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Theorem polid2i 26052
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 4cn 10620 . 2  |-  4  e.  CC
2 polid2.1 . . 3  |-  A  e. 
~H
3 polid2.2 . . 3  |-  B  e. 
~H
42, 3hicli 25976 . 2  |-  ( A 
.ih  B )  e.  CC
5 4ne0 10639 . 2  |-  4  =/=  0
6 2cn 10613 . . . 4  |-  2  e.  CC
7 polid2.3 . . . . . 6  |-  C  e. 
~H
8 polid2.4 . . . . . 6  |-  D  e. 
~H
97, 8hicli 25976 . . . . 5  |-  ( C 
.ih  D )  e.  CC
104, 9addcli 9603 . . . 4  |-  ( ( A  .ih  B )  +  ( C  .ih  D ) )  e.  CC
114, 9subcli 9900 . . . 4  |-  ( ( A  .ih  B )  -  ( C  .ih  D ) )  e.  CC
126, 10, 11adddii 9609 . . 3  |-  ( 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 ) ) ) )
13 ppncan 9866 . . . . . . 7  |-  ( ( ( 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 ) ) )
144, 9, 4, 13mp3an 1325 . . . . . 6  |-  ( ( ( A  .ih  B
)  +  ( C 
.ih  D ) )  +  ( ( A 
.ih  B )  -  ( C  .ih  D ) ) )  =  ( ( A  .ih  B
)  +  ( A 
.ih  B ) )
1542timesi 10663 . . . . . 6  |-  ( 2  x.  ( A  .ih  B ) )  =  ( ( A  .ih  B
)  +  ( A 
.ih  B ) )
1614, 15eqtr4i 2475 . . . . 5  |-  ( ( ( A  .ih  B
)  +  ( C 
.ih  D ) )  +  ( ( A 
.ih  B )  -  ( C  .ih  D ) ) )  =  ( 2  x.  ( A 
.ih  B ) )
1716oveq2i 6292 . . . 4  |-  ( 2  x.  ( ( ( A  .ih  B )  +  ( C  .ih  D ) )  +  ( ( A  .ih  B
)  -  ( C 
.ih  D ) ) ) )  =  ( 2  x.  ( 2  x.  ( A  .ih  B ) ) )
186, 6, 4mulassi 9608 . . . 4  |-  ( ( 2  x.  2 )  x.  ( A  .ih  B ) )  =  ( 2  x.  ( 2  x.  ( A  .ih  B ) ) )
19 2t2e4 10692 . . . . 5  |-  ( 2  x.  2 )  =  4
2019oveq1i 6291 . . . 4  |-  ( ( 2  x.  2 )  x.  ( A  .ih  B ) )  =  ( 4  x.  ( A 
.ih  B ) )
2117, 18, 203eqtr2ri 2479 . . 3  |-  ( 4  x.  ( A  .ih  B ) )  =  ( 2  x.  ( ( ( A  .ih  B
)  +  ( C 
.ih  D ) )  +  ( ( A 
.ih  B )  -  ( C  .ih  D ) ) ) )
222, 8hicli 25976 . . . . . . 7  |-  ( A 
.ih  D )  e.  CC
237, 3hicli 25976 . . . . . . 7  |-  ( C 
.ih  B )  e.  CC
2422, 23addcli 9603 . . . . . 6  |-  ( ( A  .ih  D )  +  ( C  .ih  B ) )  e.  CC
2524, 10, 10pnncani 9920 . . . . 5  |-  ( ( ( ( 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
) ) )
262, 7, 8, 3normlem8 26012 . . . . . 6  |-  ( ( A  +h  C ) 
.ih  ( D  +h  B ) )  =  ( ( ( A 
.ih  D )  +  ( C  .ih  B
) )  +  ( ( A  .ih  B
)  +  ( C 
.ih  D ) ) )
272, 7, 8, 3normlem9 26013 . . . . . 6  |-  ( ( A  -h  C ) 
.ih  ( D  -h  B ) )  =  ( ( ( A 
.ih  D )  +  ( C  .ih  B
) )  -  (
( A  .ih  B
)  +  ( C 
.ih  D ) ) )
2826, 27oveq12i 6293 . . . . 5  |-  ( ( ( 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 ) ) ) )
29102timesi 10663 . . . . 5  |-  ( 2  x.  ( ( A 
.ih  B )  +  ( C  .ih  D
) ) )  =  ( ( ( A 
.ih  B )  +  ( C  .ih  D
) )  +  ( ( A  .ih  B
)  +  ( C 
.ih  D ) ) )
3025, 28, 293eqtr4i 2482 . . . 4  |-  ( ( ( A  +h  C
)  .ih  ( D  +h  B ) )  -  ( ( A  -h  C )  .ih  ( D  -h  B ) ) )  =  ( 2  x.  ( ( A 
.ih  B )  +  ( C  .ih  D
) ) )
31 ax-icn 9554 . . . . . . . . . 10  |-  _i  e.  CC
3231, 7hvmulcli 25909 . . . . . . . . 9  |-  ( _i  .h  C )  e. 
~H
3331, 3hvmulcli 25909 . . . . . . . . 9  |-  ( _i  .h  B )  e. 
~H
342, 32, 8, 33normlem8 26012 . . . . . . . 8  |-  ( ( 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 ) ) )
352, 32, 8, 33normlem9 26013 . . . . . . . 8  |-  ( ( 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 ) ) )
3634, 35oveq12i 6293 . . . . . . 7  |-  ( ( ( 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 ) ) ) )
3732, 33hicli 25976 . . . . . . . . 9  |-  ( ( _i  .h  C ) 
.ih  ( _i  .h  B ) )  e.  CC
3822, 37addcli 9603 . . . . . . . 8  |-  ( ( A  .ih  D )  +  ( ( _i  .h  C )  .ih  ( _i  .h  B
) ) )  e.  CC
392, 33hicli 25976 . . . . . . . . 9  |-  ( A 
.ih  ( _i  .h  B ) )  e.  CC
4032, 8hicli 25976 . . . . . . . . 9  |-  ( ( _i  .h  C ) 
.ih  D )  e.  CC
4139, 40addcli 9603 . . . . . . . 8  |-  ( ( A  .ih  ( _i  .h  B ) )  +  ( ( _i  .h  C )  .ih  D ) )  e.  CC
4238, 41, 41pnncani 9920 . . . . . . 7  |-  ( ( ( ( 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 ) ) )
43412timesi 10663 . . . . . . . 8  |-  ( 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 ) ) )
44 his5 25981 . . . . . . . . . . . 12  |-  ( ( _i  e.  CC  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  ( _i  .h  B ) )  =  ( ( * `  _i )  x.  ( A  .ih  B ) ) )
4531, 2, 3, 44mp3an 1325 . . . . . . . . . . 11  |-  ( A 
.ih  ( _i  .h  B ) )  =  ( ( * `  _i )  x.  ( A  .ih  B ) )
46 cji 12974 . . . . . . . . . . . 12  |-  ( * `
 _i )  = 
-u _i
4746oveq1i 6291 . . . . . . . . . . 11  |-  ( ( * `  _i )  x.  ( A  .ih  B ) )  =  (
-u _i  x.  ( A  .ih  B ) )
4845, 47eqtri 2472 . . . . . . . . . 10  |-  ( A 
.ih  ( _i  .h  B ) )  =  ( -u _i  x.  ( A  .ih  B ) )
49 ax-his3 25979 . . . . . . . . . . 11  |-  ( ( _i  e.  CC  /\  C  e.  ~H  /\  D  e.  ~H )  ->  (
( _i  .h  C
)  .ih  D )  =  ( _i  x.  ( C  .ih  D ) ) )
5031, 7, 8, 49mp3an 1325 . . . . . . . . . 10  |-  ( ( _i  .h  C ) 
.ih  D )  =  ( _i  x.  ( C  .ih  D ) )
5148, 50oveq12i 6293 . . . . . . . . 9  |-  ( ( A  .ih  ( _i  .h  B ) )  +  ( ( _i  .h  C )  .ih  D ) )  =  ( ( -u _i  x.  ( A  .ih  B ) )  +  ( _i  x.  ( C  .ih  D ) ) )
5251oveq2i 6292 . . . . . . . 8  |-  ( 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 ) ) ) )
5343, 52eqtr3i 2474 . . . . . . 7  |-  ( ( ( 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 ) ) ) )
5436, 42, 533eqtri 2476 . . . . . 6  |-  ( ( ( 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 ) ) ) )
5554oveq2i 6292 . . . . 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 ) ) ) ) )  =  ( _i  x.  ( 2  x.  ( ( -u _i  x.  ( A  .ih  B ) )  +  ( _i  x.  ( C 
.ih  D ) ) ) ) )
56 negicn 9826 . . . . . . . 8  |-  -u _i  e.  CC
5756, 4mulcli 9604 . . . . . . 7  |-  ( -u _i  x.  ( A  .ih  B ) )  e.  CC
5831, 9mulcli 9604 . . . . . . 7  |-  ( _i  x.  ( C  .ih  D ) )  e.  CC
5957, 58addcli 9603 . . . . . 6  |-  ( (
-u _i  x.  ( A  .ih  B ) )  +  ( _i  x.  ( C  .ih  D ) ) )  e.  CC
606, 31, 59mul12i 9778 . . . . 5  |-  ( 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 ) ) ) ) )
6131, 57, 58adddii 9609 . . . . . . 7  |-  ( _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 ) ) ) )
6231, 31mulneg2i 10010 . . . . . . . . . . 11  |-  ( _i  x.  -u _i )  = 
-u ( _i  x.  _i )
63 ixi 10185 . . . . . . . . . . . 12  |-  ( _i  x.  _i )  = 
-u 1
6463negeqi 9818 . . . . . . . . . . 11  |-  -u (
_i  x.  _i )  =  -u -u 1
65 negneg1e1 10650 . . . . . . . . . . 11  |-  -u -u 1  =  1
6662, 64, 653eqtri 2476 . . . . . . . . . 10  |-  ( _i  x.  -u _i )  =  1
6766oveq1i 6291 . . . . . . . . 9  |-  ( ( _i  x.  -u _i )  x.  ( A  .ih  B ) )  =  ( 1  x.  ( A  .ih  B ) )
6831, 56, 4mulassi 9608 . . . . . . . . 9  |-  ( ( _i  x.  -u _i )  x.  ( A  .ih  B ) )  =  ( _i  x.  ( -u _i  x.  ( A 
.ih  B ) ) )
694mulid2i 9602 . . . . . . . . 9  |-  ( 1  x.  ( A  .ih  B ) )  =  ( A  .ih  B )
7067, 68, 693eqtr3i 2480 . . . . . . . 8  |-  ( _i  x.  ( -u _i  x.  ( A  .ih  B
) ) )  =  ( A  .ih  B
)
7163oveq1i 6291 . . . . . . . . 9  |-  ( ( _i  x.  _i )  x.  ( C  .ih  D ) )  =  (
-u 1  x.  ( C  .ih  D ) )
7231, 31, 9mulassi 9608 . . . . . . . . 9  |-  ( ( _i  x.  _i )  x.  ( C  .ih  D ) )  =  ( _i  x.  ( _i  x.  ( C  .ih  D ) ) )
739mulm1i 10008 . . . . . . . . 9  |-  ( -u
1  x.  ( C 
.ih  D ) )  =  -u ( C  .ih  D )
7471, 72, 733eqtr3i 2480 . . . . . . . 8  |-  ( _i  x.  ( _i  x.  ( C  .ih  D ) ) )  =  -u ( C  .ih  D )
7570, 74oveq12i 6293 . . . . . . 7  |-  ( ( _i  x.  ( -u _i  x.  ( A  .ih  B ) ) )  +  ( _i  x.  (
_i  x.  ( C  .ih  D ) ) ) )  =  ( ( A  .ih  B )  +  -u ( C  .ih  D ) )
764, 9negsubi 9902 . . . . . . 7  |-  ( ( A  .ih  B )  +  -u ( C  .ih  D ) )  =  ( ( A  .ih  B
)  -  ( C 
.ih  D ) )
7761, 75, 763eqtri 2476 . . . . . 6  |-  ( _i  x.  ( ( -u _i  x.  ( A  .ih  B ) )  +  ( _i  x.  ( C 
.ih  D ) ) ) )  =  ( ( A  .ih  B
)  -  ( C 
.ih  D ) )
7877oveq2i 6292 . . . . 5  |-  ( 2  x.  ( _i  x.  ( ( -u _i  x.  ( A  .ih  B
) )  +  ( _i  x.  ( C 
.ih  D ) ) ) ) )  =  ( 2  x.  (
( A  .ih  B
)  -  ( C 
.ih  D ) ) )
7955, 60, 783eqtr2i 2478 . . . 4  |-  ( _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 ) ) )
8030, 79oveq12i 6293 . . 3  |-  ( ( ( ( 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 ) ) ) )
8112, 21, 803eqtr4i 2482 . 2  |-  ( 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
) ) ) ) ) )
821, 4, 5, 81mvllmuli 10384 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 setvar class
Syntax hints:    = wceq 1383    e. wcel 1804   ` cfv 5578  (class class class)co 6281   CCcc 9493   1c1 9496   _ici 9497    + caddc 9498    x. cmul 9500    - cmin 9810   -ucneg 9811    / cdiv 10213   2c2 10592   4c4 10594   *ccj 12911   ~Hchil 25814    +h cva 25815    .h csm 25816    .ih csp 25817    -h cmv 25820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-hfvadd 25895  ax-hfvmul 25900  ax-hfi 25974  ax-his1 25977  ax-his2 25978  ax-his3 25979
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-po 4790  df-so 4791  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-2 10601  df-3 10602  df-4 10603  df-cj 12914  df-re 12915  df-im 12916  df-hvsub 25866
This theorem is referenced by:  polidi  26053  lnopeq0lem1  26902  lnophmlem2  26914
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