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Theorem sigaradd 31550
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 1008 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
31simp3d 1010 . . . . . . 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 9960 . . . . . 6  |-  ( ph  ->  ( ( A  -  C )  -  ( A  -  D )
)  =  ( D  -  C ) )
76oveq2d 6298 . . . . 5  |-  ( ph  ->  ( ( B  -  D ) G ( ( A  -  C
)  -  ( A  -  D ) ) )  =  ( ( B  -  D ) G ( D  -  C ) ) )
81simp2d 1009 . . . . . . 7  |-  ( ph  ->  B  e.  CC )
98, 5subcld 9926 . . . . . 6  |-  ( ph  ->  ( B  -  D
)  e.  CC )
102, 3subcld 9926 . . . . . 6  |-  ( ph  ->  ( A  -  C
)  e.  CC )
112, 5subcld 9926 . . . . . 6  |-  ( ph  ->  ( A  -  D
)  e.  CC )
12 sharhght.sigar . . . . . . 7  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
1312sigarms 31540 . . . . . 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 1228 . . . . 5  |-  ( ph  ->  ( ( B  -  D ) G ( ( A  -  C
)  -  ( A  -  D ) ) )  =  ( ( ( B  -  D
) G ( A  -  C ) )  -  ( ( B  -  D ) G ( A  -  D
) ) ) )
157, 14eqtr3d 2510 . . . 4  |-  ( ph  ->  ( ( B  -  D ) G ( D  -  C ) )  =  ( ( ( B  -  D
) G ( A  -  C ) )  -  ( ( B  -  D ) G ( A  -  D
) ) ) )
1612sigarac 31536 . . . . . . . . 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 2510 . . . . . . 7  |-  ( ph  -> 
-u ( ( B  -  D ) G ( A  -  D
) )  =  0 )
2019negeqd 9810 . . . . . 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 31546 . . . . . . 7  |-  ( ph  ->  ( ( B  -  D ) G ( A  -  D ) )  e.  CC )
2322negnegd 9917 . . . . . 6  |-  ( ph  -> 
-u -u ( ( B  -  D ) G ( A  -  D
) )  =  ( ( B  -  D
) G ( A  -  D ) ) )
24 neg0 9861 . . . . . . 7  |-  -u 0  =  0
2524a1i 11 . . . . . 6  |-  ( ph  -> 
-u 0  =  0 )
2620, 23, 253eqtr3d 2516 . . . . 5  |-  ( ph  ->  ( ( B  -  D ) G ( A  -  D ) )  =  0 )
2726oveq2d 6298 . . . 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 31546 . . . . 5  |-  ( ph  ->  ( ( B  -  D ) G ( A  -  C ) )  e.  CC )
3029subid1d 9915 . . . 4  |-  ( ph  ->  ( ( ( B  -  D ) G ( A  -  C
) )  -  0 )  =  ( ( B  -  D ) G ( A  -  C ) ) )
3115, 27, 303eqtrd 2512 . . 3  |-  ( ph  ->  ( ( B  -  D ) G ( D  -  C ) )  =  ( ( B  -  D ) G ( A  -  C ) ) )
328, 5, 3nnncan2d 9961 . . . 4  |-  ( ph  ->  ( ( B  -  C )  -  ( D  -  C )
)  =  ( B  -  D ) )
3332oveq1d 6297 . . 3  |-  ( ph  ->  ( ( ( B  -  C )  -  ( D  -  C
) ) G ( A  -  C ) )  =  ( ( B  -  D ) G ( A  -  C ) ) )
348, 3subcld 9926 . . . 4  |-  ( ph  ->  ( B  -  C
)  e.  CC )
355, 3subcld 9926 . . . 4  |-  ( ph  ->  ( D  -  C
)  e.  CC )
3612sigarmf 31538 . . . 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 1228 . . 3  |-  ( ph  ->  ( ( ( B  -  C )  -  ( D  -  C
) ) G ( A  -  C ) )  =  ( ( ( B  -  C
) G ( A  -  C ) )  -  ( ( D  -  C ) G ( A  -  C
) ) ) )
3831, 33, 373eqtr2rd 2515 . 2  |-  ( ph  ->  ( ( ( B  -  C ) G ( A  -  C
) )  -  (
( D  -  C
) G ( A  -  C ) ) )  =  ( ( B  -  D ) G ( D  -  C ) ) )
393, 5subcld 9926 . . . 4  |-  ( ph  ->  ( C  -  D
)  e.  CC )
40 1re 9591 . . . . . 6  |-  1  e.  RR
4140a1i 11 . . . . 5  |-  ( ph  ->  1  e.  RR )
4241renegcld 9982 . . . 4  |-  ( ph  -> 
-u 1  e.  RR )
4312sigarls 31541 . . . 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 1228 . . 3  |-  ( ph  ->  ( ( B  -  D ) G ( ( C  -  D
)  x.  -u 1
) )  =  ( ( ( B  -  D ) G ( C  -  D ) )  x.  -u 1
) )
4539mulm1d 10004 . . . . 5  |-  ( ph  ->  ( -u 1  x.  ( C  -  D
) )  =  -u ( C  -  D
) )
46 ax-1cn 9546 . . . . . . . 8  |-  1  e.  CC
4746a1i 11 . . . . . . 7  |-  ( ph  ->  1  e.  CC )
4847negcld 9913 . . . . . 6  |-  ( ph  -> 
-u 1  e.  CC )
4948, 39mulcomd 9613 . . . . 5  |-  ( ph  ->  ( -u 1  x.  ( C  -  D
) )  =  ( ( C  -  D
)  x.  -u 1
) )
503, 5negsubdi2d 9942 . . . . 5  |-  ( ph  -> 
-u ( C  -  D )  =  ( D  -  C ) )
5145, 49, 503eqtr3d 2516 . . . 4  |-  ( ph  ->  ( ( C  -  D )  x.  -u 1
)  =  ( D  -  C ) )
5251oveq2d 6298 . . 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 31546 . . . . 5  |-  ( ph  ->  ( ( B  -  D ) G ( C  -  D ) )  e.  CC )
5554mulm1d 10004 . . . 4  |-  ( ph  ->  ( -u 1  x.  ( ( B  -  D ) G ( C  -  D ) ) )  =  -u ( ( B  -  D ) G ( C  -  D ) ) )
5654, 48mulcomd 9613 . . . 4  |-  ( ph  ->  ( ( ( B  -  D ) G ( C  -  D
) )  x.  -u 1
)  =  ( -u
1  x.  ( ( B  -  D ) G ( C  -  D ) ) ) )
5712sigarac 31536 . . . . 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 2518 . . 3  |-  ( ph  ->  ( ( ( B  -  D ) G ( C  -  D
) )  x.  -u 1
)  =  ( ( C  -  D ) G ( B  -  D ) ) )
6044, 52, 593eqtr3d 2516 . 2  |-  ( ph  ->  ( ( B  -  D ) G ( D  -  C ) )  =  ( ( C  -  D ) G ( B  -  D ) ) )
6112sigarperm 31544 . . 3  |-  ( ( C  e.  CC  /\  B  e.  CC  /\  D  e.  CC )  ->  (
( C  -  D
) G ( B  -  D ) )  =  ( ( B  -  C ) G ( D  -  C
) ) )
623, 8, 5, 61syl3anc 1228 . 2  |-  ( ph  ->  ( ( C  -  D ) G ( B  -  D ) )  =  ( ( B  -  C ) G ( D  -  C ) ) )
6338, 60, 623eqtrd 2512 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 973    = wceq 1379    e. wcel 1767   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489    x. cmul 9493    - cmin 9801   -ucneg 9802   *ccj 12888   Imcim 12890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-2 10590  df-cj 12891  df-re 12892  df-im 12893
This theorem is referenced by:  cevathlem2  31552
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