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Theorem polid2i 21736
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 21660 . . 3  |-  ( A 
.ih  B )  e.  CC
4 4cn 9820 . . 3  |-  4  e.  CC
5 4re 9819 . . . 4  |-  4  e.  RR
6 4pos 9832 . . . 4  |-  0  <  4
75, 6gt0ne0ii 9309 . . 3  |-  4  =/=  0
83, 4, 7divcan3i 9506 . 2  |-  ( ( 4  x.  ( A 
.ih  B ) )  /  4 )  =  ( A  .ih  B
)
9 2cn 9816 . . . . 5  |-  2  e.  CC
10 polid2.3 . . . . . . 7  |-  C  e. 
~H
11 polid2.4 . . . . . . 7  |-  D  e. 
~H
1210, 11hicli 21660 . . . . . 6  |-  ( C 
.ih  D )  e.  CC
133, 12addcli 8841 . . . . 5  |-  ( ( A  .ih  B )  +  ( C  .ih  D ) )  e.  CC
143, 12subcli 9122 . . . . 5  |-  ( ( A  .ih  B )  -  ( C  .ih  D ) )  e.  CC
159, 13, 14adddii 8847 . . . 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 9089 . . . . . . . 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 9845 . . . . . . 7  |-  ( 2  x.  ( A  .ih  B ) )  =  ( ( A  .ih  B
)  +  ( A 
.ih  B ) )
1917, 18eqtr4i 2306 . . . . . 6  |-  ( ( ( A  .ih  B
)  +  ( C 
.ih  D ) )  +  ( ( A 
.ih  B )  -  ( C  .ih  D ) ) )  =  ( 2  x.  ( A 
.ih  B ) )
2019oveq2i 5869 . . . . 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 8846 . . . . 5  |-  ( ( 2  x.  2 )  x.  ( A  .ih  B ) )  =  ( 2  x.  ( 2  x.  ( A  .ih  B ) ) )
22 2t2e4 9871 . . . . . 6  |-  ( 2  x.  2 )  =  4
2322oveq1i 5868 . . . . 5  |-  ( ( 2  x.  2 )  x.  ( A  .ih  B ) )  =  ( 4  x.  ( A 
.ih  B ) )
2420, 21, 233eqtr2ri 2310 . . . 4  |-  ( 4  x.  ( A  .ih  B ) )  =  ( 2  x.  ( ( ( A  .ih  B
)  +  ( C 
.ih  D ) )  +  ( ( A 
.ih  B )  -  ( C  .ih  D ) ) ) )
251, 11hicli 21660 . . . . . . . 8  |-  ( A 
.ih  D )  e.  CC
2610, 2hicli 21660 . . . . . . . 8  |-  ( C 
.ih  B )  e.  CC
2725, 26addcli 8841 . . . . . . 7  |-  ( ( A  .ih  D )  +  ( C  .ih  B ) )  e.  CC
2827, 13, 13pnncani 9141 . . . . . 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 21696 . . . . . . 7  |-  ( ( A  +h  C ) 
.ih  ( D  +h  B ) )  =  ( ( ( A 
.ih  D )  +  ( C  .ih  B
) )  +  ( ( A  .ih  B
)  +  ( C 
.ih  D ) ) )
301, 10, 11, 2normlem9 21697 . . . . . . 7  |-  ( ( A  -h  C ) 
.ih  ( D  -h  B ) )  =  ( ( ( A 
.ih  D )  +  ( C  .ih  B
) )  -  (
( A  .ih  B
)  +  ( C 
.ih  D ) ) )
3129, 30oveq12i 5870 . . . . . 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 9845 . . . . . 6  |-  ( 2  x.  ( ( A 
.ih  B )  +  ( C  .ih  D
) ) )  =  ( ( ( A 
.ih  B )  +  ( C  .ih  D
) )  +  ( ( A  .ih  B
)  +  ( C 
.ih  D ) ) )
3328, 31, 323eqtr4i 2313 . . . . 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 8796 . . . . . . . . . . 11  |-  _i  e.  CC
3534, 10hvmulcli 21594 . . . . . . . . . 10  |-  ( _i  .h  C )  e. 
~H
3634, 2hvmulcli 21594 . . . . . . . . . 10  |-  ( _i  .h  B )  e. 
~H
371, 35, 11, 36normlem8 21696 . . . . . . . . 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 21697 . . . . . . . . 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 5870 . . . . . . . 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 21660 . . . . . . . . . 10  |-  ( ( _i  .h  C ) 
.ih  ( _i  .h  B ) )  e.  CC
4125, 40addcli 8841 . . . . . . . . 9  |-  ( ( A  .ih  D )  +  ( ( _i  .h  C )  .ih  ( _i  .h  B
) ) )  e.  CC
421, 36hicli 21660 . . . . . . . . . 10  |-  ( A 
.ih  ( _i  .h  B ) )  e.  CC
4335, 11hicli 21660 . . . . . . . . . 10  |-  ( ( _i  .h  C ) 
.ih  D )  e.  CC
4442, 43addcli 8841 . . . . . . . . 9  |-  ( ( A  .ih  ( _i  .h  B ) )  +  ( ( _i  .h  C )  .ih  D ) )  e.  CC
4541, 44, 44pnncani 9141 . . . . . . . 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 9845 . . . . . . . . 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 21665 . . . . . . . . . . . . 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 11644 . . . . . . . . . . . . 13  |-  ( * `
 _i )  = 
-u _i
5049oveq1i 5868 . . . . . . . . . . . 12  |-  ( ( * `  _i )  x.  ( A  .ih  B ) )  =  (
-u _i  x.  ( A  .ih  B ) )
5148, 50eqtri 2303 . . . . . . . . . . 11  |-  ( A 
.ih  ( _i  .h  B ) )  =  ( -u _i  x.  ( A  .ih  B ) )
52 ax-his3 21663 . . . . . . . . . . . 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 5870 . . . . . . . . . 10  |-  ( ( A  .ih  ( _i  .h  B ) )  +  ( ( _i  .h  C )  .ih  D ) )  =  ( ( -u _i  x.  ( A  .ih  B ) )  +  ( _i  x.  ( C  .ih  D ) ) )
5554oveq2i 5869 . . . . . . . . 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 2305 . . . . . . . 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 2307 . . . . . . 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 5869 . . . . . 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 9114 . . . . . . . . 9  |-  -u _i  e.  CC
6059, 3mulcli 8842 . . . . . . . 8  |-  ( -u _i  x.  ( A  .ih  B ) )  e.  CC
6134, 12mulcli 8842 . . . . . . . 8  |-  ( _i  x.  ( C  .ih  D ) )  e.  CC
6260, 61addcli 8841 . . . . . . 7  |-  ( (
-u _i  x.  ( A  .ih  B ) )  +  ( _i  x.  ( C  .ih  D ) ) )  e.  CC
639, 34, 62mul12i 9007 . . . . . 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 8847 . . . . . . . 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 9226 . . . . . . . . . . . 12  |-  ( _i  x.  -u _i )  = 
-u ( _i  x.  _i )
66 ixi 9397 . . . . . . . . . . . . 13  |-  ( _i  x.  _i )  = 
-u 1
6766negeqi 9045 . . . . . . . . . . . 12  |-  -u (
_i  x.  _i )  =  -u -u 1
68 ax-1cn 8795 . . . . . . . . . . . . 13  |-  1  e.  CC
6968negnegi 9116 . . . . . . . . . . . 12  |-  -u -u 1  =  1
7065, 67, 693eqtri 2307 . . . . . . . . . . 11  |-  ( _i  x.  -u _i )  =  1
7170oveq1i 5868 . . . . . . . . . 10  |-  ( ( _i  x.  -u _i )  x.  ( A  .ih  B ) )  =  ( 1  x.  ( A  .ih  B ) )
7234, 59, 3mulassi 8846 . . . . . . . . . 10  |-  ( ( _i  x.  -u _i )  x.  ( A  .ih  B ) )  =  ( _i  x.  ( -u _i  x.  ( A 
.ih  B ) ) )
733mulid2i 8840 . . . . . . . . . 10  |-  ( 1  x.  ( A  .ih  B ) )  =  ( A  .ih  B )
7471, 72, 733eqtr3i 2311 . . . . . . . . 9  |-  ( _i  x.  ( -u _i  x.  ( A  .ih  B
) ) )  =  ( A  .ih  B
)
7566oveq1i 5868 . . . . . . . . . 10  |-  ( ( _i  x.  _i )  x.  ( C  .ih  D ) )  =  (
-u 1  x.  ( C  .ih  D ) )
7634, 34, 12mulassi 8846 . . . . . . . . . 10  |-  ( ( _i  x.  _i )  x.  ( C  .ih  D ) )  =  ( _i  x.  ( _i  x.  ( C  .ih  D ) ) )
7712mulm1i 9224 . . . . . . . . . 10  |-  ( -u
1  x.  ( C 
.ih  D ) )  =  -u ( C  .ih  D )
7875, 76, 773eqtr3i 2311 . . . . . . . . 9  |-  ( _i  x.  ( _i  x.  ( C  .ih  D ) ) )  =  -u ( C  .ih  D )
7974, 78oveq12i 5870 . . . . . . . 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 9124 . . . . . . . 8  |-  ( ( A  .ih  B )  +  -u ( C  .ih  D ) )  =  ( ( A  .ih  B
)  -  ( C 
.ih  D ) )
8164, 79, 803eqtri 2307 . . . . . . 7  |-  ( _i  x.  ( ( -u _i  x.  ( A  .ih  B ) )  +  ( _i  x.  ( C 
.ih  D ) ) ) )  =  ( ( A  .ih  B
)  -  ( C 
.ih  D ) )
8281oveq2i 5869 . . . . . 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 2309 . . . . 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 5870 . . . 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 2313 . . 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 5868 . 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 2305 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 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   CCcc 8735   1c1 8738   _ici 8739    + caddc 8740    x. cmul 8742    - cmin 9037   -ucneg 9038    / cdiv 9423   2c2 9795   4c4 9797   *ccj 11581   ~Hchil 21499    +h cva 21500    .h csm 21501    .ih csp 21502    -h cmv 21505
This theorem is referenced by:  polidi  21737  lnopeq0lem1  22585  lnophmlem2  22597
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-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  ax-hfvadd 21580  ax-hfvmul 21585  ax-hfi 21658  ax-his1 21661  ax-his2 21662  ax-his3 21663
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-rmo 2551  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-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  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-riota 6304  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-div 9424  df-2 9804  df-3 9805  df-4 9806  df-cj 11584  df-re 11585  df-im 11586  df-hvsub 21551
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