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Theorem sigaradd 27856
Description: Subtracting (double) area of  A D C from  A B C yields the (double) area of  D B C. (Contributed by Saveliy Skresanov, 23-Sep-2017.)
Hypotheses
Ref Expression
sharhght.sigar  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
sharhght.a  |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
)
sharhght.b  |-  ( ph  ->  ( D  e.  CC  /\  ( ( A  -  D ) G ( B  -  D ) )  =  0 ) )
Assertion
Ref Expression
sigaradd  |-  ( ph  ->  ( ( ( B  -  C ) G ( A  -  C
) )  -  (
( D  -  C
) G ( A  -  C ) ) )  =  ( ( B  -  C ) G ( D  -  C ) ) )
Distinct variable groups:    x, y, A    x, B, y    x, C, y    x, D, y
Allowed substitution hints:    ph( x, y)    G( x, y)

Proof of Theorem sigaradd
StepHypRef Expression
1 sharhght.a . . . . . . 7  |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
)
21simp2d 968 . . . . . 6  |-  ( ph  ->  B  e.  CC )
31simp3d 969 . . . . . 6  |-  ( ph  ->  C  e.  CC )
42, 3subcld 9157 . . . . 5  |-  ( ph  ->  ( B  -  C
)  e.  CC )
51simp1d 967 . . . . . 6  |-  ( ph  ->  A  e.  CC )
65, 3subcld 9157 . . . . 5  |-  ( ph  ->  ( A  -  C
)  e.  CC )
7 sharhght.b . . . . . . 7  |-  ( ph  ->  ( D  e.  CC  /\  ( ( A  -  D ) G ( B  -  D ) )  =  0 ) )
87simpld 445 . . . . . 6  |-  ( ph  ->  D  e.  CC )
98, 3subcld 9157 . . . . 5  |-  ( ph  ->  ( D  -  C
)  e.  CC )
10 sharhght.sigar . . . . . 6  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
1110sigarmf 27844 . . . . 5  |-  ( ( ( B  -  C
)  e.  CC  /\  ( A  -  C
)  e.  CC  /\  ( D  -  C
)  e.  CC )  ->  ( ( ( B  -  C )  -  ( D  -  C ) ) G ( A  -  C
) )  =  ( ( ( B  -  C ) G ( A  -  C ) )  -  ( ( D  -  C ) G ( A  -  C ) ) ) )
124, 6, 9, 11syl3anc 1182 . . . 4  |-  ( ph  ->  ( ( ( B  -  C )  -  ( D  -  C
) ) G ( A  -  C ) )  =  ( ( ( B  -  C
) G ( A  -  C ) )  -  ( ( D  -  C ) G ( A  -  C
) ) ) )
132, 8, 3nnncan2d 9192 . . . . 5  |-  ( ph  ->  ( ( B  -  C )  -  ( D  -  C )
)  =  ( B  -  D ) )
1413oveq1d 5873 . . . 4  |-  ( ph  ->  ( ( ( B  -  C )  -  ( D  -  C
) ) G ( A  -  C ) )  =  ( ( B  -  D ) G ( A  -  C ) ) )
1512, 14eqtr3d 2317 . . 3  |-  ( ph  ->  ( ( ( B  -  C ) G ( A  -  C
) )  -  (
( D  -  C
) G ( A  -  C ) ) )  =  ( ( B  -  D ) G ( A  -  C ) ) )
165, 3, 8nnncan1d 9191 . . . . . 6  |-  ( ph  ->  ( ( A  -  C )  -  ( A  -  D )
)  =  ( D  -  C ) )
1716oveq2d 5874 . . . . 5  |-  ( ph  ->  ( ( B  -  D ) G ( ( A  -  C
)  -  ( A  -  D ) ) )  =  ( ( B  -  D ) G ( D  -  C ) ) )
182, 8subcld 9157 . . . . . 6  |-  ( ph  ->  ( B  -  D
)  e.  CC )
195, 8subcld 9157 . . . . . 6  |-  ( ph  ->  ( A  -  D
)  e.  CC )
2010sigarms 27846 . . . . . 6  |-  ( ( ( B  -  D
)  e.  CC  /\  ( A  -  C
)  e.  CC  /\  ( A  -  D
)  e.  CC )  ->  ( ( B  -  D ) G ( ( A  -  C )  -  ( A  -  D )
) )  =  ( ( ( B  -  D ) G ( A  -  C ) )  -  ( ( B  -  D ) G ( A  -  D ) ) ) )
2118, 6, 19, 20syl3anc 1182 . . . . 5  |-  ( ph  ->  ( ( B  -  D ) G ( ( A  -  C
)  -  ( A  -  D ) ) )  =  ( ( ( B  -  D
) G ( A  -  C ) )  -  ( ( B  -  D ) G ( A  -  D
) ) ) )
2217, 21eqtr3d 2317 . . . 4  |-  ( ph  ->  ( ( B  -  D ) G ( D  -  C ) )  =  ( ( ( B  -  D
) G ( A  -  C ) )  -  ( ( B  -  D ) G ( A  -  D
) ) ) )
2310sigarac 27842 . . . . . . . . 9  |-  ( ( ( A  -  D
)  e.  CC  /\  ( B  -  D
)  e.  CC )  ->  ( ( A  -  D ) G ( B  -  D
) )  =  -u ( ( B  -  D ) G ( A  -  D ) ) )
2419, 18, 23syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( ( A  -  D ) G ( B  -  D ) )  =  -u (
( B  -  D
) G ( A  -  D ) ) )
257simprd 449 . . . . . . . 8  |-  ( ph  ->  ( ( A  -  D ) G ( B  -  D ) )  =  0 )
2624, 25eqtr3d 2317 . . . . . . 7  |-  ( ph  -> 
-u ( ( B  -  D ) G ( A  -  D
) )  =  0 )
2726negeqd 9046 . . . . . 6  |-  ( ph  -> 
-u -u ( ( B  -  D ) G ( A  -  D
) )  =  -u
0 )
2818, 19jca 518 . . . . . . . 8  |-  ( ph  ->  ( ( B  -  D )  e.  CC  /\  ( A  -  D
)  e.  CC ) )
2910, 28sigarimcd 27852 . . . . . . 7  |-  ( ph  ->  ( ( B  -  D ) G ( A  -  D ) )  e.  CC )
3029negnegd 9148 . . . . . 6  |-  ( ph  -> 
-u -u ( ( B  -  D ) G ( A  -  D
) )  =  ( ( B  -  D
) G ( A  -  D ) ) )
31 neg0 9093 . . . . . . 7  |-  -u 0  =  0
3231a1i 10 . . . . . 6  |-  ( ph  -> 
-u 0  =  0 )
3327, 30, 323eqtr3d 2323 . . . . 5  |-  ( ph  ->  ( ( B  -  D ) G ( A  -  D ) )  =  0 )
3433oveq2d 5874 . . . 4  |-  ( ph  ->  ( ( ( B  -  D ) G ( A  -  C
) )  -  (
( B  -  D
) G ( A  -  D ) ) )  =  ( ( ( B  -  D
) G ( A  -  C ) )  -  0 ) )
3518, 6jca 518 . . . . . 6  |-  ( ph  ->  ( ( B  -  D )  e.  CC  /\  ( A  -  C
)  e.  CC ) )
3610, 35sigarimcd 27852 . . . . 5  |-  ( ph  ->  ( ( B  -  D ) G ( A  -  C ) )  e.  CC )
3736subid1d 9146 . . . 4  |-  ( ph  ->  ( ( ( B  -  D ) G ( A  -  C
) )  -  0 )  =  ( ( B  -  D ) G ( A  -  C ) ) )
3822, 34, 373eqtrd 2319 . . 3  |-  ( ph  ->  ( ( B  -  D ) G ( D  -  C ) )  =  ( ( B  -  D ) G ( A  -  C ) ) )
3915, 38eqtr4d 2318 . 2  |-  ( ph  ->  ( ( ( B  -  C ) G ( A  -  C
) )  -  (
( D  -  C
) G ( A  -  C ) ) )  =  ( ( B  -  D ) G ( D  -  C ) ) )
403, 8subcld 9157 . . . 4  |-  ( ph  ->  ( C  -  D
)  e.  CC )
41 1re 8837 . . . . . 6  |-  1  e.  RR
4241a1i 10 . . . . 5  |-  ( ph  ->  1  e.  RR )
4342renegcld 9210 . . . 4  |-  ( ph  -> 
-u 1  e.  RR )
4410sigarls 27847 . . . 4  |-  ( ( ( B  -  D
)  e.  CC  /\  ( C  -  D
)  e.  CC  /\  -u 1  e.  RR )  ->  ( ( B  -  D ) G ( ( C  -  D )  x.  -u 1
) )  =  ( ( ( B  -  D ) G ( C  -  D ) )  x.  -u 1
) )
4518, 40, 43, 44syl3anc 1182 . . 3  |-  ( ph  ->  ( ( B  -  D ) G ( ( C  -  D
)  x.  -u 1
) )  =  ( ( ( B  -  D ) G ( C  -  D ) )  x.  -u 1
) )
4640mulm1d 9231 . . . . 5  |-  ( ph  ->  ( -u 1  x.  ( C  -  D
) )  =  -u ( C  -  D
) )
47 ax-1cn 8795 . . . . . . . 8  |-  1  e.  CC
4847a1i 10 . . . . . . 7  |-  ( ph  ->  1  e.  CC )
4948negcld 9144 . . . . . 6  |-  ( ph  -> 
-u 1  e.  CC )
5049, 40mulcomd 8856 . . . . 5  |-  ( ph  ->  ( -u 1  x.  ( C  -  D
) )  =  ( ( C  -  D
)  x.  -u 1
) )
513, 8negsubdi2d 9173 . . . . 5  |-  ( ph  -> 
-u ( C  -  D )  =  ( D  -  C ) )
5246, 50, 513eqtr3d 2323 . . . 4  |-  ( ph  ->  ( ( C  -  D )  x.  -u 1
)  =  ( D  -  C ) )
5352oveq2d 5874 . . 3  |-  ( ph  ->  ( ( B  -  D ) G ( ( C  -  D
)  x.  -u 1
) )  =  ( ( B  -  D
) G ( D  -  C ) ) )
5418, 40jca 518 . . . . . 6  |-  ( ph  ->  ( ( B  -  D )  e.  CC  /\  ( C  -  D
)  e.  CC ) )
5510, 54sigarimcd 27852 . . . . 5  |-  ( ph  ->  ( ( B  -  D ) G ( C  -  D ) )  e.  CC )
5655mulm1d 9231 . . . 4  |-  ( ph  ->  ( -u 1  x.  ( ( B  -  D ) G ( C  -  D ) ) )  =  -u ( ( B  -  D ) G ( C  -  D ) ) )
5755, 49mulcomd 8856 . . . 4  |-  ( ph  ->  ( ( ( B  -  D ) G ( C  -  D
) )  x.  -u 1
)  =  ( -u
1  x.  ( ( B  -  D ) G ( C  -  D ) ) ) )
5810sigarac 27842 . . . . 5  |-  ( ( ( C  -  D
)  e.  CC  /\  ( B  -  D
)  e.  CC )  ->  ( ( C  -  D ) G ( B  -  D
) )  =  -u ( ( B  -  D ) G ( C  -  D ) ) )
5940, 18, 58syl2anc 642 . . . 4  |-  ( ph  ->  ( ( C  -  D ) G ( B  -  D ) )  =  -u (
( B  -  D
) G ( C  -  D ) ) )
6056, 57, 593eqtr4d 2325 . . 3  |-  ( ph  ->  ( ( ( B  -  D ) G ( C  -  D
) )  x.  -u 1
)  =  ( ( C  -  D ) G ( B  -  D ) ) )
6145, 53, 603eqtr3d 2323 . 2  |-  ( ph  ->  ( ( B  -  D ) G ( D  -  C ) )  =  ( ( C  -  D ) G ( B  -  D ) ) )
6210sigarperm 27850 . . 3  |-  ( ( C  e.  CC  /\  B  e.  CC  /\  D  e.  CC )  ->  (
( C  -  D
) G ( B  -  D ) )  =  ( ( B  -  C ) G ( D  -  C
) ) )
633, 2, 8, 62syl3anc 1182 . 2  |-  ( ph  ->  ( ( C  -  D ) G ( B  -  D ) )  =  ( ( B  -  C ) G ( D  -  C ) ) )
6439, 61, 633eqtrd 2319 1  |-  ( ph  ->  ( ( ( B  -  C ) G ( A  -  C
) )  -  (
( D  -  C
) G ( A  -  C ) ) )  =  ( ( B  -  C ) G ( D  -  C ) ) )
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   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742    - cmin 9037   -ucneg 9038   *ccj 11581   Imcim 11583
This theorem is referenced by:  cevathlem2  27858
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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