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Theorem sigarexp 27172
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 956 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  B  e.  CC )
2 simp3 957 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  C  e.  CC )
31, 2subcld 9247 . . 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 27167 . . 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 1229 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  C
) G ( B  -  C ) )  =  ( ( A G ( B  -  C ) )  -  ( C G ( B  -  C ) ) ) )
74sigarms 27169 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G ( B  -  C ) )  =  ( ( A G B )  -  ( A G C ) ) )
87oveq1d 5960 . 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 27169 . . . . 5  |-  ( ( C  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C G ( B  -  C ) )  =  ( ( C G B )  -  ( C G C ) ) )
102, 9syld3an1 1228 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C G ( B  -  C ) )  =  ( ( C G B )  -  ( C G C ) ) )
114sigarid 27171 . . . . . 6  |-  ( C  e.  CC  ->  ( C G C )  =  0 )
122, 11syl 15 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C G C )  =  0 )
1312oveq2d 5961 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( C G B )  -  ( C G C ) )  =  ( ( C G B )  - 
0 ) )
144sigarim 27164 . . . . . . 7  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C G B )  e.  RR )
1514recnd 8951 . . . . . 6  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C G B )  e.  CC )
162, 1, 15syl2anc 642 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C G B )  e.  CC )
1716subid1d 9236 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( C G B )  -  0 )  =  ( C G B ) )
1810, 13, 173eqtrd 2394 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C G ( B  -  C ) )  =  ( C G B ) )
1918oveq2d 5961 . 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 2394 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 set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   ` cfv 5337  (class class class)co 5945    e. cmpt2 5947   CCcc 8825   0cc0 8827    x. cmul 8832    - cmin 9127   *ccj 11677   Imcim 11679
This theorem is referenced by:  sigarperm  27173
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-po 4396  df-so 4397  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-riota 6391  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-2 9894  df-cj 11680  df-re 11681  df-im 11682
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