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Theorem sigarms 29911
Description: Signed area is additive (with respect to subtraction) 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
sigarms  |-  ( ( 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 sigarms
StepHypRef Expression
1 simp1 988 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  A  e.  CC )
2 simp2 989 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  B  e.  CC )
3 simp3 990 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  C  e.  CC )
42, 3subcld 9734 . . 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 29907 . . 3  |-  ( ( A  e.  CC  /\  ( B  -  C
)  e.  CC )  ->  ( A G ( B  -  C
) )  =  -u ( ( B  -  C ) G A ) )
71, 4, 6syl2anc 661 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G ( B  -  C ) )  = 
-u ( ( B  -  C ) G A ) )
85sigarmf 29909 . . . . 5  |-  ( ( B  e.  CC  /\  A  e.  CC  /\  C  e.  CC )  ->  (
( B  -  C
) G A )  =  ( ( B G A )  -  ( C G A ) ) )
98negeqd 9619 . . . 4  |-  ( ( B  e.  CC  /\  A  e.  CC  /\  C  e.  CC )  ->  -u (
( B  -  C
) G A )  =  -u ( ( B G A )  -  ( C G A ) ) )
1093com12 1191 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  -u (
( B  -  C
) G A )  =  -u ( ( B G A )  -  ( C G A ) ) )
11 3simpa 985 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  e.  CC  /\  B  e.  CC ) )
1211ancomd 451 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  e.  CC  /\  A  e.  CC ) )
135sigarim 29906 . . . . . 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 9427 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B G A )  e.  CC )
16 3simpb 986 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  e.  CC  /\  C  e.  CC ) )
1716ancomd 451 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C  e.  CC  /\  A  e.  CC ) )
185sigarim 29906 . . . . . 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 9427 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C G A )  e.  CC )
21 negsubdi 9680 . . . . 5  |-  ( ( ( B G A )  e.  CC  /\  ( C G A )  e.  CC )  ->  -u ( ( B G A )  -  ( C G A ) )  =  ( -u ( B G A )  +  ( C G A ) ) )
22 simpl 457 . . . . . . 7  |-  ( ( ( B G A )  e.  CC  /\  ( C G A )  e.  CC )  -> 
( B G A )  e.  CC )
2322negcld 9721 . . . . . 6  |-  ( ( ( B G A )  e.  CC  /\  ( C G A )  e.  CC )  ->  -u ( B G A )  e.  CC )
24 simpr 461 . . . . . 6  |-  ( ( ( B G A )  e.  CC  /\  ( C G A )  e.  CC )  -> 
( C G A )  e.  CC )
2523, 24subnegd 9741 . . . . 5  |-  ( ( ( B G A )  e.  CC  /\  ( C G A )  e.  CC )  -> 
( -u ( B G A )  -  -u ( C G A ) )  =  ( -u ( B G A )  +  ( C G A ) ) )
2621, 25eqtr4d 2478 . . . 4  |-  ( ( ( B G A )  e.  CC  /\  ( C G A )  e.  CC )  ->  -u ( ( B G A )  -  ( C G A ) )  =  ( -u ( B G A )  -  -u ( C G A ) ) )
2715, 20, 26syl2anc 661 . . 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 ) ) )
2810, 27eqtrd 2475 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  -u (
( B  -  C
) G A )  =  ( -u ( B G A )  -  -u ( C G A ) ) )
295sigarac 29907 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A G B )  =  -u ( B G A ) )
301, 2, 29syl2anc 661 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G B )  = 
-u ( B G A ) )
3130eqcomd 2448 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  -u ( B G A )  =  ( A G B ) )
325sigarac 29907 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A G C )  =  -u ( C G A ) )
331, 3, 32syl2anc 661 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G C )  = 
-u ( C G A ) )
3433eqcomd 2448 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  -u ( C G A )  =  ( A G C ) )
3531, 34oveq12d 6124 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( -u ( B G A )  -  -u ( C G A ) )  =  ( ( A G B )  -  ( A G C ) ) )
367, 28, 353eqtrd 2479 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G ( B  -  C ) )  =  ( ( A G B )  -  ( A G C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   ` cfv 5433  (class class class)co 6106    e. cmpt2 6108   CCcc 9295   RRcr 9296    + caddc 9300    x. cmul 9302    - cmin 9610   -ucneg 9611   *ccj 12600   Imcim 12602
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 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374
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 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-op 3899  df-uni 4107  df-br 4308  df-opab 4366  df-mpt 4367  df-id 4651  df-po 4656  df-so 4657  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-er 7116  df-en 7326  df-dom 7327  df-sdom 7328  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-div 10009  df-2 10395  df-cj 12603  df-re 12604  df-im 12605
This theorem is referenced by:  sigarexp  29914  sigaradd  29921
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