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Theorem sharhght 29904
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 1002 . . . . . . . 8  |-  ( ph  ->  C  e.  CC )
31simp1d 1000 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
42, 3subcld 9722 . . . . . . 7  |-  ( ph  ->  ( C  -  A
)  e.  CC )
54adantr 465 . . . . . 6  |-  ( (
ph  /\  B  =  D )  ->  ( C  -  A )  e.  CC )
6 sharhght.b . . . . . . . . 9  |-  ( ph  ->  ( D  e.  CC  /\  ( ( A  -  D ) G ( B  -  D ) )  =  0 ) )
76simpld 459 . . . . . . . 8  |-  ( ph  ->  D  e.  CC )
87, 3subcld 9722 . . . . . . 7  |-  ( ph  ->  ( D  -  A
)  e.  CC )
98adantr 465 . . . . . 6  |-  ( (
ph  /\  B  =  D )  ->  ( D  -  A )  e.  CC )
10 sharhght.sigar . . . . . . 7  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
1110sigarim 29890 . . . . . 6  |-  ( ( ( C  -  A
)  e.  CC  /\  ( D  -  A
)  e.  CC )  ->  ( ( C  -  A ) G ( D  -  A
) )  e.  RR )
125, 9, 11syl2anc 661 . . . . 5  |-  ( (
ph  /\  B  =  D )  ->  (
( C  -  A
) G ( D  -  A ) )  e.  RR )
1312recnd 9415 . . . 4  |-  ( (
ph  /\  B  =  D )  ->  (
( C  -  A
) G ( D  -  A ) )  e.  CC )
1413mul01d 9571 . . 3  |-  ( (
ph  /\  B  =  D )  ->  (
( ( C  -  A ) G ( D  -  A ) )  x.  0 )  =  0 )
151simp2d 1001 . . . . . 6  |-  ( ph  ->  B  e.  CC )
1615adantr 465 . . . . 5  |-  ( (
ph  /\  B  =  D )  ->  B  e.  CC )
17 simpr 461 . . . . 5  |-  ( (
ph  /\  B  =  D )  ->  B  =  D )
1816, 17subeq0bd 9777 . . . 4  |-  ( (
ph  /\  B  =  D )  ->  ( B  -  D )  =  0 )
1918oveq2d 6110 . . 3  |-  ( (
ph  /\  B  =  D )  ->  (
( ( C  -  A ) G ( D  -  A ) )  x.  ( B  -  D ) )  =  ( ( ( C  -  A ) G ( D  -  A ) )  x.  0 ) )
202, 15subcld 9722 . . . . . . . 8  |-  ( ph  ->  ( C  -  B
)  e.  CC )
2120adantr 465 . . . . . . 7  |-  ( (
ph  /\  B  =  D )  ->  ( C  -  B )  e.  CC )
227, 15subcld 9722 . . . . . . . 8  |-  ( ph  ->  ( D  -  B
)  e.  CC )
2322adantr 465 . . . . . . 7  |-  ( (
ph  /\  B  =  D )  ->  ( D  -  B )  e.  CC )
2410sigarval 29889 . . . . . . 7  |-  ( ( ( C  -  B
)  e.  CC  /\  ( D  -  B
)  e.  CC )  ->  ( ( C  -  B ) G ( D  -  B
) )  =  ( Im `  ( ( * `  ( C  -  B ) )  x.  ( D  -  B ) ) ) )
2521, 23, 24syl2anc 661 . . . . . 6  |-  ( (
ph  /\  B  =  D )  ->  (
( C  -  B
) G ( D  -  B ) )  =  ( Im `  ( ( * `  ( C  -  B
) )  x.  ( D  -  B )
) ) )
267adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  B  =  D )  ->  D  e.  CC )
2717eqcomd 2448 . . . . . . . . . 10  |-  ( (
ph  /\  B  =  D )  ->  D  =  B )
2826, 27subeq0bd 9777 . . . . . . . . 9  |-  ( (
ph  /\  B  =  D )  ->  ( D  -  B )  =  0 )
2928oveq2d 6110 . . . . . . . 8  |-  ( (
ph  /\  B  =  D )  ->  (
( * `  ( C  -  B )
)  x.  ( D  -  B ) )  =  ( ( * `
 ( C  -  B ) )  x.  0 ) )
3021cjcld 12688 . . . . . . . . 9  |-  ( (
ph  /\  B  =  D )  ->  (
* `  ( C  -  B ) )  e.  CC )
3130mul01d 9571 . . . . . . . 8  |-  ( (
ph  /\  B  =  D )  ->  (
( * `  ( C  -  B )
)  x.  0 )  =  0 )
3229, 31eqtrd 2475 . . . . . . 7  |-  ( (
ph  /\  B  =  D )  ->  (
( * `  ( C  -  B )
)  x.  ( D  -  B ) )  =  0 )
3332fveq2d 5698 . . . . . 6  |-  ( (
ph  /\  B  =  D )  ->  (
Im `  ( (
* `  ( C  -  B ) )  x.  ( D  -  B
) ) )  =  ( Im `  0
) )
34 0re 9389 . . . . . . . 8  |-  0  e.  RR
3534a1i 11 . . . . . . 7  |-  ( (
ph  /\  B  =  D )  ->  0  e.  RR )
3635reim0d 12717 . . . . . 6  |-  ( (
ph  /\  B  =  D )  ->  (
Im `  0 )  =  0 )
3725, 33, 363eqtrd 2479 . . . . 5  |-  ( (
ph  /\  B  =  D )  ->  (
( C  -  B
) G ( D  -  B ) )  =  0 )
3837oveq1d 6109 . . . 4  |-  ( (
ph  /\  B  =  D )  ->  (
( ( C  -  B ) G ( D  -  B ) )  x.  ( A  -  D ) )  =  ( 0  x.  ( A  -  D
) ) )
393adantr 465 . . . . . 6  |-  ( (
ph  /\  B  =  D )  ->  A  e.  CC )
4039, 26subcld 9722 . . . . 5  |-  ( (
ph  /\  B  =  D )  ->  ( A  -  D )  e.  CC )
4140mul02d 9570 . . . 4  |-  ( (
ph  /\  B  =  D )  ->  (
0  x.  ( A  -  D ) )  =  0 )
4238, 41eqtrd 2475 . . 3  |-  ( (
ph  /\  B  =  D )  ->  (
( ( C  -  B ) G ( D  -  B ) )  x.  ( A  -  D ) )  =  0 )
4314, 19, 423eqtr4d 2485 . 2  |-  ( (
ph  /\  B  =  D )  ->  (
( ( C  -  A ) G ( D  -  A ) )  x.  ( B  -  D ) )  =  ( ( ( C  -  B ) G ( D  -  B ) )  x.  ( A  -  D
) ) )
442adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  -.  B  =  D )  ->  C  e.  CC )
4515adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  -.  B  =  D )  ->  B  e.  CC )
463adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  -.  B  =  D )  ->  A  e.  CC )
4744, 45, 46npncand 9746 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  B
)  +  ( B  -  A ) )  =  ( C  -  A ) )
4847oveq1d 6109 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( C  -  B )  +  ( B  -  A ) ) G ( D  -  A ) )  =  ( ( C  -  A ) G ( D  -  A
) ) )
4944, 45subcld 9722 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  ( C  -  B )  e.  CC )
508adantr 465 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  ( D  -  A )  e.  CC )
5145, 46subcld 9722 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  ( B  -  A )  e.  CC )
5210sigaraf 29892 . . . . . . . 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 1218 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( C  -  B )  +  ( B  -  A ) ) G ( D  -  A ) )  =  ( ( ( C  -  B ) G ( D  -  A ) )  +  ( ( B  -  A ) G ( D  -  A ) ) ) )
5448, 53eqtr3d 2477 . . . . . 6  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  A
) G ( D  -  A ) )  =  ( ( ( C  -  B ) G ( D  -  A ) )  +  ( ( B  -  A ) G ( D  -  A ) ) ) )
556simprd 463 . . . . . . . . 9  |-  ( ph  ->  ( ( A  -  D ) G ( B  -  D ) )  =  0 )
5655adantr 465 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( A  -  D
) G ( B  -  D ) )  =  0 )
577adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  -.  B  =  D )  ->  D  e.  CC )
5810sigarperm 29899 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  D  e.  CC )  ->  (
( A  -  D
) G ( B  -  D ) )  =  ( ( B  -  A ) G ( D  -  A
) ) )
5946, 45, 57, 58syl3anc 1218 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( A  -  D
) G ( B  -  D ) )  =  ( ( B  -  A ) G ( D  -  A
) ) )
6056, 59eqtr3d 2477 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  0  =  ( ( B  -  A ) G ( D  -  A
) ) )
6160oveq2d 6110 . . . . . 6  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( C  -  B ) G ( D  -  A ) )  +  0 )  =  ( ( ( C  -  B ) G ( D  -  A ) )  +  ( ( B  -  A ) G ( D  -  A ) ) ) )
6210sigarim 29890 . . . . . . . . 9  |-  ( ( ( C  -  B
)  e.  CC  /\  ( D  -  A
)  e.  CC )  ->  ( ( C  -  B ) G ( D  -  A
) )  e.  RR )
6349, 50, 62syl2anc 661 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  B
) G ( D  -  A ) )  e.  RR )
6463recnd 9415 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  B
) G ( D  -  A ) )  e.  CC )
6564addid1d 9572 . . . . . 6  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( C  -  B ) G ( D  -  A ) )  +  0 )  =  ( ( C  -  B ) G ( D  -  A
) ) )
6654, 61, 653eqtr2d 2481 . . . . 5  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  A
) G ( D  -  A ) )  =  ( ( C  -  B ) G ( D  -  A
) ) )
6745, 57negsubdi2d 9738 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  B  =  D )  ->  -u ( B  -  D )  =  ( D  -  B ) )
6867eqcomd 2448 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  B  =  D )  ->  ( D  -  B )  =  -u ( B  -  D ) )
6968oveq1d 6109 . . . . . . . . . 10  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( D  -  B
)  /  ( B  -  D ) )  =  ( -u ( B  -  D )  /  ( B  -  D ) ) )
7045, 57subcld 9722 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  B  =  D )  ->  ( B  -  D )  e.  CC )
71 simpr 461 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  B  =  D )  ->  -.  B  =  D )
7271neneqad 2684 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  B  =  D )  ->  B  =/=  D )
7345, 57, 72subne0d 9731 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  B  =  D )  ->  ( B  -  D )  =/=  0 )
7470, 70, 73divnegd 10123 . . . . . . . . . 10  |-  ( (
ph  /\  -.  B  =  D )  ->  -u (
( B  -  D
)  /  ( B  -  D ) )  =  ( -u ( B  -  D )  /  ( B  -  D ) ) )
7570, 73dividd 10108 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( B  -  D
)  /  ( B  -  D ) )  =  1 )
7675negeqd 9607 . . . . . . . . . 10  |-  ( (
ph  /\  -.  B  =  D )  ->  -u (
( B  -  D
)  /  ( B  -  D ) )  =  -u 1 )
7769, 74, 763eqtr2d 2481 . . . . . . . . 9  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( D  -  B
)  /  ( B  -  D ) )  =  -u 1 )
7877oveq1d 6109 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( D  -  B )  /  ( B  -  D )
)  x.  ( A  -  D ) )  =  ( -u 1  x.  ( A  -  D
) ) )
7946, 57subcld 9722 . . . . . . . . 9  |-  ( (
ph  /\  -.  B  =  D )  ->  ( A  -  D )  e.  CC )
8079mulm1d 9799 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  ( -u 1  x.  ( A  -  D ) )  =  -u ( A  -  D ) )
8146, 57negsubdi2d 9738 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  -u ( A  -  D )  =  ( D  -  A ) )
8278, 80, 813eqtrd 2479 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( D  -  B )  /  ( B  -  D )
)  x.  ( A  -  D ) )  =  ( D  -  A ) )
8357, 45subcld 9722 . . . . . . . 8  |-  ( (
ph  /\  -.  B  =  D )  ->  ( D  -  B )  e.  CC )
8483, 70, 79, 73div32d 10133 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( D  -  B )  /  ( B  -  D )
)  x.  ( A  -  D ) )  =  ( ( D  -  B )  x.  ( ( A  -  D )  /  ( B  -  D )
) ) )
8582, 84eqtr3d 2477 . . . . . 6  |-  ( (
ph  /\  -.  B  =  D )  ->  ( D  -  A )  =  ( ( D  -  B )  x.  ( ( A  -  D )  /  ( B  -  D )
) ) )
8685oveq2d 6110 . . . . 5  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  B
) G ( D  -  A ) )  =  ( ( C  -  B ) G ( ( D  -  B )  x.  (
( A  -  D
)  /  ( B  -  D ) ) ) ) )
8757, 46, 453jca 1168 . . . . . . 7  |-  ( (
ph  /\  -.  B  =  D )  ->  ( D  e.  CC  /\  A  e.  CC  /\  B  e.  CC ) )
8810, 87, 71, 56sigardiv 29900 . . . . . 6  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( A  -  D
)  /  ( B  -  D ) )  e.  RR )
8910sigarls 29896 . . . . . 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 1218 . . . . 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 2479 . . . 4  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  A
) G ( D  -  A ) )  =  ( ( ( C  -  B ) G ( D  -  B ) )  x.  ( ( A  -  D )  /  ( B  -  D )
) ) )
9291oveq1d 6109 . . 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 29890 . . . . . 6  |-  ( ( ( C  -  B
)  e.  CC  /\  ( D  -  B
)  e.  CC )  ->  ( ( C  -  B ) G ( D  -  B
) )  e.  RR )
9493recnd 9415 . . . . 5  |-  ( ( ( C  -  B
)  e.  CC  /\  ( D  -  B
)  e.  CC )  ->  ( ( C  -  B ) G ( D  -  B
) )  e.  CC )
9549, 83, 94syl2anc 661 . . . 4  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( C  -  B
) G ( D  -  B ) )  e.  CC )
9679, 70, 73divcld 10110 . . . 4  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( A  -  D
)  /  ( B  -  D ) )  e.  CC )
9795, 96, 70mulassd 9412 . . 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 10111 . . . 4  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( A  -  D )  /  ( B  -  D )
)  x.  ( B  -  D ) )  =  ( A  -  D ) )
9998oveq2d 6110 . . 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 2479 . 2  |-  ( (
ph  /\  -.  B  =  D )  ->  (
( ( C  -  A ) G ( D  -  A ) )  x.  ( B  -  D ) )  =  ( ( ( C  -  B ) G ( D  -  B ) )  x.  ( A  -  D
) ) )
10143, 100pm2.61dan 789 1  |-  ( ph  ->  ( ( ( C  -  A ) G ( D  -  A
) )  x.  ( B  -  D )
)  =  ( ( ( C  -  B
) G ( D  -  B ) )  x.  ( A  -  D ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   ` cfv 5421  (class class class)co 6094    e. cmpt2 6096   CCcc 9283   RRcr 9284   0cc0 9285   1c1 9286    + caddc 9288    x. cmul 9290    - cmin 9598   -ucneg 9599    / cdiv 9996   *ccj 12588   Imcim 12590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-op 3887  df-uni 4095  df-br 4296  df-opab 4354  df-mpt 4355  df-id 4639  df-po 4644  df-so 4645  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-er 7104  df-en 7314  df-dom 7315  df-sdom 7316  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-div 9997  df-2 10383  df-cj 12591  df-re 12592  df-im 12593
This theorem is referenced by:  cevathlem2  29907
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