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Theorem polid2i 26212
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 10548 . 2  |-  4  e.  CC
2 polid2.1 . . 3  |-  A  e. 
~H
3 polid2.2 . . 3  |-  B  e. 
~H
42, 3hicli 26136 . 2  |-  ( A 
.ih  B )  e.  CC
5 4ne0 10567 . 2  |-  4  =/=  0
6 2cn 10541 . . . 4  |-  2  e.  CC
7 polid2.3 . . . . . 6  |-  C  e. 
~H
8 polid2.4 . . . . . 6  |-  D  e. 
~H
97, 8hicli 26136 . . . . 5  |-  ( C 
.ih  D )  e.  CC
104, 9addcli 9529 . . . 4  |-  ( ( A  .ih  B )  +  ( C  .ih  D ) )  e.  CC
114, 9subcli 9826 . . . 4  |-  ( ( A  .ih  B )  -  ( C  .ih  D ) )  e.  CC
126, 10, 11adddii 9535 . . 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 9792 . . . . . . 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 1322 . . . . . 6  |-  ( ( ( A  .ih  B
)  +  ( C 
.ih  D ) )  +  ( ( A 
.ih  B )  -  ( C  .ih  D ) ) )  =  ( ( A  .ih  B
)  +  ( A 
.ih  B ) )
1542timesi 10591 . . . . . 6  |-  ( 2  x.  ( A  .ih  B ) )  =  ( ( A  .ih  B
)  +  ( A 
.ih  B ) )
1614, 15eqtr4i 2424 . . . . 5  |-  ( ( ( A  .ih  B
)  +  ( C 
.ih  D ) )  +  ( ( A 
.ih  B )  -  ( C  .ih  D ) ) )  =  ( 2  x.  ( A 
.ih  B ) )
1716oveq2i 6225 . . . 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 9534 . . . 4  |-  ( ( 2  x.  2 )  x.  ( A  .ih  B ) )  =  ( 2  x.  ( 2  x.  ( A  .ih  B ) ) )
19 2t2e4 10620 . . . . 5  |-  ( 2  x.  2 )  =  4
2019oveq1i 6224 . . . 4  |-  ( ( 2  x.  2 )  x.  ( A  .ih  B ) )  =  ( 4  x.  ( A 
.ih  B ) )
2117, 18, 203eqtr2ri 2428 . . 3  |-  ( 4  x.  ( A  .ih  B ) )  =  ( 2  x.  ( ( ( A  .ih  B
)  +  ( C 
.ih  D ) )  +  ( ( A 
.ih  B )  -  ( C  .ih  D ) ) ) )
222, 8hicli 26136 . . . . . . 7  |-  ( A 
.ih  D )  e.  CC
237, 3hicli 26136 . . . . . . 7  |-  ( C 
.ih  B )  e.  CC
2422, 23addcli 9529 . . . . . 6  |-  ( ( A  .ih  D )  +  ( C  .ih  B ) )  e.  CC
2524, 10, 10pnncani 9846 . . . . 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 26172 . . . . . 6  |-  ( ( A  +h  C ) 
.ih  ( D  +h  B ) )  =  ( ( ( A 
.ih  D )  +  ( C  .ih  B
) )  +  ( ( A  .ih  B
)  +  ( C 
.ih  D ) ) )
272, 7, 8, 3normlem9 26173 . . . . . 6  |-  ( ( A  -h  C ) 
.ih  ( D  -h  B ) )  =  ( ( ( A 
.ih  D )  +  ( C  .ih  B
) )  -  (
( A  .ih  B
)  +  ( C 
.ih  D ) ) )
2826, 27oveq12i 6226 . . . . 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 10591 . . . . 5  |-  ( 2  x.  ( ( A 
.ih  B )  +  ( C  .ih  D
) ) )  =  ( ( ( A 
.ih  B )  +  ( C  .ih  D
) )  +  ( ( A  .ih  B
)  +  ( C 
.ih  D ) ) )
3025, 28, 293eqtr4i 2431 . . . 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 9480 . . . . . . . . . 10  |-  _i  e.  CC
3231, 7hvmulcli 26069 . . . . . . . . 9  |-  ( _i  .h  C )  e. 
~H
3331, 3hvmulcli 26069 . . . . . . . . 9  |-  ( _i  .h  B )  e. 
~H
342, 32, 8, 33normlem8 26172 . . . . . . . 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 26173 . . . . . . . 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 6226 . . . . . . 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 26136 . . . . . . . . 9  |-  ( ( _i  .h  C ) 
.ih  ( _i  .h  B ) )  e.  CC
3822, 37addcli 9529 . . . . . . . 8  |-  ( ( A  .ih  D )  +  ( ( _i  .h  C )  .ih  ( _i  .h  B
) ) )  e.  CC
392, 33hicli 26136 . . . . . . . . 9  |-  ( A 
.ih  ( _i  .h  B ) )  e.  CC
4032, 8hicli 26136 . . . . . . . . 9  |-  ( ( _i  .h  C ) 
.ih  D )  e.  CC
4139, 40addcli 9529 . . . . . . . 8  |-  ( ( A  .ih  ( _i  .h  B ) )  +  ( ( _i  .h  C )  .ih  D ) )  e.  CC
4238, 41, 41pnncani 9846 . . . . . . 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 10591 . . . . . . . 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 26141 . . . . . . . . . . . 12  |-  ( ( _i  e.  CC  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  ( _i  .h  B ) )  =  ( ( * `  _i )  x.  ( A  .ih  B ) ) )
4531, 2, 3, 44mp3an 1322 . . . . . . . . . . 11  |-  ( A 
.ih  ( _i  .h  B ) )  =  ( ( * `  _i )  x.  ( A  .ih  B ) )
46 cji 13013 . . . . . . . . . . . 12  |-  ( * `
 _i )  = 
-u _i
4746oveq1i 6224 . . . . . . . . . . 11  |-  ( ( * `  _i )  x.  ( A  .ih  B ) )  =  (
-u _i  x.  ( A  .ih  B ) )
4845, 47eqtri 2421 . . . . . . . . . 10  |-  ( A 
.ih  ( _i  .h  B ) )  =  ( -u _i  x.  ( A  .ih  B ) )
49 ax-his3 26139 . . . . . . . . . . 11  |-  ( ( _i  e.  CC  /\  C  e.  ~H  /\  D  e.  ~H )  ->  (
( _i  .h  C
)  .ih  D )  =  ( _i  x.  ( C  .ih  D ) ) )
5031, 7, 8, 49mp3an 1322 . . . . . . . . . 10  |-  ( ( _i  .h  C ) 
.ih  D )  =  ( _i  x.  ( C  .ih  D ) )
5148, 50oveq12i 6226 . . . . . . . . 9  |-  ( ( A  .ih  ( _i  .h  B ) )  +  ( ( _i  .h  C )  .ih  D ) )  =  ( ( -u _i  x.  ( A  .ih  B ) )  +  ( _i  x.  ( C  .ih  D ) ) )
5251oveq2i 6225 . . . . . . . 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 2423 . . . . . . 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 2425 . . . . . 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 6225 . . . . 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 9752 . . . . . . . 8  |-  -u _i  e.  CC
5756, 4mulcli 9530 . . . . . . 7  |-  ( -u _i  x.  ( A  .ih  B ) )  e.  CC
5831, 9mulcli 9530 . . . . . . 7  |-  ( _i  x.  ( C  .ih  D ) )  e.  CC
5957, 58addcli 9529 . . . . . 6  |-  ( (
-u _i  x.  ( A  .ih  B ) )  +  ( _i  x.  ( C  .ih  D ) ) )  e.  CC
606, 31, 59mul12i 9704 . . . . 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 9535 . . . . . . 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 9939 . . . . . . . . . . 11  |-  ( _i  x.  -u _i )  = 
-u ( _i  x.  _i )
63 ixi 10113 . . . . . . . . . . . 12  |-  ( _i  x.  _i )  = 
-u 1
6463negeqi 9744 . . . . . . . . . . 11  |-  -u (
_i  x.  _i )  =  -u -u 1
65 negneg1e1 10578 . . . . . . . . . . 11  |-  -u -u 1  =  1
6662, 64, 653eqtri 2425 . . . . . . . . . 10  |-  ( _i  x.  -u _i )  =  1
6766oveq1i 6224 . . . . . . . . 9  |-  ( ( _i  x.  -u _i )  x.  ( A  .ih  B ) )  =  ( 1  x.  ( A  .ih  B ) )
6831, 56, 4mulassi 9534 . . . . . . . . 9  |-  ( ( _i  x.  -u _i )  x.  ( A  .ih  B ) )  =  ( _i  x.  ( -u _i  x.  ( A 
.ih  B ) ) )
694mulid2i 9528 . . . . . . . . 9  |-  ( 1  x.  ( A  .ih  B ) )  =  ( A  .ih  B )
7067, 68, 693eqtr3i 2429 . . . . . . . 8  |-  ( _i  x.  ( -u _i  x.  ( A  .ih  B
) ) )  =  ( A  .ih  B
)
7163oveq1i 6224 . . . . . . . . 9  |-  ( ( _i  x.  _i )  x.  ( C  .ih  D ) )  =  (
-u 1  x.  ( C  .ih  D ) )
7231, 31, 9mulassi 9534 . . . . . . . . 9  |-  ( ( _i  x.  _i )  x.  ( C  .ih  D ) )  =  ( _i  x.  ( _i  x.  ( C  .ih  D ) ) )
739mulm1i 9937 . . . . . . . . 9  |-  ( -u
1  x.  ( C 
.ih  D ) )  =  -u ( C  .ih  D )
7471, 72, 733eqtr3i 2429 . . . . . . . 8  |-  ( _i  x.  ( _i  x.  ( C  .ih  D ) ) )  =  -u ( C  .ih  D )
7570, 74oveq12i 6226 . . . . . . 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 9828 . . . . . . 7  |-  ( ( A  .ih  B )  +  -u ( C  .ih  D ) )  =  ( ( A  .ih  B
)  -  ( C 
.ih  D ) )
7761, 75, 763eqtri 2425 . . . . . 6  |-  ( _i  x.  ( ( -u _i  x.  ( A  .ih  B ) )  +  ( _i  x.  ( C 
.ih  D ) ) ) )  =  ( ( A  .ih  B
)  -  ( C 
.ih  D ) )
7877oveq2i 6225 . . . . 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 2427 . . . 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 6226 . . 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 2431 . 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 10312 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 1399    e. wcel 1836   ` cfv 5509  (class class class)co 6214   CCcc 9419   1c1 9422   _ici 9423    + caddc 9424    x. cmul 9426    - cmin 9736   -ucneg 9737    / cdiv 10141   2c2 10520   4c4 10522   *ccj 12950   ~Hchil 25974    +h cva 25975    .h csm 25976    .ih csp 25977    -h cmv 25980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370  ax-sep 4501  ax-nul 4509  ax-pow 4556  ax-pr 4614  ax-un 6509  ax-resscn 9478  ax-1cn 9479  ax-icn 9480  ax-addcl 9481  ax-addrcl 9482  ax-mulcl 9483  ax-mulrcl 9484  ax-mulcom 9485  ax-addass 9486  ax-mulass 9487  ax-distr 9488  ax-i2m1 9489  ax-1ne0 9490  ax-1rid 9491  ax-rnegex 9492  ax-rrecex 9493  ax-cnre 9494  ax-pre-lttri 9495  ax-pre-lttrn 9496  ax-pre-ltadd 9497  ax-pre-mulgt0 9498  ax-hfvadd 26055  ax-hfvmul 26060  ax-hfi 26134  ax-his1 26137  ax-his2 26138  ax-his3 26139
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-nel 2590  df-ral 2747  df-rex 2748  df-reu 2749  df-rmo 2750  df-rab 2751  df-v 3049  df-sbc 3266  df-csb 3362  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-nul 3725  df-if 3871  df-pw 3942  df-sn 3958  df-pr 3960  df-op 3964  df-uni 4177  df-iun 4258  df-br 4381  df-opab 4439  df-mpt 4440  df-id 4722  df-po 4727  df-so 4728  df-xp 4932  df-rel 4933  df-cnv 4934  df-co 4935  df-dm 4936  df-rn 4937  df-res 4938  df-ima 4939  df-iota 5473  df-fun 5511  df-fn 5512  df-f 5513  df-f1 5514  df-fo 5515  df-f1o 5516  df-fv 5517  df-riota 6176  df-ov 6217  df-oprab 6218  df-mpt2 6219  df-er 7247  df-en 7454  df-dom 7455  df-sdom 7456  df-pnf 9559  df-mnf 9560  df-xr 9561  df-ltxr 9562  df-le 9563  df-sub 9738  df-neg 9739  df-div 10142  df-2 10529  df-3 10530  df-4 10531  df-cj 12953  df-re 12954  df-im 12955  df-hvsub 26026
This theorem is referenced by:  polidi  26213  lnopeq0lem1  27061  lnophmlem2  27073
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