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Theorem sigarcol 27821
Description: Given three points  A,  B and  C such that  -.  A  =  B, the point  C lies on the line going through  A and  B iff the corresponding signed area is zero. That justifies the usage of signed area as a collinearity indicator. (Contributed by Saveliy Skresanov, 22-Sep-2017.)
Hypotheses
Ref Expression
sigarcol.sigar  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
sigarcol.a  |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
)
sigarcol.b  |-  ( ph  ->  -.  A  =  B )
Assertion
Ref Expression
sigarcol  |-  ( ph  ->  ( ( ( A  -  C ) G ( B  -  C
) )  =  0  <->  E. t  e.  RR  C  =  ( B  +  ( t  x.  ( A  -  B
) ) ) ) )
Distinct variable groups:    x, t,
y, A    t, B, x, y    t, C, x, y    t, G    ph, t
Allowed substitution hints:    ph( x, y)    G( x, y)

Proof of Theorem sigarcol
StepHypRef Expression
1 sigarcol.sigar . . . . 5  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
2 sigarcol.a . . . . . . . 8  |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
)
32simp2d 970 . . . . . . 7  |-  ( ph  ->  B  e.  CC )
42simp3d 971 . . . . . . 7  |-  ( ph  ->  C  e.  CC )
52simp1d 969 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
63, 4, 53jca 1134 . . . . . 6  |-  ( ph  ->  ( B  e.  CC  /\  C  e.  CC  /\  A  e.  CC )
)
76adantr 452 . . . . 5  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  ( B  e.  CC  /\  C  e.  CC  /\  A  e.  CC ) )
8 sigarcol.b . . . . . 6  |-  ( ph  ->  -.  A  =  B )
98adantr 452 . . . . 5  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  -.  A  =  B )
101sigarperm 27817 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  C
) G ( B  -  C ) )  =  ( ( B  -  A ) G ( C  -  A
) ) )
112, 10syl 16 . . . . . . . 8  |-  ( ph  ->  ( ( A  -  C ) G ( B  -  C ) )  =  ( ( B  -  A ) G ( C  -  A ) ) )
121sigarperm 27817 . . . . . . . . 9  |-  ( ( B  e.  CC  /\  C  e.  CC  /\  A  e.  CC )  ->  (
( B  -  A
) G ( C  -  A ) )  =  ( ( C  -  B ) G ( A  -  B
) ) )
136, 12syl 16 . . . . . . . 8  |-  ( ph  ->  ( ( B  -  A ) G ( C  -  A ) )  =  ( ( C  -  B ) G ( A  -  B ) ) )
1411, 13eqtrd 2467 . . . . . . 7  |-  ( ph  ->  ( ( A  -  C ) G ( B  -  C ) )  =  ( ( C  -  B ) G ( A  -  B ) ) )
1514eqeq1d 2443 . . . . . 6  |-  ( ph  ->  ( ( ( A  -  C ) G ( B  -  C
) )  =  0  <-> 
( ( C  -  B ) G ( A  -  B ) )  =  0 ) )
1615biimpa 471 . . . . 5  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  (
( C  -  B
) G ( A  -  B ) )  =  0 )
171, 7, 9, 16sigardiv 27818 . . . 4  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  (
( C  -  B
)  /  ( A  -  B ) )  e.  RR )
184, 3subcld 9403 . . . . . . . 8  |-  ( ph  ->  ( C  -  B
)  e.  CC )
1918adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  ( C  -  B )  e.  CC )
205, 3subcld 9403 . . . . . . . 8  |-  ( ph  ->  ( A  -  B
)  e.  CC )
2120adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  ( A  -  B )  e.  CC )
225adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  A  e.  CC )
233adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  B  e.  CC )
249neneqad 2668 . . . . . . . 8  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  A  =/=  B )
2522, 23, 24subne0d 9412 . . . . . . 7  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  ( A  -  B )  =/=  0 )
2619, 21, 25divcan1d 9783 . . . . . 6  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  (
( ( C  -  B )  /  ( A  -  B )
)  x.  ( A  -  B ) )  =  ( C  -  B ) )
2726oveq2d 6089 . . . . 5  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  ( B  +  ( (
( C  -  B
)  /  ( A  -  B ) )  x.  ( A  -  B ) ) )  =  ( B  +  ( C  -  B
) ) )
284adantr 452 . . . . . 6  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  C  e.  CC )
2923, 28pncan3d 9406 . . . . 5  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  ( B  +  ( C  -  B ) )  =  C )
3027, 29eqtr2d 2468 . . . 4  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  C  =  ( B  +  ( ( ( C  -  B )  / 
( A  -  B
) )  x.  ( A  -  B )
) ) )
31 oveq1 6080 . . . . . . 7  |-  ( t  =  ( ( C  -  B )  / 
( A  -  B
) )  ->  (
t  x.  ( A  -  B ) )  =  ( ( ( C  -  B )  /  ( A  -  B ) )  x.  ( A  -  B
) ) )
3231oveq2d 6089 . . . . . 6  |-  ( t  =  ( ( C  -  B )  / 
( A  -  B
) )  ->  ( B  +  ( t  x.  ( A  -  B
) ) )  =  ( B  +  ( ( ( C  -  B )  /  ( A  -  B )
)  x.  ( A  -  B ) ) ) )
3332eqeq2d 2446 . . . . 5  |-  ( t  =  ( ( C  -  B )  / 
( A  -  B
) )  ->  ( C  =  ( B  +  ( t  x.  ( A  -  B
) ) )  <->  C  =  ( B  +  (
( ( C  -  B )  /  ( A  -  B )
)  x.  ( A  -  B ) ) ) ) )
3433rspcev 3044 . . . 4  |-  ( ( ( ( C  -  B )  /  ( A  -  B )
)  e.  RR  /\  C  =  ( B  +  ( ( ( C  -  B )  /  ( A  -  B ) )  x.  ( A  -  B
) ) ) )  ->  E. t  e.  RR  C  =  ( B  +  ( t  x.  ( A  -  B
) ) ) )
3517, 30, 34syl2anc 643 . . 3  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  E. t  e.  RR  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )
3635ex 424 . 2  |-  ( ph  ->  ( ( ( A  -  C ) G ( B  -  C
) )  =  0  ->  E. t  e.  RR  C  =  ( B  +  ( t  x.  ( A  -  B
) ) ) ) )
37143ad2ant1 978 . . . 4  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( ( A  -  C ) G ( B  -  C ) )  =  ( ( C  -  B ) G ( A  -  B ) ) )
38 simp3 959 . . . . . . . 8  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  C  =  ( B  +  (
t  x.  ( A  -  B ) ) ) )
3938oveq1d 6088 . . . . . . 7  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( C  -  B )  =  ( ( B  +  ( t  x.  ( A  -  B ) ) )  -  B ) )
4033ad2ant1 978 . . . . . . . 8  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  B  e.  CC )
41 simp2 958 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  t  e.  RR )
4241recnd 9106 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  t  e.  CC )
4353ad2ant1 978 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  A  e.  CC )
4443, 40subcld 9403 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( A  -  B )  e.  CC )
4542, 44mulcld 9100 . . . . . . . 8  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( t  x.  ( A  -  B
) )  e.  CC )
4640, 45pncan2d 9405 . . . . . . 7  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( ( B  +  ( t  x.  ( A  -  B
) ) )  -  B )  =  ( t  x.  ( A  -  B ) ) )
4739, 46eqtrd 2467 . . . . . 6  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( C  -  B )  =  ( t  x.  ( A  -  B ) ) )
4847oveq1d 6088 . . . . 5  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( ( C  -  B ) G ( A  -  B ) )  =  ( ( t  x.  ( A  -  B
) ) G ( A  -  B ) ) )
4942, 44mulcomd 9101 . . . . . 6  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( t  x.  ( A  -  B
) )  =  ( ( A  -  B
)  x.  t ) )
5049oveq1d 6088 . . . . 5  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( (
t  x.  ( A  -  B ) ) G ( A  -  B ) )  =  ( ( ( A  -  B )  x.  t ) G ( A  -  B ) ) )
5148, 50eqtrd 2467 . . . 4  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( ( C  -  B ) G ( A  -  B ) )  =  ( ( ( A  -  B )  x.  t ) G ( A  -  B ) ) )
5244, 42mulcld 9100 . . . . . 6  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( ( A  -  B )  x.  t )  e.  CC )
531sigarac 27809 . . . . . 6  |-  ( ( ( ( A  -  B )  x.  t
)  e.  CC  /\  ( A  -  B
)  e.  CC )  ->  ( ( ( A  -  B )  x.  t ) G ( A  -  B
) )  =  -u ( ( A  -  B ) G ( ( A  -  B
)  x.  t ) ) )
5452, 44, 53syl2anc 643 . . . . 5  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( (
( A  -  B
)  x.  t ) G ( A  -  B ) )  = 
-u ( ( A  -  B ) G ( ( A  -  B )  x.  t
) ) )
551sigarls 27814 . . . . . . . 8  |-  ( ( ( A  -  B
)  e.  CC  /\  ( A  -  B
)  e.  CC  /\  t  e.  RR )  ->  ( ( A  -  B ) G ( ( A  -  B
)  x.  t ) )  =  ( ( ( A  -  B
) G ( A  -  B ) )  x.  t ) )
5644, 44, 41, 55syl3anc 1184 . . . . . . 7  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( ( A  -  B ) G ( ( A  -  B )  x.  t ) )  =  ( ( ( A  -  B ) G ( A  -  B
) )  x.  t
) )
571sigarid 27815 . . . . . . . . 9  |-  ( ( A  -  B )  e.  CC  ->  (
( A  -  B
) G ( A  -  B ) )  =  0 )
5844, 57syl 16 . . . . . . . 8  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( ( A  -  B ) G ( A  -  B ) )  =  0 )
5958oveq1d 6088 . . . . . . 7  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( (
( A  -  B
) G ( A  -  B ) )  x.  t )  =  ( 0  x.  t
) )
6042mul02d 9256 . . . . . . 7  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( 0  x.  t )  =  0 )
6156, 59, 603eqtrd 2471 . . . . . 6  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( ( A  -  B ) G ( ( A  -  B )  x.  t ) )  =  0 )
6261negeqd 9292 . . . . 5  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  -u ( ( A  -  B ) G ( ( A  -  B )  x.  t ) )  = 
-u 0 )
63 neg0 9339 . . . . . 6  |-  -u 0  =  0
6463a1i 11 . . . . 5  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  -u 0  =  0 )
6554, 62, 643eqtrd 2471 . . . 4  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( (
( A  -  B
)  x.  t ) G ( A  -  B ) )  =  0 )
6637, 51, 653eqtrd 2471 . . 3  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( ( A  -  C ) G ( B  -  C ) )  =  0 )
6766rexlimdv3a 2824 . 2  |-  ( ph  ->  ( E. t  e.  RR  C  =  ( B  +  ( t  x.  ( A  -  B ) ) )  ->  ( ( A  -  C ) G ( B  -  C
) )  =  0 ) )
6836, 67impbid 184 1  |-  ( ph  ->  ( ( ( A  -  C ) G ( B  -  C
) )  =  0  <->  E. t  e.  RR  C  =  ( B  +  ( t  x.  ( A  -  B
) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   E.wrex 2698   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   CCcc 8980   RRcr 8981   0cc0 8982    + caddc 8985    x. cmul 8987    - cmin 9283   -ucneg 9284    / cdiv 9669   *ccj 11893   Imcim 11895
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-2 10050  df-cj 11896  df-re 11897  df-im 11898
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