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Theorem sigaradd 37451
Description: Subtracting (double) area of  A D C from  A B C yields the (double) area of  D B C. (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
sigaradd  |-  ( ph  ->  ( ( ( B  -  C ) G ( A  -  C
) )  -  (
( D  -  C
) G ( A  -  C ) ) )  =  ( ( B  -  C ) G ( D  -  C ) ) )
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 sigaradd
StepHypRef Expression
1 sharhght.a . . . . . . . 8  |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
)
21simp1d 1009 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
31simp3d 1011 . . . . . . 7  |-  ( ph  ->  C  e.  CC )
4 sharhght.b . . . . . . . 8  |-  ( ph  ->  ( D  e.  CC  /\  ( ( A  -  D ) G ( B  -  D ) )  =  0 ) )
54simpld 457 . . . . . . 7  |-  ( ph  ->  D  e.  CC )
62, 3, 5nnncan1d 10001 . . . . . 6  |-  ( ph  ->  ( ( A  -  C )  -  ( A  -  D )
)  =  ( D  -  C ) )
76oveq2d 6294 . . . . 5  |-  ( ph  ->  ( ( B  -  D ) G ( ( A  -  C
)  -  ( A  -  D ) ) )  =  ( ( B  -  D ) G ( D  -  C ) ) )
81simp2d 1010 . . . . . . 7  |-  ( ph  ->  B  e.  CC )
98, 5subcld 9967 . . . . . 6  |-  ( ph  ->  ( B  -  D
)  e.  CC )
102, 3subcld 9967 . . . . . 6  |-  ( ph  ->  ( A  -  C
)  e.  CC )
112, 5subcld 9967 . . . . . 6  |-  ( ph  ->  ( A  -  D
)  e.  CC )
12 sharhght.sigar . . . . . . 7  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
1312sigarms 37441 . . . . . 6  |-  ( ( ( B  -  D
)  e.  CC  /\  ( A  -  C
)  e.  CC  /\  ( A  -  D
)  e.  CC )  ->  ( ( B  -  D ) G ( ( A  -  C )  -  ( A  -  D )
) )  =  ( ( ( B  -  D ) G ( A  -  C ) )  -  ( ( B  -  D ) G ( A  -  D ) ) ) )
149, 10, 11, 13syl3anc 1230 . . . . 5  |-  ( ph  ->  ( ( B  -  D ) G ( ( A  -  C
)  -  ( A  -  D ) ) )  =  ( ( ( B  -  D
) G ( A  -  C ) )  -  ( ( B  -  D ) G ( A  -  D
) ) ) )
157, 14eqtr3d 2445 . . . 4  |-  ( ph  ->  ( ( B  -  D ) G ( D  -  C ) )  =  ( ( ( B  -  D
) G ( A  -  C ) )  -  ( ( B  -  D ) G ( A  -  D
) ) ) )
1612sigarac 37437 . . . . . . . . 9  |-  ( ( ( A  -  D
)  e.  CC  /\  ( B  -  D
)  e.  CC )  ->  ( ( A  -  D ) G ( B  -  D
) )  =  -u ( ( B  -  D ) G ( A  -  D ) ) )
1711, 9, 16syl2anc 659 . . . . . . . 8  |-  ( ph  ->  ( ( A  -  D ) G ( B  -  D ) )  =  -u (
( B  -  D
) G ( A  -  D ) ) )
184simprd 461 . . . . . . . 8  |-  ( ph  ->  ( ( A  -  D ) G ( B  -  D ) )  =  0 )
1917, 18eqtr3d 2445 . . . . . . 7  |-  ( ph  -> 
-u ( ( B  -  D ) G ( A  -  D
) )  =  0 )
2019negeqd 9850 . . . . . 6  |-  ( ph  -> 
-u -u ( ( B  -  D ) G ( A  -  D
) )  =  -u
0 )
219, 11jca 530 . . . . . . . 8  |-  ( ph  ->  ( ( B  -  D )  e.  CC  /\  ( A  -  D
)  e.  CC ) )
2212, 21sigarimcd 37447 . . . . . . 7  |-  ( ph  ->  ( ( B  -  D ) G ( A  -  D ) )  e.  CC )
2322negnegd 9958 . . . . . 6  |-  ( ph  -> 
-u -u ( ( B  -  D ) G ( A  -  D
) )  =  ( ( B  -  D
) G ( A  -  D ) ) )
24 neg0 9901 . . . . . . 7  |-  -u 0  =  0
2524a1i 11 . . . . . 6  |-  ( ph  -> 
-u 0  =  0 )
2620, 23, 253eqtr3d 2451 . . . . 5  |-  ( ph  ->  ( ( B  -  D ) G ( A  -  D ) )  =  0 )
2726oveq2d 6294 . . . 4  |-  ( ph  ->  ( ( ( B  -  D ) G ( A  -  C
) )  -  (
( B  -  D
) G ( A  -  D ) ) )  =  ( ( ( B  -  D
) G ( A  -  C ) )  -  0 ) )
289, 10jca 530 . . . . . 6  |-  ( ph  ->  ( ( B  -  D )  e.  CC  /\  ( A  -  C
)  e.  CC ) )
2912, 28sigarimcd 37447 . . . . 5  |-  ( ph  ->  ( ( B  -  D ) G ( A  -  C ) )  e.  CC )
3029subid1d 9956 . . . 4  |-  ( ph  ->  ( ( ( B  -  D ) G ( A  -  C
) )  -  0 )  =  ( ( B  -  D ) G ( A  -  C ) ) )
3115, 27, 303eqtrd 2447 . . 3  |-  ( ph  ->  ( ( B  -  D ) G ( D  -  C ) )  =  ( ( B  -  D ) G ( A  -  C ) ) )
328, 5, 3nnncan2d 10002 . . . 4  |-  ( ph  ->  ( ( B  -  C )  -  ( D  -  C )
)  =  ( B  -  D ) )
3332oveq1d 6293 . . 3  |-  ( ph  ->  ( ( ( B  -  C )  -  ( D  -  C
) ) G ( A  -  C ) )  =  ( ( B  -  D ) G ( A  -  C ) ) )
348, 3subcld 9967 . . . 4  |-  ( ph  ->  ( B  -  C
)  e.  CC )
355, 3subcld 9967 . . . 4  |-  ( ph  ->  ( D  -  C
)  e.  CC )
3612sigarmf 37439 . . . 4  |-  ( ( ( B  -  C
)  e.  CC  /\  ( A  -  C
)  e.  CC  /\  ( D  -  C
)  e.  CC )  ->  ( ( ( B  -  C )  -  ( D  -  C ) ) G ( A  -  C
) )  =  ( ( ( B  -  C ) G ( A  -  C ) )  -  ( ( D  -  C ) G ( A  -  C ) ) ) )
3734, 10, 35, 36syl3anc 1230 . . 3  |-  ( ph  ->  ( ( ( B  -  C )  -  ( D  -  C
) ) G ( A  -  C ) )  =  ( ( ( B  -  C
) G ( A  -  C ) )  -  ( ( D  -  C ) G ( A  -  C
) ) ) )
3831, 33, 373eqtr2rd 2450 . 2  |-  ( ph  ->  ( ( ( B  -  C ) G ( A  -  C
) )  -  (
( D  -  C
) G ( A  -  C ) ) )  =  ( ( B  -  D ) G ( D  -  C ) ) )
393, 5subcld 9967 . . . 4  |-  ( ph  ->  ( C  -  D
)  e.  CC )
40 1red 9641 . . . . 5  |-  ( ph  ->  1  e.  RR )
4140renegcld 10027 . . . 4  |-  ( ph  -> 
-u 1  e.  RR )
4212sigarls 37442 . . . 4  |-  ( ( ( B  -  D
)  e.  CC  /\  ( C  -  D
)  e.  CC  /\  -u 1  e.  RR )  ->  ( ( B  -  D ) G ( ( C  -  D )  x.  -u 1
) )  =  ( ( ( B  -  D ) G ( C  -  D ) )  x.  -u 1
) )
439, 39, 41, 42syl3anc 1230 . . 3  |-  ( ph  ->  ( ( B  -  D ) G ( ( C  -  D
)  x.  -u 1
) )  =  ( ( ( B  -  D ) G ( C  -  D ) )  x.  -u 1
) )
4439mulm1d 10049 . . . . 5  |-  ( ph  ->  ( -u 1  x.  ( C  -  D
) )  =  -u ( C  -  D
) )
45 1cnd 9642 . . . . . . 7  |-  ( ph  ->  1  e.  CC )
4645negcld 9954 . . . . . 6  |-  ( ph  -> 
-u 1  e.  CC )
4746, 39mulcomd 9647 . . . . 5  |-  ( ph  ->  ( -u 1  x.  ( C  -  D
) )  =  ( ( C  -  D
)  x.  -u 1
) )
483, 5negsubdi2d 9983 . . . . 5  |-  ( ph  -> 
-u ( C  -  D )  =  ( D  -  C ) )
4944, 47, 483eqtr3d 2451 . . . 4  |-  ( ph  ->  ( ( C  -  D )  x.  -u 1
)  =  ( D  -  C ) )
5049oveq2d 6294 . . 3  |-  ( ph  ->  ( ( B  -  D ) G ( ( C  -  D
)  x.  -u 1
) )  =  ( ( B  -  D
) G ( D  -  C ) ) )
519, 39jca 530 . . . . . 6  |-  ( ph  ->  ( ( B  -  D )  e.  CC  /\  ( C  -  D
)  e.  CC ) )
5212, 51sigarimcd 37447 . . . . 5  |-  ( ph  ->  ( ( B  -  D ) G ( C  -  D ) )  e.  CC )
5352mulm1d 10049 . . . 4  |-  ( ph  ->  ( -u 1  x.  ( ( B  -  D ) G ( C  -  D ) ) )  =  -u ( ( B  -  D ) G ( C  -  D ) ) )
5452, 46mulcomd 9647 . . . 4  |-  ( ph  ->  ( ( ( B  -  D ) G ( C  -  D
) )  x.  -u 1
)  =  ( -u
1  x.  ( ( B  -  D ) G ( C  -  D ) ) ) )
5512sigarac 37437 . . . . 5  |-  ( ( ( C  -  D
)  e.  CC  /\  ( B  -  D
)  e.  CC )  ->  ( ( C  -  D ) G ( B  -  D
) )  =  -u ( ( B  -  D ) G ( C  -  D ) ) )
5639, 9, 55syl2anc 659 . . . 4  |-  ( ph  ->  ( ( C  -  D ) G ( B  -  D ) )  =  -u (
( B  -  D
) G ( C  -  D ) ) )
5753, 54, 563eqtr4d 2453 . . 3  |-  ( ph  ->  ( ( ( B  -  D ) G ( C  -  D
) )  x.  -u 1
)  =  ( ( C  -  D ) G ( B  -  D ) ) )
5843, 50, 573eqtr3d 2451 . 2  |-  ( ph  ->  ( ( B  -  D ) G ( D  -  C ) )  =  ( ( C  -  D ) G ( B  -  D ) ) )
5912sigarperm 37445 . . 3  |-  ( ( C  e.  CC  /\  B  e.  CC  /\  D  e.  CC )  ->  (
( C  -  D
) G ( B  -  D ) )  =  ( ( B  -  C ) G ( D  -  C
) ) )
603, 8, 5, 59syl3anc 1230 . 2  |-  ( ph  ->  ( ( C  -  D ) G ( B  -  D ) )  =  ( ( B  -  C ) G ( D  -  C ) ) )
6138, 58, 603eqtrd 2447 1  |-  ( ph  ->  ( ( ( B  -  C ) G ( A  -  C
) )  -  (
( D  -  C
) G ( A  -  C ) ) )  =  ( ( B  -  C ) G ( D  -  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   ` cfv 5569  (class class class)co 6278    |-> cmpt2 6280   CCcc 9520   RRcr 9521   0cc0 9522   1c1 9523    x. cmul 9527    - cmin 9841   -ucneg 9842   *ccj 13078   Imcim 13080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-po 4744  df-so 4745  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-2 10635  df-cj 13081  df-re 13082  df-im 13083
This theorem is referenced by:  cevathlem2  37453
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