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Theorem sigarls 37899
Description: Signed area is linear 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
sigarls  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  RR )  ->  ( A G ( B  x.  C ) )  =  ( ( A G B )  x.  C
) )
Distinct variable groups:    x, y, A    x, B, y    x, C, y
Allowed substitution hints:    G( x, y)

Proof of Theorem sigarls
StepHypRef Expression
1 simp1 1005 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  RR )  ->  A  e.  CC )
21cjcld 13227 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  RR )  ->  (
* `  A )  e.  CC )
3 simp2 1006 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  RR )  ->  B  e.  CC )
4 simpr 462 . . . . . . 7  |-  ( ( B  e.  CC  /\  C  e.  RR )  ->  C  e.  RR )
54recnd 9658 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  RR )  ->  C  e.  CC )
653adant1 1023 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  RR )  ->  C  e.  CC )
72, 3, 6mulassd 9655 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  RR )  ->  (
( ( * `  A )  x.  B
)  x.  C )  =  ( ( * `
 A )  x.  ( B  x.  C
) ) )
87fveq2d 5876 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  RR )  ->  (
Im `  ( (
( * `  A
)  x.  B )  x.  C ) )  =  ( Im `  ( ( * `  A )  x.  ( B  x.  C )
) ) )
9 simp3 1007 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  RR )  ->  C  e.  RR )
102, 3mulcld 9652 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  RR )  ->  (
( * `  A
)  x.  B )  e.  CC )
119, 10immul2d 13259 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  RR )  ->  (
Im `  ( C  x.  ( ( * `  A )  x.  B
) ) )  =  ( C  x.  (
Im `  ( (
* `  A )  x.  B ) ) ) )
1210, 6mulcomd 9653 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  RR )  ->  (
( ( * `  A )  x.  B
)  x.  C )  =  ( C  x.  ( ( * `  A )  x.  B
) ) )
1312fveq2d 5876 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  RR )  ->  (
Im `  ( (
( * `  A
)  x.  B )  x.  C ) )  =  ( Im `  ( C  x.  (
( * `  A
)  x.  B ) ) ) )
14 imcl 13142 . . . . . . 7  |-  ( ( ( * `  A
)  x.  B )  e.  CC  ->  (
Im `  ( (
* `  A )  x.  B ) )  e.  RR )
1514recnd 9658 . . . . . 6  |-  ( ( ( * `  A
)  x.  B )  e.  CC  ->  (
Im `  ( (
* `  A )  x.  B ) )  e.  CC )
1610, 15syl 17 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  RR )  ->  (
Im `  ( (
* `  A )  x.  B ) )  e.  CC )
1716, 6mulcomd 9653 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  RR )  ->  (
( Im `  (
( * `  A
)  x.  B ) )  x.  C )  =  ( C  x.  ( Im `  ( ( * `  A )  x.  B ) ) ) )
1811, 13, 173eqtr4d 2471 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  RR )  ->  (
Im `  ( (
( * `  A
)  x.  B )  x.  C ) )  =  ( ( Im
`  ( ( * `
 A )  x.  B ) )  x.  C ) )
198, 18eqtr3d 2463 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  RR )  ->  (
Im `  ( (
* `  A )  x.  ( B  x.  C
) ) )  =  ( ( Im `  ( ( * `  A )  x.  B
) )  x.  C
) )
20 simpl 458 . . . . 5  |-  ( ( B  e.  CC  /\  C  e.  RR )  ->  B  e.  CC )
2120, 5mulcld 9652 . . . 4  |-  ( ( B  e.  CC  /\  C  e.  RR )  ->  ( B  x.  C
)  e.  CC )
22213adant1 1023 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  RR )  ->  ( B  x.  C )  e.  CC )
23 sigar . . . 4  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
2423sigarval 37892 . . 3  |-  ( ( A  e.  CC  /\  ( B  x.  C
)  e.  CC )  ->  ( A G ( B  x.  C
) )  =  ( Im `  ( ( * `  A )  x.  ( B  x.  C ) ) ) )
251, 22, 24syl2anc 665 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  RR )  ->  ( A G ( B  x.  C ) )  =  ( Im `  (
( * `  A
)  x.  ( B  x.  C ) ) ) )
2623sigarval 37892 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A G B )  =  ( Im
`  ( ( * `
 A )  x.  B ) ) )
27263adant3 1025 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  RR )  ->  ( A G B )  =  ( Im `  (
( * `  A
)  x.  B ) ) )
2827oveq1d 6311 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  RR )  ->  (
( A G B )  x.  C )  =  ( ( Im
`  ( ( * `
 A )  x.  B ) )  x.  C ) )
2919, 25, 283eqtr4d 2471 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  RR )  ->  ( A G ( B  x.  C ) )  =  ( ( A G B )  x.  C
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   ` cfv 5592  (class class class)co 6296    |-> cmpt2 6298   CCcc 9526   RRcr 9527    x. cmul 9533   *ccj 13127   Imcim 13129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-po 4766  df-so 4767  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-2 10657  df-cj 13130  df-re 13131  df-im 13132
This theorem is referenced by:  sigarcol  37906  sharhght  37907  sigaradd  37908
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