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Theorem polid2i 21566
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 21490 . . 3  |-  ( A 
.ih  B )  e.  CC
4 4cn 9700 . . 3  |-  4  e.  CC
5 4re 9699 . . . 4  |-  4  e.  RR
6 4pos 9712 . . . 4  |-  0  <  4
75, 6gt0ne0ii 9189 . . 3  |-  4  =/=  0
83, 4, 7divcan3i 9386 . 2  |-  ( ( 4  x.  ( A 
.ih  B ) )  /  4 )  =  ( A  .ih  B
)
9 2cn 9696 . . . . 5  |-  2  e.  CC
10 polid2.3 . . . . . . 7  |-  C  e. 
~H
11 polid2.4 . . . . . . 7  |-  D  e. 
~H
1210, 11hicli 21490 . . . . . 6  |-  ( C 
.ih  D )  e.  CC
133, 12addcli 8721 . . . . 5  |-  ( ( A  .ih  B )  +  ( C  .ih  D ) )  e.  CC
143, 12subcli 9002 . . . . 5  |-  ( ( A  .ih  B )  -  ( C  .ih  D ) )  e.  CC
159, 13, 14adddii 8727 . . . 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 8969 . . . . . . . 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 1282 . . . . . . 7  |-  ( ( ( A  .ih  B
)  +  ( C 
.ih  D ) )  +  ( ( A 
.ih  B )  -  ( C  .ih  D ) ) )  =  ( ( A  .ih  B
)  +  ( A 
.ih  B ) )
1832timesi 9724 . . . . . . 7  |-  ( 2  x.  ( A  .ih  B ) )  =  ( ( A  .ih  B
)  +  ( A 
.ih  B ) )
1917, 18eqtr4i 2276 . . . . . 6  |-  ( ( ( A  .ih  B
)  +  ( C 
.ih  D ) )  +  ( ( A 
.ih  B )  -  ( C  .ih  D ) ) )  =  ( 2  x.  ( A 
.ih  B ) )
2019oveq2i 5721 . . . . 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 8726 . . . . 5  |-  ( ( 2  x.  2 )  x.  ( A  .ih  B ) )  =  ( 2  x.  ( 2  x.  ( A  .ih  B ) ) )
22 2t2e4 9750 . . . . . 6  |-  ( 2  x.  2 )  =  4
2322oveq1i 5720 . . . . 5  |-  ( ( 2  x.  2 )  x.  ( A  .ih  B ) )  =  ( 4  x.  ( A 
.ih  B ) )
2420, 21, 233eqtr2ri 2280 . . . 4  |-  ( 4  x.  ( A  .ih  B ) )  =  ( 2  x.  ( ( ( A  .ih  B
)  +  ( C 
.ih  D ) )  +  ( ( A 
.ih  B )  -  ( C  .ih  D ) ) ) )
251, 11hicli 21490 . . . . . . . 8  |-  ( A 
.ih  D )  e.  CC
2610, 2hicli 21490 . . . . . . . 8  |-  ( C 
.ih  B )  e.  CC
2725, 26addcli 8721 . . . . . . 7  |-  ( ( A  .ih  D )  +  ( C  .ih  B ) )  e.  CC
2827, 13, 13pnncani 9021 . . . . . 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 21526 . . . . . . 7  |-  ( ( A  +h  C ) 
.ih  ( D  +h  B ) )  =  ( ( ( A 
.ih  D )  +  ( C  .ih  B
) )  +  ( ( A  .ih  B
)  +  ( C 
.ih  D ) ) )
301, 10, 11, 2normlem9 21527 . . . . . . 7  |-  ( ( A  -h  C ) 
.ih  ( D  -h  B ) )  =  ( ( ( A 
.ih  D )  +  ( C  .ih  B
) )  -  (
( A  .ih  B
)  +  ( C 
.ih  D ) ) )
3129, 30oveq12i 5722 . . . . . 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 9724 . . . . . 6  |-  ( 2  x.  ( ( A 
.ih  B )  +  ( C  .ih  D
) ) )  =  ( ( ( A 
.ih  B )  +  ( C  .ih  D
) )  +  ( ( A  .ih  B
)  +  ( C 
.ih  D ) ) )
3328, 31, 323eqtr4i 2283 . . . . 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 8676 . . . . . . . . . . 11  |-  _i  e.  CC
3534, 10hvmulcli 21424 . . . . . . . . . 10  |-  ( _i  .h  C )  e. 
~H
3634, 2hvmulcli 21424 . . . . . . . . . 10  |-  ( _i  .h  B )  e. 
~H
371, 35, 11, 36normlem8 21526 . . . . . . . . 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 21527 . . . . . . . . 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 5722 . . . . . . . 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 21490 . . . . . . . . . 10  |-  ( ( _i  .h  C ) 
.ih  ( _i  .h  B ) )  e.  CC
4125, 40addcli 8721 . . . . . . . . 9  |-  ( ( A  .ih  D )  +  ( ( _i  .h  C )  .ih  ( _i  .h  B
) ) )  e.  CC
421, 36hicli 21490 . . . . . . . . . 10  |-  ( A 
.ih  ( _i  .h  B ) )  e.  CC
4335, 11hicli 21490 . . . . . . . . . 10  |-  ( ( _i  .h  C ) 
.ih  D )  e.  CC
4442, 43addcli 8721 . . . . . . . . 9  |-  ( ( A  .ih  ( _i  .h  B ) )  +  ( ( _i  .h  C )  .ih  D ) )  e.  CC
4541, 44, 44pnncani 9021 . . . . . . . 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 9724 . . . . . . . . 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 21495 . . . . . . . . . . . . 13  |-  ( ( _i  e.  CC  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  ( _i  .h  B ) )  =  ( ( * `  _i )  x.  ( A  .ih  B ) ) )
4834, 1, 2, 47mp3an 1282 . . . . . . . . . . . 12  |-  ( A 
.ih  ( _i  .h  B ) )  =  ( ( * `  _i )  x.  ( A  .ih  B ) )
49 cji 11521 . . . . . . . . . . . . 13  |-  ( * `
 _i )  = 
-u _i
5049oveq1i 5720 . . . . . . . . . . . 12  |-  ( ( * `  _i )  x.  ( A  .ih  B ) )  =  (
-u _i  x.  ( A  .ih  B ) )
5148, 50eqtri 2273 . . . . . . . . . . 11  |-  ( A 
.ih  ( _i  .h  B ) )  =  ( -u _i  x.  ( A  .ih  B ) )
52 ax-his3 21493 . . . . . . . . . . . 12  |-  ( ( _i  e.  CC  /\  C  e.  ~H  /\  D  e.  ~H )  ->  (
( _i  .h  C
)  .ih  D )  =  ( _i  x.  ( C  .ih  D ) ) )
5334, 10, 11, 52mp3an 1282 . . . . . . . . . . 11  |-  ( ( _i  .h  C ) 
.ih  D )  =  ( _i  x.  ( C  .ih  D ) )
5451, 53oveq12i 5722 . . . . . . . . . 10  |-  ( ( A  .ih  ( _i  .h  B ) )  +  ( ( _i  .h  C )  .ih  D ) )  =  ( ( -u _i  x.  ( A  .ih  B ) )  +  ( _i  x.  ( C  .ih  D ) ) )
5554oveq2i 5721 . . . . . . . . 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 2275 . . . . . . . 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 2277 . . . . . . 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 5721 . . . . . 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 8994 . . . . . . . . 9  |-  -u _i  e.  CC
6059, 3mulcli 8722 . . . . . . . 8  |-  ( -u _i  x.  ( A  .ih  B ) )  e.  CC
6134, 12mulcli 8722 . . . . . . . 8  |-  ( _i  x.  ( C  .ih  D ) )  e.  CC
6260, 61addcli 8721 . . . . . . 7  |-  ( (
-u _i  x.  ( A  .ih  B ) )  +  ( _i  x.  ( C  .ih  D ) ) )  e.  CC
639, 34, 62mul12i 8887 . . . . . 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 8727 . . . . . . . 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 9106 . . . . . . . . . . . 12  |-  ( _i  x.  -u _i )  = 
-u ( _i  x.  _i )
66 ixi 9277 . . . . . . . . . . . . 13  |-  ( _i  x.  _i )  = 
-u 1
6766negeqi 8925 . . . . . . . . . . . 12  |-  -u (
_i  x.  _i )  =  -u -u 1
68 ax-1cn 8675 . . . . . . . . . . . . 13  |-  1  e.  CC
6968negnegi 8996 . . . . . . . . . . . 12  |-  -u -u 1  =  1
7065, 67, 693eqtri 2277 . . . . . . . . . . 11  |-  ( _i  x.  -u _i )  =  1
7170oveq1i 5720 . . . . . . . . . 10  |-  ( ( _i  x.  -u _i )  x.  ( A  .ih  B ) )  =  ( 1  x.  ( A  .ih  B ) )
7234, 59, 3mulassi 8726 . . . . . . . . . 10  |-  ( ( _i  x.  -u _i )  x.  ( A  .ih  B ) )  =  ( _i  x.  ( -u _i  x.  ( A 
.ih  B ) ) )
733mulid2i 8720 . . . . . . . . . 10  |-  ( 1  x.  ( A  .ih  B ) )  =  ( A  .ih  B )
7471, 72, 733eqtr3i 2281 . . . . . . . . 9  |-  ( _i  x.  ( -u _i  x.  ( A  .ih  B
) ) )  =  ( A  .ih  B
)
7566oveq1i 5720 . . . . . . . . . 10  |-  ( ( _i  x.  _i )  x.  ( C  .ih  D ) )  =  (
-u 1  x.  ( C  .ih  D ) )
7634, 34, 12mulassi 8726 . . . . . . . . . 10  |-  ( ( _i  x.  _i )  x.  ( C  .ih  D ) )  =  ( _i  x.  ( _i  x.  ( C  .ih  D ) ) )
7712mulm1i 9104 . . . . . . . . . 10  |-  ( -u
1  x.  ( C 
.ih  D ) )  =  -u ( C  .ih  D )
7875, 76, 773eqtr3i 2281 . . . . . . . . 9  |-  ( _i  x.  ( _i  x.  ( C  .ih  D ) ) )  =  -u ( C  .ih  D )
7974, 78oveq12i 5722 . . . . . . . 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 9004 . . . . . . . 8  |-  ( ( A  .ih  B )  +  -u ( C  .ih  D ) )  =  ( ( A  .ih  B
)  -  ( C 
.ih  D ) )
8164, 79, 803eqtri 2277 . . . . . . 7  |-  ( _i  x.  ( ( -u _i  x.  ( A  .ih  B ) )  +  ( _i  x.  ( C 
.ih  D ) ) ) )  =  ( ( A  .ih  B
)  -  ( C 
.ih  D ) )
8281oveq2i 5721 . . . . . 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 2279 . . . . 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 5722 . . . 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 2283 . . 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 5720 . 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 2275 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 1619    e. wcel 1621   ` cfv 4592  (class class class)co 5710   CCcc 8615   1c1 8618   _ici 8619    + caddc 8620    x. cmul 8622    - cmin 8917   -ucneg 8918    / cdiv 9303   2c2 9675   4c4 9677   *ccj 11458   ~Hchil 21329    +h cva 21330    .h csm 21331    .ih csp 21332    -h cmv 21335
This theorem is referenced by:  polidi  21567  lnopeq0lem1  22415  lnophmlem2  22427
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-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-hfvadd 21410  ax-hfvmul 21415  ax-hfi 21488  ax-his1 21491  ax-his2 21492  ax-his3 21493
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-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-po 4207  df-so 4208  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-iota 6143  df-riota 6190  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  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-2 9684  df-3 9685  df-4 9686  df-cj 11461  df-re 11462  df-im 11463  df-hvsub 21381
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