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Theorem sigarmf 27844
Description: Signed area is additive (with respect to subtraction) by the first 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
sigarmf  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  C
) G B )  =  ( ( A G B )  -  ( C G B ) ) )
Distinct variable groups:    x, y, A    x, B, y    x, C, y
Allowed substitution hints:    G( x, y)

Proof of Theorem sigarmf
StepHypRef Expression
1 cjsub 11634 . . . . . . 7  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( * `  ( A  -  C )
)  =  ( ( * `  A )  -  ( * `  C ) ) )
21oveq1d 5873 . . . . . 6  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( ( * `  ( A  -  C
) )  x.  B
)  =  ( ( ( * `  A
)  -  ( * `
 C ) )  x.  B ) )
323adant2 974 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( * `  ( A  -  C )
)  x.  B )  =  ( ( ( * `  A )  -  ( * `  C ) )  x.  B ) )
4 simp1 955 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  A  e.  CC )
54cjcld 11681 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
* `  A )  e.  CC )
6 simp3 957 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  C  e.  CC )
76cjcld 11681 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
* `  C )  e.  CC )
8 simp2 956 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  B  e.  CC )
95, 7, 8subdird 9236 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( * `  A )  -  (
* `  C )
)  x.  B )  =  ( ( ( * `  A )  x.  B )  -  ( ( * `  C )  x.  B
) ) )
103, 9eqtrd 2315 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( * `  ( A  -  C )
)  x.  B )  =  ( ( ( * `  A )  x.  B )  -  ( ( * `  C )  x.  B
) ) )
1110fveq2d 5529 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
Im `  ( (
* `  ( A  -  C ) )  x.  B ) )  =  ( Im `  (
( ( * `  A )  x.  B
)  -  ( ( * `  C )  x.  B ) ) ) )
125, 8mulcld 8855 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( * `  A
)  x.  B )  e.  CC )
137, 8mulcld 8855 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( * `  C
)  x.  B )  e.  CC )
1412, 13imsubd 11702 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
Im `  ( (
( * `  A
)  x.  B )  -  ( ( * `
 C )  x.  B ) ) )  =  ( ( Im
`  ( ( * `
 A )  x.  B ) )  -  ( Im `  ( ( * `  C )  x.  B ) ) ) )
1511, 14eqtrd 2315 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
Im `  ( (
* `  ( A  -  C ) )  x.  B ) )  =  ( ( Im `  ( ( * `  A )  x.  B
) )  -  (
Im `  ( (
* `  C )  x.  B ) ) ) )
164, 6subcld 9157 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  C )  e.  CC )
17 sigar . . . 4  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
1817sigarval 27840 . . 3  |-  ( ( ( A  -  C
)  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  C ) G B )  =  ( Im
`  ( ( * `
 ( A  -  C ) )  x.  B ) ) )
1916, 8, 18syl2anc 642 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  C
) G B )  =  ( Im `  ( ( * `  ( A  -  C
) )  x.  B
) ) )
2017sigarval 27840 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A G B )  =  ( Im
`  ( ( * `
 A )  x.  B ) ) )
21203adant3 975 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G B )  =  ( Im `  (
( * `  A
)  x.  B ) ) )
22 3simpc 954 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  e.  CC  /\  C  e.  CC ) )
2322ancomd 438 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C  e.  CC  /\  B  e.  CC ) )
2417sigarval 27840 . . . 4  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C G B )  =  ( Im
`  ( ( * `
 C )  x.  B ) ) )
2523, 24syl 15 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C G B )  =  ( Im `  (
( * `  C
)  x.  B ) ) )
2621, 25oveq12d 5876 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A G B )  -  ( C G B ) )  =  ( ( Im
`  ( ( * `
 A )  x.  B ) )  -  ( Im `  ( ( * `  C )  x.  B ) ) ) )
2715, 19, 263eqtr4d 2325 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  C
) G B )  =  ( ( A G B )  -  ( C G B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   CCcc 8735    x. cmul 8742    - cmin 9037   *ccj 11581   Imcim 11583
This theorem is referenced by:  sigarms  27846  sigarexp  27849  sigaradd  27856
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-2 9804  df-cj 11584  df-re 11585  df-im 11586
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