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Theorem sharhght 27958
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 969 . . . . . . . 8  |-  ( ph  ->  C  e.  CC )
31simp1d 967 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
42, 3subcld 9173 . . . . . . 7  |-  ( ph  ->  ( C  -  A
)  e.  CC )
54adantr 451 . . . . . 6  |-  ( (
ph  /\  B  =  D )  ->  ( C  -  A )  e.  CC )
6 sharhght.b . . . . . . . . 9  |-  ( ph  ->  ( D  e.  CC  /\  ( ( A  -  D ) G ( B  -  D ) )  =  0 ) )
76simpld 445 . . . . . . . 8  |-  ( ph  ->  D  e.  CC )
87, 3subcld 9173 . . . . . . 7  |-  ( ph  ->  ( D  -  A
)  e.  CC )
98adantr 451 . . . . . 6  |-  ( (
ph  /\  B  =  D )  ->  ( D  -  A )  e.  CC )
10 sharhght.sigar . . . . . . 7  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
1110sigarim 27944 . . . . . 6  |-  ( ( ( C  -  A
)  e.  CC  /\  ( D  -  A
)  e.  CC )  ->  ( ( C  -  A ) G ( D  -  A
) )  e.  RR )
125, 9, 11syl2anc 642 . . . . 5  |-  ( (
ph  /\  B  =  D )  ->  (
( C  -  A
) G ( D  -  A ) )  e.  RR )
1312recnd 8877 . . . 4  |-  ( (
ph  /\  B  =  D )  ->  (
( C  -  A
) G ( D  -  A ) )  e.  CC )
1413mul01d 9027 . . 3  |-  ( (
ph  /\  B  =  D )  ->  (
( ( C  -  A ) G ( D  -  A ) )  x.  0 )  =  0 )
151simp2d 968 . . . . . 6  |-  ( ph  ->  B  e.  CC )
1615adantr 451 . . . . 5  |-  ( (
ph  /\  B  =  D )  ->  B  e.  CC )
17 simpr 447 . . . . 5  |-  ( (
ph  /\  B  =  D )  ->  B  =  D )
1816, 17subeq0bd 9225 . . . 4  |-  ( (
ph  /\  B  =  D )  ->  ( B  -  D )  =  0 )
1918oveq2d 5890 . . 3  |-  ( (
ph  /\  B  =  D )  ->  (
( ( C  -  A ) G ( D  -  A ) )  x.  ( B  -  D ) )  =  ( ( ( C  -  A ) G ( D  -  A ) )  x.  0 ) )
202, 15subcld 9173 . . . . . . . 8  |-  ( ph  ->  ( C  -  B
)  e.  CC )
2120adantr 451 . . . . . . 7  |-  ( (
ph  /\  B  =  D )  ->  ( C  -  B )  e.  CC )
227, 15subcld 9173 . . . . . . . 8  |-  ( ph  ->  ( D  -  B
)  e.  CC )
2322adantr 451 . . . . . . 7  |-  ( (
ph  /\  B  =  D )  ->  ( D  -  B )  e.  CC )
2410sigarval 27943 . . . . . . 7  |-  ( ( ( C  -  B
)  e.  CC  /\  ( D  -  B
)  e.  CC )  ->  ( ( C  -  B ) G ( D  -  B
) )  =  ( Im `  ( ( * `  ( C  -  B ) )  x.  ( D  -  B ) ) ) )
2521, 23, 24syl2anc 642 . . . . . 6  |-  ( (
ph  /\  B  =  D )  ->  (
( C  -  B
) G ( D  -  B ) )  =  ( Im `  ( ( * `  ( C  -  B
) )  x.  ( D  -  B )
) ) )
267adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  B  =  D )  ->  D  e.  CC )
2717eqcomd 2301 . . . . . . . . . 10  |-  ( (
ph  /\  B  =  D )  ->  D  =  B )
2826, 27subeq0bd 9225 . . . . . . . . 9  |-  ( (
ph  /\  B  =  D )  ->  ( D  -  B )  =  0 )
2928oveq2d 5890 . . . . . . . 8  |-  ( (
ph  /\  B  =  D )  ->  (
( * `  ( C  -  B )
)  x.  ( D  -  B ) )  =  ( ( * `
 ( C  -  B ) )  x.  0 ) )
3021cjcld 11697 . . . . . . . . 9  |-  ( (
ph  /\  B  =  D )  ->  (
* `  ( C  -  B ) )  e.  CC )
3130mul01d 9027 . . . . . . . 8  |-  ( (
ph  /\  B  =  D )  ->  (
( * `  ( C  -  B )
)  x.  0 )  =  0 )
3229, 31eqtrd 2328 . . . . . . 7  |-  ( (
ph  /\  B  =  D )  ->  (
( * `  ( C  -  B )
)  x.  ( D  -  B ) )  =  0 )
3332fveq2d 5545 . . . . . 6  |-  ( (
ph  /\  B  =  D )  ->  (
Im `  ( (
* `  ( C  -  B ) )  x.  ( D  -  B
) ) )  =  ( Im `  0
) )
34 0re 8854 . . . . . . . 8  |-  0  e.  RR
3534a1i 10 . . . . . . 7  |-  ( (
ph  /\  B  =  D )  ->  0  e.  RR )
3635reim0d 11726 . . . . . 6  |-  ( (
ph  /\  B  =  D )  ->  (
Im `  0 )  =  0 )
3725, 33, 363eqtrd 2332 . . . . 5  |-  ( (
ph  /\  B  =  D )  ->  (
( C  -  B
) G ( D  -  B ) )  =  0 )
3837oveq1d 5889 . . . 4  |-  ( (
ph  /\  B  =  D )  ->  (
( ( C  -  B ) G ( D  -  B ) )  x.  ( A  -  D ) )  =  ( 0  x.  ( A  -  D
) ) )
393adantr 451 . . . . . 6  |-  ( (
ph  /\  B  =  D )  ->  A  e.  CC )
4039, 26subcld 9173 . . . . 5  |-  ( (
ph  /\  B  =  D )  ->  ( A  -  D )  e.  CC )
4140mul02d 9026 . . . 4  |-  ( (
ph  /\  B  =  D )  ->  (
0  x.  ( A  -  D ) )  =  0 )
4238, 41eqtrd 2328 . . 3  |-  ( (
ph  /\  B  =  D )  ->  (
( ( C  -  B ) G ( D  -  B ) )  x.  ( A  -  D ) )  =  0 )
4314, 19, 423eqtr4d 2338 . 2  |-  ( (
ph  /\  B  =  D )  ->  (
( ( C  -  A ) G ( D  -  A ) )  x.  ( B  -  D ) )  =  ( ( ( C  -  B ) G ( D  -  B ) )  x.  ( A  -  D
) ) )
442adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  -.  B  =  D )  ->  C  e.  CC )
4515adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  -.  B  =  D )  ->  B  e.  CC )
463adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  -.  B  =  D )  ->  A  e.  CC )
4744, 45, 46npncand 9197 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  B
)  +  ( B  -  A ) )  =  ( C  -  A ) )
4847oveq1d 5889 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( C  -  B )  +  ( B  -  A ) ) G ( D  -  A ) )  =  ( ( C  -  A ) G ( D  -  A
) ) )
4944, 45subcld 9173 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  ( C  -  B )  e.  CC )
508adantr 451 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  ( D  -  A )  e.  CC )
5145, 46subcld 9173 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  ( B  -  A )  e.  CC )
5210sigaraf 27946 . . . . . . . 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 1182 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( C  -  B )  +  ( B  -  A ) ) G ( D  -  A ) )  =  ( ( ( C  -  B ) G ( D  -  A ) )  +  ( ( B  -  A ) G ( D  -  A ) ) ) )
5448, 53eqtr3d 2330 . . . . . 6  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  A
) G ( D  -  A ) )  =  ( ( ( C  -  B ) G ( D  -  A ) )  +  ( ( B  -  A ) G ( D  -  A ) ) ) )
556simprd 449 . . . . . . . . 9  |-  ( ph  ->  ( ( A  -  D ) G ( B  -  D ) )  =  0 )
5655adantr 451 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( A  -  D
) G ( B  -  D ) )  =  0 )
577adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  -.  B  =  D )  ->  D  e.  CC )
5810sigarperm 27953 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  D  e.  CC )  ->  (
( A  -  D
) G ( B  -  D ) )  =  ( ( B  -  A ) G ( D  -  A
) ) )
5946, 45, 57, 58syl3anc 1182 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( A  -  D
) G ( B  -  D ) )  =  ( ( B  -  A ) G ( D  -  A
) ) )
6056, 59eqtr3d 2330 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  0  =  ( ( B  -  A ) G ( D  -  A
) ) )
6160oveq2d 5890 . . . . . 6  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( C  -  B ) G ( D  -  A ) )  +  0 )  =  ( ( ( C  -  B ) G ( D  -  A ) )  +  ( ( B  -  A ) G ( D  -  A ) ) ) )
6210sigarim 27944 . . . . . . . . 9  |-  ( ( ( C  -  B
)  e.  CC  /\  ( D  -  A
)  e.  CC )  ->  ( ( C  -  B ) G ( D  -  A
) )  e.  RR )
6349, 50, 62syl2anc 642 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  B
) G ( D  -  A ) )  e.  RR )
6463recnd 8877 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  B
) G ( D  -  A ) )  e.  CC )
6564addid1d 9028 . . . . . 6  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( C  -  B ) G ( D  -  A ) )  +  0 )  =  ( ( C  -  B ) G ( D  -  A
) ) )
6654, 61, 653eqtr2d 2334 . . . . 5  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  A
) G ( D  -  A ) )  =  ( ( C  -  B ) G ( D  -  A
) ) )
6745, 57negsubdi2d 9189 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  B  =  D )  ->  -u ( B  -  D )  =  ( D  -  B ) )
6867eqcomd 2301 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  B  =  D )  ->  ( D  -  B )  =  -u ( B  -  D ) )
6968oveq1d 5889 . . . . . . . . . 10  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( D  -  B
)  /  ( B  -  D ) )  =  ( -u ( B  -  D )  /  ( B  -  D ) ) )
7045, 57subcld 9173 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  B  =  D )  ->  ( B  -  D )  e.  CC )
71 simpr 447 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  B  =  D )  ->  -.  B  =  D )
7271neneqad 2529 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  B  =  D )  ->  B  =/=  D )
7345, 57, 72subne0d 9182 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  B  =  D )  ->  ( B  -  D )  =/=  0 )
7470, 70, 73divnegd 9565 . . . . . . . . . 10  |-  ( (
ph  /\  -.  B  =  D )  ->  -u (
( B  -  D
)  /  ( B  -  D ) )  =  ( -u ( B  -  D )  /  ( B  -  D ) ) )
7570, 73dividd 9550 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( B  -  D
)  /  ( B  -  D ) )  =  1 )
7675negeqd 9062 . . . . . . . . . 10  |-  ( (
ph  /\  -.  B  =  D )  ->  -u (
( B  -  D
)  /  ( B  -  D ) )  =  -u 1 )
7769, 74, 763eqtr2d 2334 . . . . . . . . 9  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( D  -  B
)  /  ( B  -  D ) )  =  -u 1 )
7877oveq1d 5889 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( D  -  B )  /  ( B  -  D )
)  x.  ( A  -  D ) )  =  ( -u 1  x.  ( A  -  D
) ) )
7946, 57subcld 9173 . . . . . . . . 9  |-  ( (
ph  /\  -.  B  =  D )  ->  ( A  -  D )  e.  CC )
8079mulm1d 9247 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  ( -u 1  x.  ( A  -  D ) )  =  -u ( A  -  D ) )
8146, 57negsubdi2d 9189 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  -u ( A  -  D )  =  ( D  -  A ) )
8278, 80, 813eqtrd 2332 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( D  -  B )  /  ( B  -  D )
)  x.  ( A  -  D ) )  =  ( D  -  A ) )
8357, 45subcld 9173 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  ( D  -  B )  e.  CC )
8483, 70, 79, 73div32d 9575 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( D  -  B )  /  ( B  -  D )
)  x.  ( A  -  D ) )  =  ( ( D  -  B )  x.  ( ( A  -  D )  /  ( B  -  D )
) ) )
8582, 84eqtr3d 2330 . . . . . 6  |-  ( (
ph  /\  -.  B  =  D )  ->  ( D  -  A )  =  ( ( D  -  B )  x.  ( ( A  -  D )  /  ( B  -  D )
) ) )
8685oveq2d 5890 . . . . 5  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  B
) G ( D  -  A ) )  =  ( ( C  -  B ) G ( ( D  -  B )  x.  (
( A  -  D
)  /  ( B  -  D ) ) ) ) )
8757, 46, 453jca 1132 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  ( D  e.  CC  /\  A  e.  CC  /\  B  e.  CC ) )
8810, 87, 71, 56sigardiv 27954 . . . . . 6  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( A  -  D
)  /  ( B  -  D ) )  e.  RR )
8910sigarls 27950 . . . . . 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 1182 . . . . 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 2332 . . . 4  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  A
) G ( D  -  A ) )  =  ( ( ( C  -  B ) G ( D  -  B ) )  x.  ( ( A  -  D )  /  ( B  -  D )
) ) )
9291oveq1d 5889 . . 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 27944 . . . . . 6  |-  ( ( ( C  -  B
)  e.  CC  /\  ( D  -  B
)  e.  CC )  ->  ( ( C  -  B ) G ( D  -  B
) )  e.  RR )
9493recnd 8877 . . . . 5  |-  ( ( ( C  -  B
)  e.  CC  /\  ( D  -  B
)  e.  CC )  ->  ( ( C  -  B ) G ( D  -  B
) )  e.  CC )
9549, 83, 94syl2anc 642 . . . 4  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  B
) G ( D  -  B ) )  e.  CC )
9679, 70, 73divcld 9552 . . . 4  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( A  -  D
)  /  ( B  -  D ) )  e.  CC )
9795, 96, 70mulassd 8874 . . 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 9553 . . . 4  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( A  -  D )  /  ( B  -  D )
)  x.  ( B  -  D ) )  =  ( A  -  D ) )
9998oveq2d 5890 . . 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 2332 . 2  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( C  -  A ) G ( D  -  A ) )  x.  ( B  -  D ) )  =  ( ( ( C  -  B ) G ( D  -  B ) )  x.  ( A  -  D
) ) )
10143, 100pm2.61dan 766 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    - cmin 9053   -ucneg 9054    / cdiv 9439   *ccj 11597   Imcim 11599
This theorem is referenced by:  cevathlem2  27961
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-2 9820  df-cj 11600  df-re 11601  df-im 11602
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