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Theorem sigaradd 27833
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 970 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
31simp3d 972 . . . . . . 7  |-  ( ph  ->  C  e.  CC )
4 sharhght.b . . . . . . . 8  |-  ( ph  ->  ( D  e.  CC  /\  ( ( A  -  D ) G ( B  -  D ) )  =  0 ) )
54simpld 447 . . . . . . 7  |-  ( ph  ->  D  e.  CC )
62, 3, 5nnncan1d 9446 . . . . . 6  |-  ( ph  ->  ( ( A  -  C )  -  ( A  -  D )
)  =  ( D  -  C ) )
76oveq2d 6098 . . . . 5  |-  ( ph  ->  ( ( B  -  D ) G ( ( A  -  C
)  -  ( A  -  D ) ) )  =  ( ( B  -  D ) G ( D  -  C ) ) )
81simp2d 971 . . . . . . 7  |-  ( ph  ->  B  e.  CC )
98, 5subcld 9412 . . . . . 6  |-  ( ph  ->  ( B  -  D
)  e.  CC )
102, 3subcld 9412 . . . . . 6  |-  ( ph  ->  ( A  -  C
)  e.  CC )
112, 5subcld 9412 . . . . . 6  |-  ( ph  ->  ( A  -  D
)  e.  CC )
12 sharhght.sigar . . . . . . 7  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
1312sigarms 27823 . . . . . 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 1185 . . . . 5  |-  ( ph  ->  ( ( B  -  D ) G ( ( A  -  C
)  -  ( A  -  D ) ) )  =  ( ( ( B  -  D
) G ( A  -  C ) )  -  ( ( B  -  D ) G ( A  -  D
) ) ) )
157, 14eqtr3d 2471 . . . 4  |-  ( ph  ->  ( ( B  -  D ) G ( D  -  C ) )  =  ( ( ( B  -  D
) G ( A  -  C ) )  -  ( ( B  -  D ) G ( A  -  D
) ) ) )
1612sigarac 27819 . . . . . . . . 9  |-  ( ( ( A  -  D
)  e.  CC  /\  ( B  -  D
)  e.  CC )  ->  ( ( A  -  D ) G ( B  -  D
) )  =  -u ( ( B  -  D ) G ( A  -  D ) ) )
1711, 9, 16syl2anc 644 . . . . . . . 8  |-  ( ph  ->  ( ( A  -  D ) G ( B  -  D ) )  =  -u (
( B  -  D
) G ( A  -  D ) ) )
184simprd 451 . . . . . . . 8  |-  ( ph  ->  ( ( A  -  D ) G ( B  -  D ) )  =  0 )
1917, 18eqtr3d 2471 . . . . . . 7  |-  ( ph  -> 
-u ( ( B  -  D ) G ( A  -  D
) )  =  0 )
2019negeqd 9301 . . . . . 6  |-  ( ph  -> 
-u -u ( ( B  -  D ) G ( A  -  D
) )  =  -u
0 )
219, 11jca 520 . . . . . . . 8  |-  ( ph  ->  ( ( B  -  D )  e.  CC  /\  ( A  -  D
)  e.  CC ) )
2212, 21sigarimcd 27829 . . . . . . 7  |-  ( ph  ->  ( ( B  -  D ) G ( A  -  D ) )  e.  CC )
2322negnegd 9403 . . . . . 6  |-  ( ph  -> 
-u -u ( ( B  -  D ) G ( A  -  D
) )  =  ( ( B  -  D
) G ( A  -  D ) ) )
24 neg0 9348 . . . . . . 7  |-  -u 0  =  0
2524a1i 11 . . . . . 6  |-  ( ph  -> 
-u 0  =  0 )
2620, 23, 253eqtr3d 2477 . . . . 5  |-  ( ph  ->  ( ( B  -  D ) G ( A  -  D ) )  =  0 )
2726oveq2d 6098 . . . 4  |-  ( ph  ->  ( ( ( B  -  D ) G ( A  -  C
) )  -  (
( B  -  D
) G ( A  -  D ) ) )  =  ( ( ( B  -  D
) G ( A  -  C ) )  -  0 ) )
289, 10jca 520 . . . . . 6  |-  ( ph  ->  ( ( B  -  D )  e.  CC  /\  ( A  -  C
)  e.  CC ) )
2912, 28sigarimcd 27829 . . . . 5  |-  ( ph  ->  ( ( B  -  D ) G ( A  -  C ) )  e.  CC )
3029subid1d 9401 . . . 4  |-  ( ph  ->  ( ( ( B  -  D ) G ( A  -  C
) )  -  0 )  =  ( ( B  -  D ) G ( A  -  C ) ) )
3115, 27, 303eqtrd 2473 . . 3  |-  ( ph  ->  ( ( B  -  D ) G ( D  -  C ) )  =  ( ( B  -  D ) G ( A  -  C ) ) )
328, 5, 3nnncan2d 9447 . . . 4  |-  ( ph  ->  ( ( B  -  C )  -  ( D  -  C )
)  =  ( B  -  D ) )
3332oveq1d 6097 . . 3  |-  ( ph  ->  ( ( ( B  -  C )  -  ( D  -  C
) ) G ( A  -  C ) )  =  ( ( B  -  D ) G ( A  -  C ) ) )
348, 3subcld 9412 . . . 4  |-  ( ph  ->  ( B  -  C
)  e.  CC )
355, 3subcld 9412 . . . 4  |-  ( ph  ->  ( D  -  C
)  e.  CC )
3612sigarmf 27821 . . . 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 1185 . . 3  |-  ( ph  ->  ( ( ( B  -  C )  -  ( D  -  C
) ) G ( A  -  C ) )  =  ( ( ( B  -  C
) G ( A  -  C ) )  -  ( ( D  -  C ) G ( A  -  C
) ) ) )
3831, 33, 373eqtr2rd 2476 . 2  |-  ( ph  ->  ( ( ( B  -  C ) G ( A  -  C
) )  -  (
( D  -  C
) G ( A  -  C ) ) )  =  ( ( B  -  D ) G ( D  -  C ) ) )
393, 5subcld 9412 . . . 4  |-  ( ph  ->  ( C  -  D
)  e.  CC )
40 1re 9091 . . . . . 6  |-  1  e.  RR
4140a1i 11 . . . . 5  |-  ( ph  ->  1  e.  RR )
4241renegcld 9465 . . . 4  |-  ( ph  -> 
-u 1  e.  RR )
4312sigarls 27824 . . . 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
) )
449, 39, 42, 43syl3anc 1185 . . 3  |-  ( ph  ->  ( ( B  -  D ) G ( ( C  -  D
)  x.  -u 1
) )  =  ( ( ( B  -  D ) G ( C  -  D ) )  x.  -u 1
) )
4539mulm1d 9486 . . . . 5  |-  ( ph  ->  ( -u 1  x.  ( C  -  D
) )  =  -u ( C  -  D
) )
46 ax-1cn 9049 . . . . . . . 8  |-  1  e.  CC
4746a1i 11 . . . . . . 7  |-  ( ph  ->  1  e.  CC )
4847negcld 9399 . . . . . 6  |-  ( ph  -> 
-u 1  e.  CC )
4948, 39mulcomd 9110 . . . . 5  |-  ( ph  ->  ( -u 1  x.  ( C  -  D
) )  =  ( ( C  -  D
)  x.  -u 1
) )
503, 5negsubdi2d 9428 . . . . 5  |-  ( ph  -> 
-u ( C  -  D )  =  ( D  -  C ) )
5145, 49, 503eqtr3d 2477 . . . 4  |-  ( ph  ->  ( ( C  -  D )  x.  -u 1
)  =  ( D  -  C ) )
5251oveq2d 6098 . . 3  |-  ( ph  ->  ( ( B  -  D ) G ( ( C  -  D
)  x.  -u 1
) )  =  ( ( B  -  D
) G ( D  -  C ) ) )
539, 39jca 520 . . . . . 6  |-  ( ph  ->  ( ( B  -  D )  e.  CC  /\  ( C  -  D
)  e.  CC ) )
5412, 53sigarimcd 27829 . . . . 5  |-  ( ph  ->  ( ( B  -  D ) G ( C  -  D ) )  e.  CC )
5554mulm1d 9486 . . . 4  |-  ( ph  ->  ( -u 1  x.  ( ( B  -  D ) G ( C  -  D ) ) )  =  -u ( ( B  -  D ) G ( C  -  D ) ) )
5654, 48mulcomd 9110 . . . 4  |-  ( ph  ->  ( ( ( B  -  D ) G ( C  -  D
) )  x.  -u 1
)  =  ( -u
1  x.  ( ( B  -  D ) G ( C  -  D ) ) ) )
5712sigarac 27819 . . . . 5  |-  ( ( ( C  -  D
)  e.  CC  /\  ( B  -  D
)  e.  CC )  ->  ( ( C  -  D ) G ( B  -  D
) )  =  -u ( ( B  -  D ) G ( C  -  D ) ) )
5839, 9, 57syl2anc 644 . . . 4  |-  ( ph  ->  ( ( C  -  D ) G ( B  -  D ) )  =  -u (
( B  -  D
) G ( C  -  D ) ) )
5955, 56, 583eqtr4d 2479 . . 3  |-  ( ph  ->  ( ( ( B  -  D ) G ( C  -  D
) )  x.  -u 1
)  =  ( ( C  -  D ) G ( B  -  D ) ) )
6044, 52, 593eqtr3d 2477 . 2  |-  ( ph  ->  ( ( B  -  D ) G ( D  -  C ) )  =  ( ( C  -  D ) G ( B  -  D ) ) )
6112sigarperm 27827 . . 3  |-  ( ( C  e.  CC  /\  B  e.  CC  /\  D  e.  CC )  ->  (
( C  -  D
) G ( B  -  D ) )  =  ( ( B  -  C ) G ( D  -  C
) ) )
623, 8, 5, 61syl3anc 1185 . 2  |-  ( ph  ->  ( ( C  -  D ) G ( B  -  D ) )  =  ( ( B  -  C ) G ( D  -  C ) ) )
6338, 60, 623eqtrd 2473 1  |-  ( ph  ->  ( ( ( B  -  C ) G ( A  -  C
) )  -  (
( D  -  C
) G ( A  -  C ) ) )  =  ( ( B  -  C ) G ( D  -  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   ` cfv 5455  (class class class)co 6082    e. cmpt2 6084   CCcc 8989   RRcr 8990   0cc0 8991   1c1 8992    x. cmul 8996    - cmin 9292   -ucneg 9293   *ccj 11902   Imcim 11904
This theorem is referenced by:  cevathlem2  27835
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 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-po 4504  df-so 4505  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-riota 6550  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-2 10059  df-cj 11905  df-re 11906  df-im 11907
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