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Theorem sigarms 37422
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 997 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  A  e.  CC )
2 simp2 998 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  B  e.  CC )
3 simp3 999 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  C  e.  CC )
42, 3subcld 9966 . . 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 37418 . . 3  |-  ( ( A  e.  CC  /\  ( B  -  C
)  e.  CC )  ->  ( A G ( B  -  C
) )  =  -u ( ( B  -  C ) G A ) )
71, 4, 6syl2anc 659 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G ( B  -  C ) )  = 
-u ( ( B  -  C ) G A ) )
85sigarmf 37420 . . . . 5  |-  ( ( B  e.  CC  /\  A  e.  CC  /\  C  e.  CC )  ->  (
( B  -  C
) G A )  =  ( ( B G A )  -  ( C G A ) ) )
98negeqd 9849 . . . 4  |-  ( ( B  e.  CC  /\  A  e.  CC  /\  C  e.  CC )  ->  -u (
( B  -  C
) G A )  =  -u ( ( B G A )  -  ( C G A ) ) )
1093com12 1201 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  -u (
( B  -  C
) G A )  =  -u ( ( B G A )  -  ( C G A ) ) )
11 3simpa 994 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  e.  CC  /\  B  e.  CC ) )
1211ancomd 449 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  e.  CC  /\  A  e.  CC ) )
135sigarim 37417 . . . . . 6  |-  ( ( B  e.  CC  /\  A  e.  CC )  ->  ( B G A )  e.  RR )
1412, 13syl 17 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B G A )  e.  RR )
1514recnd 9651 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B G A )  e.  CC )
16 3simpb 995 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  e.  CC  /\  C  e.  CC ) )
1716ancomd 449 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C  e.  CC  /\  A  e.  CC ) )
185sigarim 37417 . . . . . 6  |-  ( ( C  e.  CC  /\  A  e.  CC )  ->  ( C G A )  e.  RR )
1917, 18syl 17 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C G A )  e.  RR )
2019recnd 9651 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C G A )  e.  CC )
21 negsubdi 9910 . . . . 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 455 . . . . . . 7  |-  ( ( ( B G A )  e.  CC  /\  ( C G A )  e.  CC )  -> 
( B G A )  e.  CC )
2322negcld 9953 . . . . . 6  |-  ( ( ( B G A )  e.  CC  /\  ( C G A )  e.  CC )  ->  -u ( B G A )  e.  CC )
24 simpr 459 . . . . . 6  |-  ( ( ( B G A )  e.  CC  /\  ( C G A )  e.  CC )  -> 
( C G A )  e.  CC )
2523, 24subnegd 9973 . . . . 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 2446 . . . 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 659 . . 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 2443 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  -u (
( B  -  C
) G A )  =  ( -u ( B G A )  -  -u ( C G A ) ) )
295sigarac 37418 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A G B )  =  -u ( B G A ) )
301, 2, 29syl2anc 659 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G B )  = 
-u ( B G A ) )
3130eqcomd 2410 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  -u ( B G A )  =  ( A G B ) )
325sigarac 37418 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A G C )  =  -u ( C G A ) )
331, 3, 32syl2anc 659 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G C )  = 
-u ( C G A ) )
3433eqcomd 2410 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  -u ( C G A )  =  ( A G C ) )
3531, 34oveq12d 6295 . 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 2447 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842   ` cfv 5568  (class class class)co 6277    |-> cmpt2 6279   CCcc 9519   RRcr 9520    + caddc 9524    x. cmul 9526    - cmin 9840   -ucneg 9841   *ccj 13076   Imcim 13078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-po 4743  df-so 4744  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-2 10634  df-cj 13079  df-re 13080  df-im 13081
This theorem is referenced by:  sigarexp  37425  sigaradd  37432
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