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Theorem sigaras 27823
Description: Signed area is additive by the second argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)
Hypothesis
Ref Expression
sigar  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
Assertion
Ref Expression
sigaras  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G ( B  +  C ) )  =  ( ( A G B )  +  ( A G C ) ) )
Distinct variable groups:    x, y, A    x, B, y    x, C, y
Allowed substitution hints:    G( x, y)

Proof of Theorem sigaras
StepHypRef Expression
1 simp1 958 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  A  e.  CC )
2 simp2 959 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  B  e.  CC )
3 simp3 960 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  C  e.  CC )
42, 3addcld 9109 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  +  C )  e.  CC )
5 sigar . . . 4  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
65sigarac 27820 . . 3  |-  ( ( A  e.  CC  /\  ( B  +  C
)  e.  CC )  ->  ( A G ( B  +  C
) )  =  -u ( ( B  +  C ) G A ) )
71, 4, 6syl2anc 644 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G ( B  +  C ) )  = 
-u ( ( B  +  C ) G A ) )
85sigaraf 27821 . . . . 5  |-  ( ( B  e.  CC  /\  A  e.  CC  /\  C  e.  CC )  ->  (
( B  +  C
) G A )  =  ( ( B G A )  +  ( C G A ) ) )
98negeqd 9302 . . . 4  |-  ( ( B  e.  CC  /\  A  e.  CC  /\  C  e.  CC )  ->  -u (
( B  +  C
) G A )  =  -u ( ( B G A )  +  ( C G A ) ) )
1093com12 1158 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  -u (
( B  +  C
) G A )  =  -u ( ( B G A )  +  ( C G A ) ) )
11 3simpa 955 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  e.  CC  /\  B  e.  CC ) )
1211ancomd 440 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  e.  CC  /\  A  e.  CC ) )
135sigarim 27819 . . . . . 6  |-  ( ( B  e.  CC  /\  A  e.  CC )  ->  ( B G A )  e.  RR )
1412, 13syl 16 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B G A )  e.  RR )
1514recnd 9116 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B G A )  e.  CC )
16 3simpb 956 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  e.  CC  /\  C  e.  CC ) )
1716ancomd 440 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C  e.  CC  /\  A  e.  CC ) )
185sigarim 27819 . . . . . 6  |-  ( ( C  e.  CC  /\  A  e.  CC )  ->  ( C G A )  e.  RR )
1917, 18syl 16 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C G A )  e.  RR )
2019recnd 9116 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C G A )  e.  CC )
2115, 20negdid 9426 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  -u (
( B G A )  +  ( C G A ) )  =  ( -u ( B G A )  + 
-u ( C G A ) ) )
2210, 21eqtrd 2470 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  -u (
( B  +  C
) G A )  =  ( -u ( B G A )  + 
-u ( C G A ) ) )
235sigarac 27820 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A G B )  =  -u ( B G A ) )
241, 2, 23syl2anc 644 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G B )  = 
-u ( B G A ) )
2524eqcomd 2443 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  -u ( B G A )  =  ( A G B ) )
265sigarac 27820 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A G C )  =  -u ( C G A ) )
271, 3, 26syl2anc 644 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G C )  = 
-u ( C G A ) )
2827eqcomd 2443 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  -u ( C G A )  =  ( A G C ) )
2925, 28oveq12d 6101 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( -u ( B G A )  +  -u ( C G A ) )  =  ( ( A G B )  +  ( A G C ) ) )
307, 22, 293eqtrd 2474 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G ( B  +  C ) )  =  ( ( A G B )  +  ( A G C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   CCcc 8990   RRcr 8991    + caddc 8995    x. cmul 8997   -ucneg 9294   *ccj 11903   Imcim 11905
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 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-po 4505  df-so 4506  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-riota 6551  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-2 10060  df-cj 11906  df-re 11907  df-im 11908
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