .h  B ) )  =  ( if ( A  e.  ~H ,  A ,  0h )  +h  (
_i  .h  B )
) )
109fveq2d 5381 . . . . . . . 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 5725 . . . . . . 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 5717 . . . . . . . . 9  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  -h  ( _i  .h  B ) )  =  ( if ( A  e.  ~H ,  A ,  0h )  -h  (
_i  .h  B )
) )
1312fveq2d 5381 . . . . . . . 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 5725 . . . . . . 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 5728 . . . . . 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 5726 . . . . 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 5728 . . . 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 5725 . . 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 2267 . 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 5718 . . 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 5718 . . . . . . . 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 5381 . . . . . . 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 5725 . . . . . 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 5718 . . . . . . . 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 5381 . . . . . . 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 5725 . . . . . 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 5728 . . . . 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 5718 . . . . . . . . . 10  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
_i  .h  B )  =  ( _i  .h  if ( B  e.  ~H ,  B ,  0h )
) )
2928oveq2d 5726 . . . . . . . . 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 5381 . . . . . . . 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 5725 . . . . . . 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 5726 . . . . . . . . 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 5381 . . . . . . . 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 5725 . . . . . . 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 5728 . . . . . 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 5726 . . . . 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 5728 . . . 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 5725 . . 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 2267 . 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 21413 . . . 4  |-  0h  e.  ~H
4140elimel 3522 . . 3  |-  if ( A  e.  ~H ,  A ,  0h )  e.  ~H
4240elimel 3522 . . 3  |-  if ( B  e.  ~H ,  B ,  0h )  e.  ~H
4341, 42polidi 21567 . 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 3512 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 3470   ` cfv 4592  (class class class)co 5710   _ici 8619    + caddc 8620    x. cmul 8622    - cmin 8917    / cdiv 9303   2c2 9675   4c4 9677   ^cexp 10982   ~Hchil 21329    +h cva 21330    .h csm 21331    .ih csp 21332   normhcno 21333   0hc0v 21334    -h cmv 21335
This theorem is referenced by:  hhip  21586
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695  ax-hfvadd 21410  ax-hv0cl 21413  ax-hfvmul 21415  ax-hvmul0 21420  ax-hfi 21488  ax-his1 21491  ax-his2 21492  ax-his3 21493  ax-his4 21494
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-sup 7078  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-4 9686  df-n0 9845  df-z 9904  df-uz 10110  df-rp 10234  df-seq 10925  df-exp 10983  df-cj 11461  df-re 11462  df-im 11463  df-sqr 11597  df-hnorm 21378  df-hvsub 21381
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