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Theorem sigarexp 32279
Description: Expand the signed area formula by linearity. (Contributed by Saveliy Skresanov, 20-Sep-2017.)
Hypothesis
Ref Expression
sigar  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
Assertion
Ref Expression
sigarexp  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  C
) G ( B  -  C ) )  =  ( ( ( A G B )  -  ( A G C ) )  -  ( C G B ) ) )
Distinct variable groups:    x, y, A    x, B, y    x, C, y
Allowed substitution hints:    G( x, y)

Proof of Theorem sigarexp
StepHypRef Expression
1 simp2 997 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  B  e.  CC )
2 simp3 998 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  C  e.  CC )
31, 2subcld 9950 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  -  C )  e.  CC )
4 sigar . . . 4  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
54sigarmf 32274 . . 3  |-  ( ( A  e.  CC  /\  ( B  -  C
)  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  C ) G ( B  -  C ) )  =  ( ( A G ( B  -  C ) )  -  ( C G ( B  -  C
) ) ) )
63, 5syld3an2 1275 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  C
) G ( B  -  C ) )  =  ( ( A G ( B  -  C ) )  -  ( C G ( B  -  C ) ) ) )
74sigarms 32276 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G ( B  -  C ) )  =  ( ( A G B )  -  ( A G C ) ) )
87oveq1d 6311 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A G ( B  -  C ) )  -  ( C G ( B  -  C ) ) )  =  ( ( ( A G B )  -  ( A G C ) )  -  ( C G ( B  -  C ) ) ) )
94sigarms 32276 . . . . 5  |-  ( ( C  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C G ( B  -  C ) )  =  ( ( C G B )  -  ( C G C ) ) )
102, 9syld3an1 1274 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C G ( B  -  C ) )  =  ( ( C G B )  -  ( C G C ) ) )
114sigarid 32278 . . . . . 6  |-  ( C  e.  CC  ->  ( C G C )  =  0 )
122, 11syl 16 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C G C )  =  0 )
1312oveq2d 6312 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( C G B )  -  ( C G C ) )  =  ( ( C G B )  - 
0 ) )
144sigarim 32271 . . . . . . 7  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C G B )  e.  RR )
1514recnd 9639 . . . . . 6  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C G B )  e.  CC )
162, 1, 15syl2anc 661 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C G B )  e.  CC )
1716subid1d 9939 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( C G B )  -  0 )  =  ( C G B ) )
1810, 13, 173eqtrd 2502 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C G ( B  -  C ) )  =  ( C G B ) )
1918oveq2d 6312 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A G B )  -  ( A G C ) )  -  ( C G ( B  -  C
) ) )  =  ( ( ( A G B )  -  ( A G C ) )  -  ( C G B ) ) )
206, 8, 193eqtrd 2502 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  C
) G ( B  -  C ) )  =  ( ( ( A G B )  -  ( A G C ) )  -  ( C G B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   CCcc 9507   0cc0 9509    x. cmul 9514    - cmin 9824   *ccj 12941   Imcim 12943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-2 10615  df-cj 12944  df-re 12945  df-im 12946
This theorem is referenced by:  sigarperm  32280
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