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Theorem polid2i 21752
Description: Generalized polarization identity. Generalization of Exercise 4(a) of [ReedSimon] p. 63. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)
Hypotheses
Ref Expression
polid2.1  |-  A  e. 
~H
polid2.2  |-  B  e. 
~H
polid2.3  |-  C  e. 
~H
polid2.4  |-  D  e. 
~H
Assertion
Ref Expression
polid2i  |-  ( A 
.ih  B )  =  ( ( ( ( ( A  +h  C
)  .ih  ( D  +h  B ) )  -  ( ( A  -h  C )  .ih  ( D  -h  B ) ) )  +  ( _i  x.  ( ( ( A  +h  ( _i  .h  C ) ) 
.ih  ( D  +h  ( _i  .h  B
) ) )  -  ( ( A  -h  ( _i  .h  C
) )  .ih  ( D  -h  ( _i  .h  B ) ) ) ) ) )  / 
4 )

Proof of Theorem polid2i
StepHypRef Expression
1 polid2.1 . . . 4  |-  A  e. 
~H
2 polid2.2 . . . 4  |-  B  e. 
~H
31, 2hicli 21676 . . 3  |-  ( A 
.ih  B )  e.  CC
4 4cn 9836 . . 3  |-  4  e.  CC
5 4re 9835 . . . 4  |-  4  e.  RR
6 4pos 9848 . . . 4  |-  0  <  4
75, 6gt0ne0ii 9325 . . 3  |-  4  =/=  0
83, 4, 7divcan3i 9522 . 2  |-  ( ( 4  x.  ( A 
.ih  B ) )  /  4 )  =  ( A  .ih  B
)
9 2cn 9832 . . . . 5  |-  2  e.  CC
10 polid2.3 . . . . . . 7  |-  C  e. 
~H
11 polid2.4 . . . . . . 7  |-  D  e. 
~H
1210, 11hicli 21676 . . . . . 6  |-  ( C 
.ih  D )  e.  CC
133, 12addcli 8857 . . . . 5  |-  ( ( A  .ih  B )  +  ( C  .ih  D ) )  e.  CC
143, 12subcli 9138 . . . . 5  |-  ( ( A  .ih  B )  -  ( C  .ih  D ) )  e.  CC
159, 13, 14adddii 8863 . . . 4  |-  ( 2  x.  ( ( ( A  .ih  B )  +  ( C  .ih  D ) )  +  ( ( A  .ih  B
)  -  ( C 
.ih  D ) ) ) )  =  ( ( 2  x.  (
( A  .ih  B
)  +  ( C 
.ih  D ) ) )  +  ( 2  x.  ( ( A 
.ih  B )  -  ( C  .ih  D ) ) ) )
16 ppncan 9105 . . . . . . . 8  |-  ( ( ( A  .ih  B
)  e.  CC  /\  ( C  .ih  D )  e.  CC  /\  ( A  .ih  B )  e.  CC )  ->  (
( ( A  .ih  B )  +  ( C 
.ih  D ) )  +  ( ( A 
.ih  B )  -  ( C  .ih  D ) ) )  =  ( ( A  .ih  B
)  +  ( A 
.ih  B ) ) )
173, 12, 3, 16mp3an 1277 . . . . . . 7  |-  ( ( ( A  .ih  B
)  +  ( C 
.ih  D ) )  +  ( ( A 
.ih  B )  -  ( C  .ih  D ) ) )  =  ( ( A  .ih  B
)  +  ( A 
.ih  B ) )
1832timesi 9861 . . . . . . 7  |-  ( 2  x.  ( A  .ih  B ) )  =  ( ( A  .ih  B
)  +  ( A 
.ih  B ) )
1917, 18eqtr4i 2319 . . . . . 6  |-  ( ( ( A  .ih  B
)  +  ( C 
.ih  D ) )  +  ( ( A 
.ih  B )  -  ( C  .ih  D ) ) )  =  ( 2  x.  ( A 
.ih  B ) )
2019oveq2i 5885 . . . . 5  |-  ( 2  x.  ( ( ( A  .ih  B )  +  ( C  .ih  D ) )  +  ( ( A  .ih  B
)  -  ( C 
.ih  D ) ) ) )  =  ( 2  x.  ( 2  x.  ( A  .ih  B ) ) )
219, 9, 3mulassi 8862 . . . . 5  |-  ( ( 2  x.  2 )  x.  ( A  .ih  B ) )  =  ( 2  x.  ( 2  x.  ( A  .ih  B ) ) )
22 2t2e4 9887 . . . . . 6  |-  ( 2  x.  2 )  =  4
2322oveq1i 5884 . . . . 5  |-  ( ( 2  x.  2 )  x.  ( A  .ih  B ) )  =  ( 4  x.  ( A 
.ih  B ) )
2420, 21, 233eqtr2ri 2323 . . . 4  |-  ( 4  x.  ( A  .ih  B ) )  =  ( 2  x.  ( ( ( A  .ih  B
)  +  ( C 
.ih  D ) )  +  ( ( A 
.ih  B )  -  ( C  .ih  D ) ) ) )
251, 11hicli 21676 . . . . . . . 8  |-  ( A 
.ih  D )  e.  CC
2610, 2hicli 21676 . . . . . . . 8  |-  ( C 
.ih  B )  e.  CC
2725, 26addcli 8857 . . . . . . 7  |-  ( ( A  .ih  D )  +  ( C  .ih  B ) )  e.  CC
2827, 13, 13pnncani 9157 . . . . . 6  |-  ( ( ( ( A  .ih  D )  +  ( C 
.ih  B ) )  +  ( ( A 
.ih  B )  +  ( C  .ih  D
) ) )  -  ( ( ( A 
.ih  D )  +  ( C  .ih  B
) )  -  (
( A  .ih  B
)  +  ( C 
.ih  D ) ) ) )  =  ( ( ( A  .ih  B )  +  ( C 
.ih  D ) )  +  ( ( A 
.ih  B )  +  ( C  .ih  D
) ) )
291, 10, 11, 2normlem8 21712 . . . . . . 7  |-  ( ( A  +h  C ) 
.ih  ( D  +h  B ) )  =  ( ( ( A 
.ih  D )  +  ( C  .ih  B
) )  +  ( ( A  .ih  B
)  +  ( C 
.ih  D ) ) )
301, 10, 11, 2normlem9 21713 . . . . . . 7  |-  ( ( A  -h  C ) 
.ih  ( D  -h  B ) )  =  ( ( ( A 
.ih  D )  +  ( C  .ih  B
) )  -  (
( A  .ih  B
)  +  ( C 
.ih  D ) ) )
3129, 30oveq12i 5886 . . . . . 6  |-  ( ( ( A  +h  C
)  .ih  ( D  +h  B ) )  -  ( ( A  -h  C )  .ih  ( D  -h  B ) ) )  =  ( ( ( ( A  .ih  D )  +  ( C 
.ih  B ) )  +  ( ( A 
.ih  B )  +  ( C  .ih  D
) ) )  -  ( ( ( A 
.ih  D )  +  ( C  .ih  B
) )  -  (
( A  .ih  B
)  +  ( C 
.ih  D ) ) ) )
32132timesi 9861 . . . . . 6  |-  ( 2  x.  ( ( A 
.ih  B )  +  ( C  .ih  D
) ) )  =  ( ( ( A 
.ih  B )  +  ( C  .ih  D
) )  +  ( ( A  .ih  B
)  +  ( C 
.ih  D ) ) )
3328, 31, 323eqtr4i 2326 . . . . 5  |-  ( ( ( A  +h  C
)  .ih  ( D  +h  B ) )  -  ( ( A  -h  C )  .ih  ( D  -h  B ) ) )  =  ( 2  x.  ( ( A 
.ih  B )  +  ( C  .ih  D
) ) )
34 ax-icn 8812 . . . . . . . . . . 11  |-  _i  e.  CC
3534, 10hvmulcli 21610 . . . . . . . . . 10  |-  ( _i  .h  C )  e. 
~H
3634, 2hvmulcli 21610 . . . . . . . . . 10  |-  ( _i  .h  B )  e. 
~H
371, 35, 11, 36normlem8 21712 . . . . . . . . 9  |-  ( ( A  +h  ( _i  .h  C ) ) 
.ih  ( D  +h  ( _i  .h  B
) ) )  =  ( ( ( A 
.ih  D )  +  ( ( _i  .h  C )  .ih  (
_i  .h  B )
) )  +  ( ( A  .ih  (
_i  .h  B )
)  +  ( ( _i  .h  C ) 
.ih  D ) ) )
381, 35, 11, 36normlem9 21713 . . . . . . . . 9  |-  ( ( A  -h  ( _i  .h  C ) ) 
.ih  ( D  -h  ( _i  .h  B
) ) )  =  ( ( ( A 
.ih  D )  +  ( ( _i  .h  C )  .ih  (
_i  .h  B )
) )  -  (
( A  .ih  (
_i  .h  B )
)  +  ( ( _i  .h  C ) 
.ih  D ) ) )
3937, 38oveq12i 5886 . . . . . . . 8  |-  ( ( ( A  +h  (
_i  .h  C )
)  .ih  ( D  +h  ( _i  .h  B
) ) )  -  ( ( A  -h  ( _i  .h  C
) )  .ih  ( D  -h  ( _i  .h  B ) ) ) )  =  ( ( ( ( A  .ih  D )  +  ( ( _i  .h  C ) 
.ih  ( _i  .h  B ) ) )  +  ( ( A 
.ih  ( _i  .h  B ) )  +  ( ( _i  .h  C )  .ih  D
) ) )  -  ( ( ( A 
.ih  D )  +  ( ( _i  .h  C )  .ih  (
_i  .h  B )
) )  -  (
( A  .ih  (
_i  .h  B )
)  +  ( ( _i  .h  C ) 
.ih  D ) ) ) )
4035, 36hicli 21676 . . . . . . . . . 10  |-  ( ( _i  .h  C ) 
.ih  ( _i  .h  B ) )  e.  CC
4125, 40addcli 8857 . . . . . . . . 9  |-  ( ( A  .ih  D )  +  ( ( _i  .h  C )  .ih  ( _i  .h  B
) ) )  e.  CC
421, 36hicli 21676 . . . . . . . . . 10  |-  ( A 
.ih  ( _i  .h  B ) )  e.  CC
4335, 11hicli 21676 . . . . . . . . . 10  |-  ( ( _i  .h  C ) 
.ih  D )  e.  CC
4442, 43addcli 8857 . . . . . . . . 9  |-  ( ( A  .ih  ( _i  .h  B ) )  +  ( ( _i  .h  C )  .ih  D ) )  e.  CC
4541, 44, 44pnncani 9157 . . . . . . . 8  |-  ( ( ( ( A  .ih  D )  +  ( ( _i  .h  C ) 
.ih  ( _i  .h  B ) ) )  +  ( ( A 
.ih  ( _i  .h  B ) )  +  ( ( _i  .h  C )  .ih  D
) ) )  -  ( ( ( A 
.ih  D )  +  ( ( _i  .h  C )  .ih  (
_i  .h  B )
) )  -  (
( A  .ih  (
_i  .h  B )
)  +  ( ( _i  .h  C ) 
.ih  D ) ) ) )  =  ( ( ( A  .ih  ( _i  .h  B
) )  +  ( ( _i  .h  C
)  .ih  D )
)  +  ( ( A  .ih  ( _i  .h  B ) )  +  ( ( _i  .h  C )  .ih  D ) ) )
46442timesi 9861 . . . . . . . . 9  |-  ( 2  x.  ( ( A 
.ih  ( _i  .h  B ) )  +  ( ( _i  .h  C )  .ih  D
) ) )  =  ( ( ( A 
.ih  ( _i  .h  B ) )  +  ( ( _i  .h  C )  .ih  D
) )  +  ( ( A  .ih  (
_i  .h  B )
)  +  ( ( _i  .h  C ) 
.ih  D ) ) )
47 his5 21681 . . . . . . . . . . . . 13  |-  ( ( _i  e.  CC  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  ( _i  .h  B ) )  =  ( ( * `  _i )  x.  ( A  .ih  B ) ) )
4834, 1, 2, 47mp3an 1277 . . . . . . . . . . . 12  |-  ( A 
.ih  ( _i  .h  B ) )  =  ( ( * `  _i )  x.  ( A  .ih  B ) )
49 cji 11660 . . . . . . . . . . . . 13  |-  ( * `
 _i )  = 
-u _i
5049oveq1i 5884 . . . . . . . . . . . 12  |-  ( ( * `  _i )  x.  ( A  .ih  B ) )  =  (
-u _i  x.  ( A  .ih  B ) )
5148, 50eqtri 2316 . . . . . . . . . . 11  |-  ( A 
.ih  ( _i  .h  B ) )  =  ( -u _i  x.  ( A  .ih  B ) )
52 ax-his3 21679 . . . . . . . . . . . 12  |-  ( ( _i  e.  CC  /\  C  e.  ~H  /\  D  e.  ~H )  ->  (
( _i  .h  C
)  .ih  D )  =  ( _i  x.  ( C  .ih  D ) ) )
5334, 10, 11, 52mp3an 1277 . . . . . . . . . . 11  |-  ( ( _i  .h  C ) 
.ih  D )  =  ( _i  x.  ( C  .ih  D ) )
5451, 53oveq12i 5886 . . . . . . . . . 10  |-  ( ( A  .ih  ( _i  .h  B ) )  +  ( ( _i  .h  C )  .ih  D ) )  =  ( ( -u _i  x.  ( A  .ih  B ) )  +  ( _i  x.  ( C  .ih  D ) ) )
5554oveq2i 5885 . . . . . . . . 9  |-  ( 2  x.  ( ( A 
.ih  ( _i  .h  B ) )  +  ( ( _i  .h  C )  .ih  D
) ) )  =  ( 2  x.  (
( -u _i  x.  ( A  .ih  B ) )  +  ( _i  x.  ( C  .ih  D ) ) ) )
5646, 55eqtr3i 2318 . . . . . . . 8  |-  ( ( ( A  .ih  (
_i  .h  B )
)  +  ( ( _i  .h  C ) 
.ih  D ) )  +  ( ( A 
.ih  ( _i  .h  B ) )  +  ( ( _i  .h  C )  .ih  D
) ) )  =  ( 2  x.  (
( -u _i  x.  ( A  .ih  B ) )  +  ( _i  x.  ( C  .ih  D ) ) ) )
5739, 45, 563eqtri 2320 . . . . . . 7  |-  ( ( ( A  +h  (
_i  .h  C )
)  .ih  ( D  +h  ( _i  .h  B
) ) )  -  ( ( A  -h  ( _i  .h  C
) )  .ih  ( D  -h  ( _i  .h  B ) ) ) )  =  ( 2  x.  ( ( -u _i  x.  ( A  .ih  B ) )  +  ( _i  x.  ( C 
.ih  D ) ) ) )
5857oveq2i 5885 . . . . . 6  |-  ( _i  x.  ( ( ( A  +h  ( _i  .h  C ) ) 
.ih  ( D  +h  ( _i  .h  B
) ) )  -  ( ( A  -h  ( _i  .h  C
) )  .ih  ( D  -h  ( _i  .h  B ) ) ) ) )  =  ( _i  x.  ( 2  x.  ( ( -u _i  x.  ( A  .ih  B ) )  +  ( _i  x.  ( C 
.ih  D ) ) ) ) )
5934negcli 9130 . . . . . . . . 9  |-  -u _i  e.  CC
6059, 3mulcli 8858 . . . . . . . 8  |-  ( -u _i  x.  ( A  .ih  B ) )  e.  CC
6134, 12mulcli 8858 . . . . . . . 8  |-  ( _i  x.  ( C  .ih  D ) )  e.  CC
6260, 61addcli 8857 . . . . . . 7  |-  ( (
-u _i  x.  ( A  .ih  B ) )  +  ( _i  x.  ( C  .ih  D ) ) )  e.  CC
639, 34, 62mul12i 9023 . . . . . 6  |-  ( 2  x.  ( _i  x.  ( ( -u _i  x.  ( A  .ih  B
) )  +  ( _i  x.  ( C 
.ih  D ) ) ) ) )  =  ( _i  x.  (
2  x.  ( (
-u _i  x.  ( A  .ih  B ) )  +  ( _i  x.  ( C  .ih  D ) ) ) ) )
6434, 60, 61adddii 8863 . . . . . . . 8  |-  ( _i  x.  ( ( -u _i  x.  ( A  .ih  B ) )  +  ( _i  x.  ( C 
.ih  D ) ) ) )  =  ( ( _i  x.  ( -u _i  x.  ( A 
.ih  B ) ) )  +  ( _i  x.  ( _i  x.  ( C  .ih  D ) ) ) )
6534, 34mulneg2i 9242 . . . . . . . . . . . 12  |-  ( _i  x.  -u _i )  = 
-u ( _i  x.  _i )
66 ixi 9413 . . . . . . . . . . . . 13  |-  ( _i  x.  _i )  = 
-u 1
6766negeqi 9061 . . . . . . . . . . . 12  |-  -u (
_i  x.  _i )  =  -u -u 1
68 ax-1cn 8811 . . . . . . . . . . . . 13  |-  1  e.  CC
6968negnegi 9132 . . . . . . . . . . . 12  |-  -u -u 1  =  1
7065, 67, 693eqtri 2320 . . . . . . . . . . 11  |-  ( _i  x.  -u _i )  =  1
7170oveq1i 5884 . . . . . . . . . 10  |-  ( ( _i  x.  -u _i )  x.  ( A  .ih  B ) )  =  ( 1  x.  ( A  .ih  B ) )
7234, 59, 3mulassi 8862 . . . . . . . . . 10  |-  ( ( _i  x.  -u _i )  x.  ( A  .ih  B ) )  =  ( _i  x.  ( -u _i  x.  ( A 
.ih  B ) ) )
733mulid2i 8856 . . . . . . . . . 10  |-  ( 1  x.  ( A  .ih  B ) )  =  ( A  .ih  B )
7471, 72, 733eqtr3i 2324 . . . . . . . . 9  |-  ( _i  x.  ( -u _i  x.  ( A  .ih  B
) ) )  =  ( A  .ih  B
)
7566oveq1i 5884 . . . . . . . . . 10  |-  ( ( _i  x.  _i )  x.  ( C  .ih  D ) )  =  (
-u 1  x.  ( C  .ih  D ) )
7634, 34, 12mulassi 8862 . . . . . . . . . 10  |-  ( ( _i  x.  _i )  x.  ( C  .ih  D ) )  =  ( _i  x.  ( _i  x.  ( C  .ih  D ) ) )
7712mulm1i 9240 . . . . . . . . . 10  |-  ( -u
1  x.  ( C 
.ih  D ) )  =  -u ( C  .ih  D )
7875, 76, 773eqtr3i 2324 . . . . . . . . 9  |-  ( _i  x.  ( _i  x.  ( C  .ih  D ) ) )  =  -u ( C  .ih  D )
7974, 78oveq12i 5886 . . . . . . . 8  |-  ( ( _i  x.  ( -u _i  x.  ( A  .ih  B ) ) )  +  ( _i  x.  (
_i  x.  ( C  .ih  D ) ) ) )  =  ( ( A  .ih  B )  +  -u ( C  .ih  D ) )
803, 12negsubi 9140 . . . . . . . 8  |-  ( ( A  .ih  B )  +  -u ( C  .ih  D ) )  =  ( ( A  .ih  B
)  -  ( C 
.ih  D ) )
8164, 79, 803eqtri 2320 . . . . . . 7  |-  ( _i  x.  ( ( -u _i  x.  ( A  .ih  B ) )  +  ( _i  x.  ( C 
.ih  D ) ) ) )  =  ( ( A  .ih  B
)  -  ( C 
.ih  D ) )
8281oveq2i 5885 . . . . . 6  |-  ( 2  x.  ( _i  x.  ( ( -u _i  x.  ( A  .ih  B
) )  +  ( _i  x.  ( C 
.ih  D ) ) ) ) )  =  ( 2  x.  (
( A  .ih  B
)  -  ( C 
.ih  D ) ) )
8358, 63, 823eqtr2i 2322 . . . . 5  |-  ( _i  x.  ( ( ( A  +h  ( _i  .h  C ) ) 
.ih  ( D  +h  ( _i  .h  B
) ) )  -  ( ( A  -h  ( _i  .h  C
) )  .ih  ( D  -h  ( _i  .h  B ) ) ) ) )  =  ( 2  x.  ( ( A  .ih  B )  -  ( C  .ih  D ) ) )
8433, 83oveq12i 5886 . . . 4  |-  ( ( ( ( A  +h  C )  .ih  ( D  +h  B ) )  -  ( ( A  -h  C )  .ih  ( D  -h  B
) ) )  +  ( _i  x.  (
( ( A  +h  ( _i  .h  C
) )  .ih  ( D  +h  ( _i  .h  B ) ) )  -  ( ( A  -h  ( _i  .h  C ) )  .ih  ( D  -h  (
_i  .h  B )
) ) ) ) )  =  ( ( 2  x.  ( ( A  .ih  B )  +  ( C  .ih  D ) ) )  +  ( 2  x.  (
( A  .ih  B
)  -  ( C 
.ih  D ) ) ) )
8515, 24, 843eqtr4i 2326 . . 3  |-  ( 4  x.  ( A  .ih  B ) )  =  ( ( ( ( A  +h  C )  .ih  ( D  +h  B
) )  -  (
( A  -h  C
)  .ih  ( D  -h  B ) ) )  +  ( _i  x.  ( ( ( A  +h  ( _i  .h  C ) )  .ih  ( D  +h  (
_i  .h  B )
) )  -  (
( A  -h  (
_i  .h  C )
)  .ih  ( D  -h  ( _i  .h  B
) ) ) ) ) )
8685oveq1i 5884 . 2  |-  ( ( 4  x.  ( A 
.ih  B ) )  /  4 )  =  ( ( ( ( ( A  +h  C
)  .ih  ( D  +h  B ) )  -  ( ( A  -h  C )  .ih  ( D  -h  B ) ) )  +  ( _i  x.  ( ( ( A  +h  ( _i  .h  C ) ) 
.ih  ( D  +h  ( _i  .h  B
) ) )  -  ( ( A  -h  ( _i  .h  C
) )  .ih  ( D  -h  ( _i  .h  B ) ) ) ) ) )  / 
4 )
878, 86eqtr3i 2318 1  |-  ( A 
.ih  B )  =  ( ( ( ( ( A  +h  C
)  .ih  ( D  +h  B ) )  -  ( ( A  -h  C )  .ih  ( D  -h  B ) ) )  +  ( _i  x.  ( ( ( A  +h  ( _i  .h  C ) ) 
.ih  ( D  +h  ( _i  .h  B
) ) )  -  ( ( A  -h  ( _i  .h  C
) )  .ih  ( D  -h  ( _i  .h  B ) ) ) ) ) )  / 
4 )
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   CCcc 8751   1c1 8754   _ici 8755    + caddc 8756    x. cmul 8758    - cmin 9053   -ucneg 9054    / cdiv 9439   2c2 9811   4c4 9813   *ccj 11597   ~Hchil 21515    +h cva 21516    .h csm 21517    .ih csp 21518    -h cmv 21521
This theorem is referenced by:  polidi  21753  lnopeq0lem1  22601  lnophmlem2  22613
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-hfvadd 21596  ax-hfvmul 21601  ax-hfi 21674  ax-his1 21677  ax-his2 21678  ax-his3 21679
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-2 9820  df-3 9821  df-4 9822  df-cj 11600  df-re 11601  df-im 11602  df-hvsub 21567
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