Users' Mathboxes Mathbox for Saveliy Skresanov < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sigarexp Structured version   Visualization version   Unicode version

Theorem sigarexp 38462
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 1008 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  B  e.  CC )
2 simp3 1009 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  C  e.  CC )
31, 2subcld 9983 . . 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 38457 . . 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 1314 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  C
) G ( B  -  C ) )  =  ( ( A G ( B  -  C ) )  -  ( C G ( B  -  C ) ) ) )
74sigarms 38459 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G ( B  -  C ) )  =  ( ( A G B )  -  ( A G C ) ) )
87oveq1d 6303 . 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 38459 . . . . 5  |-  ( ( C  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C G ( B  -  C ) )  =  ( ( C G B )  -  ( C G C ) ) )
102, 9syld3an1 1313 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C G ( B  -  C ) )  =  ( ( C G B )  -  ( C G C ) ) )
114sigarid 38461 . . . . . 6  |-  ( C  e.  CC  ->  ( C G C )  =  0 )
122, 11syl 17 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C G C )  =  0 )
1312oveq2d 6304 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( C G B )  -  ( C G C ) )  =  ( ( C G B )  - 
0 ) )
144sigarim 38454 . . . . . . 7  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C G B )  e.  RR )
1514recnd 9666 . . . . . 6  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C G B )  e.  CC )
162, 1, 15syl2anc 666 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C G B )  e.  CC )
1716subid1d 9972 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( C G B )  -  0 )  =  ( C G B ) )
1810, 13, 173eqtrd 2488 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C G ( B  -  C ) )  =  ( C G B ) )
1918oveq2d 6304 . 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 2488 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 371    /\ w3a 984    = wceq 1443    e. wcel 1886   ` cfv 5581  (class class class)co 6288    |-> cmpt2 6290   CCcc 9534   0cc0 9536    x. cmul 9541    - cmin 9857   *ccj 13152   Imcim 13154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-po 4754  df-so 4755  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-2 10665  df-cj 13155  df-re 13156  df-im 13157
This theorem is referenced by:  sigarperm  38463
  Copyright terms: Public domain W3C validator