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Theorem sharhght 27833
Description: Let  A B C be a triangle, and let  D lie on the line  A B. Then (doubled) areas of triangles  A D C and  C D B relate as lengths of corresponding bases  A D and  D B. (Contributed by Saveliy Skresanov, 23-Sep-2017.)
Hypotheses
Ref Expression
sharhght.sigar  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
sharhght.a  |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
)
sharhght.b  |-  ( ph  ->  ( D  e.  CC  /\  ( ( A  -  D ) G ( B  -  D ) )  =  0 ) )
Assertion
Ref Expression
sharhght  |-  ( ph  ->  ( ( ( C  -  A ) G ( D  -  A
) )  x.  ( B  -  D )
)  =  ( ( ( C  -  B
) G ( D  -  B ) )  x.  ( A  -  D ) ) )
Distinct variable groups:    x, y, A    x, B, y    x, C, y    x, D, y
Allowed substitution hints:    ph( x, y)    G( x, y)

Proof of Theorem sharhght
StepHypRef Expression
1 sharhght.a . . . . . . . . 9  |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
)
21simp3d 972 . . . . . . . 8  |-  ( ph  ->  C  e.  CC )
31simp1d 970 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
42, 3subcld 9413 . . . . . . 7  |-  ( ph  ->  ( C  -  A
)  e.  CC )
54adantr 453 . . . . . 6  |-  ( (
ph  /\  B  =  D )  ->  ( C  -  A )  e.  CC )
6 sharhght.b . . . . . . . . 9  |-  ( ph  ->  ( D  e.  CC  /\  ( ( A  -  D ) G ( B  -  D ) )  =  0 ) )
76simpld 447 . . . . . . . 8  |-  ( ph  ->  D  e.  CC )
87, 3subcld 9413 . . . . . . 7  |-  ( ph  ->  ( D  -  A
)  e.  CC )
98adantr 453 . . . . . 6  |-  ( (
ph  /\  B  =  D )  ->  ( D  -  A )  e.  CC )
10 sharhght.sigar . . . . . . 7  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
1110sigarim 27819 . . . . . 6  |-  ( ( ( C  -  A
)  e.  CC  /\  ( D  -  A
)  e.  CC )  ->  ( ( C  -  A ) G ( D  -  A
) )  e.  RR )
125, 9, 11syl2anc 644 . . . . 5  |-  ( (
ph  /\  B  =  D )  ->  (
( C  -  A
) G ( D  -  A ) )  e.  RR )
1312recnd 9116 . . . 4  |-  ( (
ph  /\  B  =  D )  ->  (
( C  -  A
) G ( D  -  A ) )  e.  CC )
1413mul01d 9267 . . 3  |-  ( (
ph  /\  B  =  D )  ->  (
( ( C  -  A ) G ( D  -  A ) )  x.  0 )  =  0 )
151simp2d 971 . . . . . 6  |-  ( ph  ->  B  e.  CC )
1615adantr 453 . . . . 5  |-  ( (
ph  /\  B  =  D )  ->  B  e.  CC )
17 simpr 449 . . . . 5  |-  ( (
ph  /\  B  =  D )  ->  B  =  D )
1816, 17subeq0bd 9465 . . . 4  |-  ( (
ph  /\  B  =  D )  ->  ( B  -  D )  =  0 )
1918oveq2d 6099 . . 3  |-  ( (
ph  /\  B  =  D )  ->  (
( ( C  -  A ) G ( D  -  A ) )  x.  ( B  -  D ) )  =  ( ( ( C  -  A ) G ( D  -  A ) )  x.  0 ) )
202, 15subcld 9413 . . . . . . . 8  |-  ( ph  ->  ( C  -  B
)  e.  CC )
2120adantr 453 . . . . . . 7  |-  ( (
ph  /\  B  =  D )  ->  ( C  -  B )  e.  CC )
227, 15subcld 9413 . . . . . . . 8  |-  ( ph  ->  ( D  -  B
)  e.  CC )
2322adantr 453 . . . . . . 7  |-  ( (
ph  /\  B  =  D )  ->  ( D  -  B )  e.  CC )
2410sigarval 27818 . . . . . . 7  |-  ( ( ( C  -  B
)  e.  CC  /\  ( D  -  B
)  e.  CC )  ->  ( ( C  -  B ) G ( D  -  B
) )  =  ( Im `  ( ( * `  ( C  -  B ) )  x.  ( D  -  B ) ) ) )
2521, 23, 24syl2anc 644 . . . . . 6  |-  ( (
ph  /\  B  =  D )  ->  (
( C  -  B
) G ( D  -  B ) )  =  ( Im `  ( ( * `  ( C  -  B
) )  x.  ( D  -  B )
) ) )
267adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  B  =  D )  ->  D  e.  CC )
2717eqcomd 2443 . . . . . . . . . 10  |-  ( (
ph  /\  B  =  D )  ->  D  =  B )
2826, 27subeq0bd 9465 . . . . . . . . 9  |-  ( (
ph  /\  B  =  D )  ->  ( D  -  B )  =  0 )
2928oveq2d 6099 . . . . . . . 8  |-  ( (
ph  /\  B  =  D )  ->  (
( * `  ( C  -  B )
)  x.  ( D  -  B ) )  =  ( ( * `
 ( C  -  B ) )  x.  0 ) )
3021cjcld 12003 . . . . . . . . 9  |-  ( (
ph  /\  B  =  D )  ->  (
* `  ( C  -  B ) )  e.  CC )
3130mul01d 9267 . . . . . . . 8  |-  ( (
ph  /\  B  =  D )  ->  (
( * `  ( C  -  B )
)  x.  0 )  =  0 )
3229, 31eqtrd 2470 . . . . . . 7  |-  ( (
ph  /\  B  =  D )  ->  (
( * `  ( C  -  B )
)  x.  ( D  -  B ) )  =  0 )
3332fveq2d 5734 . . . . . 6  |-  ( (
ph  /\  B  =  D )  ->  (
Im `  ( (
* `  ( C  -  B ) )  x.  ( D  -  B
) ) )  =  ( Im `  0
) )
34 0re 9093 . . . . . . . 8  |-  0  e.  RR
3534a1i 11 . . . . . . 7  |-  ( (
ph  /\  B  =  D )  ->  0  e.  RR )
3635reim0d 12032 . . . . . 6  |-  ( (
ph  /\  B  =  D )  ->  (
Im `  0 )  =  0 )
3725, 33, 363eqtrd 2474 . . . . 5  |-  ( (
ph  /\  B  =  D )  ->  (
( C  -  B
) G ( D  -  B ) )  =  0 )
3837oveq1d 6098 . . . 4  |-  ( (
ph  /\  B  =  D )  ->  (
( ( C  -  B ) G ( D  -  B ) )  x.  ( A  -  D ) )  =  ( 0  x.  ( A  -  D
) ) )
393adantr 453 . . . . . 6  |-  ( (
ph  /\  B  =  D )  ->  A  e.  CC )
4039, 26subcld 9413 . . . . 5  |-  ( (
ph  /\  B  =  D )  ->  ( A  -  D )  e.  CC )
4140mul02d 9266 . . . 4  |-  ( (
ph  /\  B  =  D )  ->  (
0  x.  ( A  -  D ) )  =  0 )
4238, 41eqtrd 2470 . . 3  |-  ( (
ph  /\  B  =  D )  ->  (
( ( C  -  B ) G ( D  -  B ) )  x.  ( A  -  D ) )  =  0 )
4314, 19, 423eqtr4d 2480 . 2  |-  ( (
ph  /\  B  =  D )  ->  (
( ( C  -  A ) G ( D  -  A ) )  x.  ( B  -  D ) )  =  ( ( ( C  -  B ) G ( D  -  B ) )  x.  ( A  -  D
) ) )
442adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  -.  B  =  D )  ->  C  e.  CC )
4515adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  -.  B  =  D )  ->  B  e.  CC )
463adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  -.  B  =  D )  ->  A  e.  CC )
4744, 45, 46npncand 9437 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  B
)  +  ( B  -  A ) )  =  ( C  -  A ) )
4847oveq1d 6098 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( C  -  B )  +  ( B  -  A ) ) G ( D  -  A ) )  =  ( ( C  -  A ) G ( D  -  A
) ) )
4944, 45subcld 9413 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  ( C  -  B )  e.  CC )
508adantr 453 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  ( D  -  A )  e.  CC )
5145, 46subcld 9413 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  ( B  -  A )  e.  CC )
5210sigaraf 27821 . . . . . . . 8  |-  ( ( ( C  -  B
)  e.  CC  /\  ( D  -  A
)  e.  CC  /\  ( B  -  A
)  e.  CC )  ->  ( ( ( C  -  B )  +  ( B  -  A ) ) G ( D  -  A
) )  =  ( ( ( C  -  B ) G ( D  -  A ) )  +  ( ( B  -  A ) G ( D  -  A ) ) ) )
5349, 50, 51, 52syl3anc 1185 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( C  -  B )  +  ( B  -  A ) ) G ( D  -  A ) )  =  ( ( ( C  -  B ) G ( D  -  A ) )  +  ( ( B  -  A ) G ( D  -  A ) ) ) )
5448, 53eqtr3d 2472 . . . . . 6  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  A
) G ( D  -  A ) )  =  ( ( ( C  -  B ) G ( D  -  A ) )  +  ( ( B  -  A ) G ( D  -  A ) ) ) )
556simprd 451 . . . . . . . . 9  |-  ( ph  ->  ( ( A  -  D ) G ( B  -  D ) )  =  0 )
5655adantr 453 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( A  -  D
) G ( B  -  D ) )  =  0 )
577adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  -.  B  =  D )  ->  D  e.  CC )
5810sigarperm 27828 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  D  e.  CC )  ->  (
( A  -  D
) G ( B  -  D ) )  =  ( ( B  -  A ) G ( D  -  A
) ) )
5946, 45, 57, 58syl3anc 1185 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( A  -  D
) G ( B  -  D ) )  =  ( ( B  -  A ) G ( D  -  A
) ) )
6056, 59eqtr3d 2472 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  0  =  ( ( B  -  A ) G ( D  -  A
) ) )
6160oveq2d 6099 . . . . . 6  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( C  -  B ) G ( D  -  A ) )  +  0 )  =  ( ( ( C  -  B ) G ( D  -  A ) )  +  ( ( B  -  A ) G ( D  -  A ) ) ) )
6210sigarim 27819 . . . . . . . . 9  |-  ( ( ( C  -  B
)  e.  CC  /\  ( D  -  A
)  e.  CC )  ->  ( ( C  -  B ) G ( D  -  A
) )  e.  RR )
6349, 50, 62syl2anc 644 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  B
) G ( D  -  A ) )  e.  RR )
6463recnd 9116 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  B
) G ( D  -  A ) )  e.  CC )
6564addid1d 9268 . . . . . 6  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( C  -  B ) G ( D  -  A ) )  +  0 )  =  ( ( C  -  B ) G ( D  -  A
) ) )
6654, 61, 653eqtr2d 2476 . . . . 5  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  A
) G ( D  -  A ) )  =  ( ( C  -  B ) G ( D  -  A
) ) )
6745, 57negsubdi2d 9429 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  B  =  D )  ->  -u ( B  -  D )  =  ( D  -  B ) )
6867eqcomd 2443 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  B  =  D )  ->  ( D  -  B )  =  -u ( B  -  D ) )
6968oveq1d 6098 . . . . . . . . . 10  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( D  -  B
)  /  ( B  -  D ) )  =  ( -u ( B  -  D )  /  ( B  -  D ) ) )
7045, 57subcld 9413 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  B  =  D )  ->  ( B  -  D )  e.  CC )
71 simpr 449 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  B  =  D )  ->  -.  B  =  D )
7271neneqad 2676 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  B  =  D )  ->  B  =/=  D )
7345, 57, 72subne0d 9422 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  B  =  D )  ->  ( B  -  D )  =/=  0 )
7470, 70, 73divnegd 9805 . . . . . . . . . 10  |-  ( (
ph  /\  -.  B  =  D )  ->  -u (
( B  -  D
)  /  ( B  -  D ) )  =  ( -u ( B  -  D )  /  ( B  -  D ) ) )
7570, 73dividd 9790 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( B  -  D
)  /  ( B  -  D ) )  =  1 )
7675negeqd 9302 . . . . . . . . . 10  |-  ( (
ph  /\  -.  B  =  D )  ->  -u (
( B  -  D
)  /  ( B  -  D ) )  =  -u 1 )
7769, 74, 763eqtr2d 2476 . . . . . . . . 9  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( D  -  B
)  /  ( B  -  D ) )  =  -u 1 )
7877oveq1d 6098 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( D  -  B )  /  ( B  -  D )
)  x.  ( A  -  D ) )  =  ( -u 1  x.  ( A  -  D
) ) )
7946, 57subcld 9413 . . . . . . . . 9  |-  ( (
ph  /\  -.  B  =  D )  ->  ( A  -  D )  e.  CC )
8079mulm1d 9487 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  ( -u 1  x.  ( A  -  D ) )  =  -u ( A  -  D ) )
8146, 57negsubdi2d 9429 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  -u ( A  -  D )  =  ( D  -  A ) )
8278, 80, 813eqtrd 2474 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( D  -  B )  /  ( B  -  D )
)  x.  ( A  -  D ) )  =  ( D  -  A ) )
8357, 45subcld 9413 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  ( D  -  B )  e.  CC )
8483, 70, 79, 73div32d 9815 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( D  -  B )  /  ( B  -  D )
)  x.  ( A  -  D ) )  =  ( ( D  -  B )  x.  ( ( A  -  D )  /  ( B  -  D )
) ) )
8582, 84eqtr3d 2472 . . . . . 6  |-  ( (
ph  /\  -.  B  =  D )  ->  ( D  -  A )  =  ( ( D  -  B )  x.  ( ( A  -  D )  /  ( B  -  D )
) ) )
8685oveq2d 6099 . . . . 5  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  B
) G ( D  -  A ) )  =  ( ( C  -  B ) G ( ( D  -  B )  x.  (
( A  -  D
)  /  ( B  -  D ) ) ) ) )
8757, 46, 453jca 1135 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  ( D  e.  CC  /\  A  e.  CC  /\  B  e.  CC ) )
8810, 87, 71, 56sigardiv 27829 . . . . . 6  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( A  -  D
)  /  ( B  -  D ) )  e.  RR )
8910sigarls 27825 . . . . . 6  |-  ( ( ( C  -  B
)  e.  CC  /\  ( D  -  B
)  e.  CC  /\  ( ( A  -  D )  /  ( B  -  D )
)  e.  RR )  ->  ( ( C  -  B ) G ( ( D  -  B )  x.  (
( A  -  D
)  /  ( B  -  D ) ) ) )  =  ( ( ( C  -  B ) G ( D  -  B ) )  x.  ( ( A  -  D )  /  ( B  -  D ) ) ) )
9049, 83, 88, 89syl3anc 1185 . . . . 5  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  B
) G ( ( D  -  B )  x.  ( ( A  -  D )  / 
( B  -  D
) ) ) )  =  ( ( ( C  -  B ) G ( D  -  B ) )  x.  ( ( A  -  D )  /  ( B  -  D )
) ) )
9166, 86, 903eqtrd 2474 . . . 4  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  A
) G ( D  -  A ) )  =  ( ( ( C  -  B ) G ( D  -  B ) )  x.  ( ( A  -  D )  /  ( B  -  D )
) ) )
9291oveq1d 6098 . . 3  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( C  -  A ) G ( D  -  A ) )  x.  ( B  -  D ) )  =  ( ( ( ( C  -  B
) G ( D  -  B ) )  x.  ( ( A  -  D )  / 
( B  -  D
) ) )  x.  ( B  -  D
) ) )
9310sigarim 27819 . . . . . 6  |-  ( ( ( C  -  B
)  e.  CC  /\  ( D  -  B
)  e.  CC )  ->  ( ( C  -  B ) G ( D  -  B
) )  e.  RR )
9493recnd 9116 . . . . 5  |-  ( ( ( C  -  B
)  e.  CC  /\  ( D  -  B
)  e.  CC )  ->  ( ( C  -  B ) G ( D  -  B
) )  e.  CC )
9549, 83, 94syl2anc 644 . . . 4  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  B
) G ( D  -  B ) )  e.  CC )
9679, 70, 73divcld 9792 . . . 4  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( A  -  D
)  /  ( B  -  D ) )  e.  CC )
9795, 96, 70mulassd 9113 . . 3  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( ( C  -  B ) G ( D  -  B
) )  x.  (
( A  -  D
)  /  ( B  -  D ) ) )  x.  ( B  -  D ) )  =  ( ( ( C  -  B ) G ( D  -  B ) )  x.  ( ( ( A  -  D )  / 
( B  -  D
) )  x.  ( B  -  D )
) ) )
9879, 70, 73divcan1d 9793 . . . 4  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( A  -  D )  /  ( B  -  D )
)  x.  ( B  -  D ) )  =  ( A  -  D ) )
9998oveq2d 6099 . . 3  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( C  -  B ) G ( D  -  B ) )  x.  ( ( ( A  -  D
)  /  ( B  -  D ) )  x.  ( B  -  D ) ) )  =  ( ( ( C  -  B ) G ( D  -  B ) )  x.  ( A  -  D
) ) )
10092, 97, 993eqtrd 2474 . 2  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( C  -  A ) G ( D  -  A ) )  x.  ( B  -  D ) )  =  ( ( ( C  -  B ) G ( D  -  B ) )  x.  ( A  -  D
) ) )
10143, 100pm2.61dan 768 1  |-  ( ph  ->  ( ( ( C  -  A ) G ( D  -  A
) )  x.  ( B  -  D )
)  =  ( ( ( C  -  B
) G ( D  -  B ) )  x.  ( A  -  D ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   CCcc 8990   RRcr 8991   0cc0 8992   1c1 8993    + caddc 8995    x. cmul 8997    - cmin 9293   -ucneg 9294    / cdiv 9679   *ccj 11903   Imcim 11905
This theorem is referenced by:  cevathlem2  27836
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-po 4505  df-so 4506  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-riota 6551  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-2 10060  df-cj 11906  df-re 11907  df-im 11908
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