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Theorem sigaradd 29907
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 1000 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
31simp3d 1002 . . . . . . 7  |-  ( ph  ->  C  e.  CC )
4 sharhght.b . . . . . . . 8  |-  ( ph  ->  ( D  e.  CC  /\  ( ( A  -  D ) G ( B  -  D ) )  =  0 ) )
54simpld 459 . . . . . . 7  |-  ( ph  ->  D  e.  CC )
62, 3, 5nnncan1d 9758 . . . . . 6  |-  ( ph  ->  ( ( A  -  C )  -  ( A  -  D )
)  =  ( D  -  C ) )
76oveq2d 6112 . . . . 5  |-  ( ph  ->  ( ( B  -  D ) G ( ( A  -  C
)  -  ( A  -  D ) ) )  =  ( ( B  -  D ) G ( D  -  C ) ) )
81simp2d 1001 . . . . . . 7  |-  ( ph  ->  B  e.  CC )
98, 5subcld 9724 . . . . . 6  |-  ( ph  ->  ( B  -  D
)  e.  CC )
102, 3subcld 9724 . . . . . 6  |-  ( ph  ->  ( A  -  C
)  e.  CC )
112, 5subcld 9724 . . . . . 6  |-  ( ph  ->  ( A  -  D
)  e.  CC )
12 sharhght.sigar . . . . . . 7  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
1312sigarms 29897 . . . . . 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 1218 . . . . 5  |-  ( ph  ->  ( ( B  -  D ) G ( ( A  -  C
)  -  ( A  -  D ) ) )  =  ( ( ( B  -  D
) G ( A  -  C ) )  -  ( ( B  -  D ) G ( A  -  D
) ) ) )
157, 14eqtr3d 2477 . . . 4  |-  ( ph  ->  ( ( B  -  D ) G ( D  -  C ) )  =  ( ( ( B  -  D
) G ( A  -  C ) )  -  ( ( B  -  D ) G ( A  -  D
) ) ) )
1612sigarac 29893 . . . . . . . . 9  |-  ( ( ( A  -  D
)  e.  CC  /\  ( B  -  D
)  e.  CC )  ->  ( ( A  -  D ) G ( B  -  D
) )  =  -u ( ( B  -  D ) G ( A  -  D ) ) )
1711, 9, 16syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( ( A  -  D ) G ( B  -  D ) )  =  -u (
( B  -  D
) G ( A  -  D ) ) )
184simprd 463 . . . . . . . 8  |-  ( ph  ->  ( ( A  -  D ) G ( B  -  D ) )  =  0 )
1917, 18eqtr3d 2477 . . . . . . 7  |-  ( ph  -> 
-u ( ( B  -  D ) G ( A  -  D
) )  =  0 )
2019negeqd 9609 . . . . . 6  |-  ( ph  -> 
-u -u ( ( B  -  D ) G ( A  -  D
) )  =  -u
0 )
219, 11jca 532 . . . . . . . 8  |-  ( ph  ->  ( ( B  -  D )  e.  CC  /\  ( A  -  D
)  e.  CC ) )
2212, 21sigarimcd 29903 . . . . . . 7  |-  ( ph  ->  ( ( B  -  D ) G ( A  -  D ) )  e.  CC )
2322negnegd 9715 . . . . . 6  |-  ( ph  -> 
-u -u ( ( B  -  D ) G ( A  -  D
) )  =  ( ( B  -  D
) G ( A  -  D ) ) )
24 neg0 9660 . . . . . . 7  |-  -u 0  =  0
2524a1i 11 . . . . . 6  |-  ( ph  -> 
-u 0  =  0 )
2620, 23, 253eqtr3d 2483 . . . . 5  |-  ( ph  ->  ( ( B  -  D ) G ( A  -  D ) )  =  0 )
2726oveq2d 6112 . . . 4  |-  ( ph  ->  ( ( ( B  -  D ) G ( A  -  C
) )  -  (
( B  -  D
) G ( A  -  D ) ) )  =  ( ( ( B  -  D
) G ( A  -  C ) )  -  0 ) )
289, 10jca 532 . . . . . 6  |-  ( ph  ->  ( ( B  -  D )  e.  CC  /\  ( A  -  C
)  e.  CC ) )
2912, 28sigarimcd 29903 . . . . 5  |-  ( ph  ->  ( ( B  -  D ) G ( A  -  C ) )  e.  CC )
3029subid1d 9713 . . . 4  |-  ( ph  ->  ( ( ( B  -  D ) G ( A  -  C
) )  -  0 )  =  ( ( B  -  D ) G ( A  -  C ) ) )
3115, 27, 303eqtrd 2479 . . 3  |-  ( ph  ->  ( ( B  -  D ) G ( D  -  C ) )  =  ( ( B  -  D ) G ( A  -  C ) ) )
328, 5, 3nnncan2d 9759 . . . 4  |-  ( ph  ->  ( ( B  -  C )  -  ( D  -  C )
)  =  ( B  -  D ) )
3332oveq1d 6111 . . 3  |-  ( ph  ->  ( ( ( B  -  C )  -  ( D  -  C
) ) G ( A  -  C ) )  =  ( ( B  -  D ) G ( A  -  C ) ) )
348, 3subcld 9724 . . . 4  |-  ( ph  ->  ( B  -  C
)  e.  CC )
355, 3subcld 9724 . . . 4  |-  ( ph  ->  ( D  -  C
)  e.  CC )
3612sigarmf 29895 . . . 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 1218 . . 3  |-  ( ph  ->  ( ( ( B  -  C )  -  ( D  -  C
) ) G ( A  -  C ) )  =  ( ( ( B  -  C
) G ( A  -  C ) )  -  ( ( D  -  C ) G ( A  -  C
) ) ) )
3831, 33, 373eqtr2rd 2482 . 2  |-  ( ph  ->  ( ( ( B  -  C ) G ( A  -  C
) )  -  (
( D  -  C
) G ( A  -  C ) ) )  =  ( ( B  -  D ) G ( D  -  C ) ) )
393, 5subcld 9724 . . . 4  |-  ( ph  ->  ( C  -  D
)  e.  CC )
40 1re 9390 . . . . . 6  |-  1  e.  RR
4140a1i 11 . . . . 5  |-  ( ph  ->  1  e.  RR )
4241renegcld 9780 . . . 4  |-  ( ph  -> 
-u 1  e.  RR )
4312sigarls 29898 . . . 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 1218 . . 3  |-  ( ph  ->  ( ( B  -  D ) G ( ( C  -  D
)  x.  -u 1
) )  =  ( ( ( B  -  D ) G ( C  -  D ) )  x.  -u 1
) )
4539mulm1d 9801 . . . . 5  |-  ( ph  ->  ( -u 1  x.  ( C  -  D
) )  =  -u ( C  -  D
) )
46 ax-1cn 9345 . . . . . . . 8  |-  1  e.  CC
4746a1i 11 . . . . . . 7  |-  ( ph  ->  1  e.  CC )
4847negcld 9711 . . . . . 6  |-  ( ph  -> 
-u 1  e.  CC )
4948, 39mulcomd 9412 . . . . 5  |-  ( ph  ->  ( -u 1  x.  ( C  -  D
) )  =  ( ( C  -  D
)  x.  -u 1
) )
503, 5negsubdi2d 9740 . . . . 5  |-  ( ph  -> 
-u ( C  -  D )  =  ( D  -  C ) )
5145, 49, 503eqtr3d 2483 . . . 4  |-  ( ph  ->  ( ( C  -  D )  x.  -u 1
)  =  ( D  -  C ) )
5251oveq2d 6112 . . 3  |-  ( ph  ->  ( ( B  -  D ) G ( ( C  -  D
)  x.  -u 1
) )  =  ( ( B  -  D
) G ( D  -  C ) ) )
539, 39jca 532 . . . . . 6  |-  ( ph  ->  ( ( B  -  D )  e.  CC  /\  ( C  -  D
)  e.  CC ) )
5412, 53sigarimcd 29903 . . . . 5  |-  ( ph  ->  ( ( B  -  D ) G ( C  -  D ) )  e.  CC )
5554mulm1d 9801 . . . 4  |-  ( ph  ->  ( -u 1  x.  ( ( B  -  D ) G ( C  -  D ) ) )  =  -u ( ( B  -  D ) G ( C  -  D ) ) )
5654, 48mulcomd 9412 . . . 4  |-  ( ph  ->  ( ( ( B  -  D ) G ( C  -  D
) )  x.  -u 1
)  =  ( -u
1  x.  ( ( B  -  D ) G ( C  -  D ) ) ) )
5712sigarac 29893 . . . . 5  |-  ( ( ( C  -  D
)  e.  CC  /\  ( B  -  D
)  e.  CC )  ->  ( ( C  -  D ) G ( B  -  D
) )  =  -u ( ( B  -  D ) G ( C  -  D ) ) )
5839, 9, 57syl2anc 661 . . . 4  |-  ( ph  ->  ( ( C  -  D ) G ( B  -  D ) )  =  -u (
( B  -  D
) G ( C  -  D ) ) )
5955, 56, 583eqtr4d 2485 . . 3  |-  ( ph  ->  ( ( ( B  -  D ) G ( C  -  D
) )  x.  -u 1
)  =  ( ( C  -  D ) G ( B  -  D ) ) )
6044, 52, 593eqtr3d 2483 . 2  |-  ( ph  ->  ( ( B  -  D ) G ( D  -  C ) )  =  ( ( C  -  D ) G ( B  -  D ) ) )
6112sigarperm 29901 . . 3  |-  ( ( C  e.  CC  /\  B  e.  CC  /\  D  e.  CC )  ->  (
( C  -  D
) G ( B  -  D ) )  =  ( ( B  -  C ) G ( D  -  C
) ) )
623, 8, 5, 61syl3anc 1218 . 2  |-  ( ph  ->  ( ( C  -  D ) G ( B  -  D ) )  =  ( ( B  -  C ) G ( D  -  C ) ) )
6338, 60, 623eqtrd 2479 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098   CCcc 9285   RRcr 9286   0cc0 9287   1c1 9288    x. cmul 9292    - cmin 9600   -ucneg 9601   *ccj 12590   Imcim 12592
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 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
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 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-2 10385  df-cj 12593  df-re 12594  df-im 12595
This theorem is referenced by:  cevathlem2  29909
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