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

Theorem sigarcol 27522
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 27518 . . . . . . . . 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 27518 . . . . . . . . 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 2419 . . . . . . 7  |-  ( ph  ->  ( ( A  -  C ) G ( B  -  C ) )  =  ( ( C  -  B ) G ( A  -  B ) ) )
1514eqeq1d 2395 . . . . . 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 27519 . . . 4  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  (
( C  -  B
)  /  ( A  -  B ) )  e.  RR )
184, 3subcld 9343 . . . . . . . 8  |-  ( ph  ->  ( C  -  B
)  e.  CC )
1918adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  ( C  -  B )  e.  CC )
205, 3subcld 9343 . . . . . . . 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 2620 . . . . . . . 8  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  A  =/=  B )
2522, 23, 24subne0d 9352 . . . . . . 7  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  ( A  -  B )  =/=  0 )
2619, 21, 25divcan1d 9723 . . . . . 6  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  (
( ( C  -  B )  /  ( A  -  B )
)  x.  ( A  -  B ) )  =  ( C  -  B ) )
2726oveq2d 6036 . . . . 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 9346 . . . . 5  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  ( B  +  ( C  -  B ) )  =  C )
3027, 29eqtr2d 2420 . . . 4  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  C  =  ( B  +  ( ( ( C  -  B )  / 
( A  -  B
) )  x.  ( A  -  B )
) ) )
31 oveq1 6027 . . . . . . 7  |-  ( t  =  ( ( C  -  B )  / 
( A  -  B
) )  ->  (
t  x.  ( A  -  B ) )  =  ( ( ( C  -  B )  /  ( A  -  B ) )  x.  ( A  -  B
) ) )
3231oveq2d 6036 . . . . . 6  |-  ( t  =  ( ( C  -  B )  / 
( A  -  B
) )  ->  ( B  +  ( t  x.  ( A  -  B
) ) )  =  ( B  +  ( ( ( C  -  B )  /  ( A  -  B )
)  x.  ( A  -  B ) ) ) )
3332eqeq2d 2398 . . . . 5  |-  ( t  =  ( ( C  -  B )  / 
( A  -  B
) )  ->  ( C  =  ( B  +  ( t  x.  ( A  -  B
) ) )  <->  C  =  ( B  +  (
( ( C  -  B )  /  ( A  -  B )
)  x.  ( A  -  B ) ) ) ) )
3433rspcev 2995 . . . 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 6035 . . . . . . 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 9047 . . . . . . . . 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 9343 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( A  -  B )  e.  CC )
4542, 44mulcld 9041 . . . . . . . 8  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( t  x.  ( A  -  B
) )  e.  CC )
4640, 45pncan2d 9345 . . . . . . 7  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( ( B  +  ( t  x.  ( A  -  B
) ) )  -  B )  =  ( t  x.  ( A  -  B ) ) )
4739, 46eqtrd 2419 . . . . . 6  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( C  -  B )  =  ( t  x.  ( A  -  B ) ) )
4847oveq1d 6035 . . . . 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 9042 . . . . . 6  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( t  x.  ( A  -  B
) )  =  ( ( A  -  B
)  x.  t ) )
5049oveq1d 6035 . . . . 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 2419 . . . 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 9041 . . . . . 6  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( ( A  -  B )  x.  t )  e.  CC )
531sigarac 27510 . . . . . 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 27515 . . . . . . . 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 27516 . . . . . . . . 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 6035 . . . . . . 7  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( (
( A  -  B
) G ( A  -  B ) )  x.  t )  =  ( 0  x.  t
) )
6042mul02d 9196 . . . . . . 7  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( 0  x.  t )  =  0 )
6156, 59, 603eqtrd 2423 . . . . . 6  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( ( A  -  B ) G ( ( A  -  B )  x.  t ) )  =  0 )
6261negeqd 9232 . . . . 5  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  -u ( ( A  -  B ) G ( ( A  -  B )  x.  t ) )  = 
-u 0 )
63 neg0 9279 . . . . . 6  |-  -u 0  =  0
6463a1i 11 . . . . 5  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  -u 0  =  0 )
6554, 62, 643eqtrd 2423 . . . 4  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( (
( A  -  B
)  x.  t ) G ( A  -  B ) )  =  0 )
6637, 51, 653eqtrd 2423 . . 3  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( ( A  -  C ) G ( B  -  C ) )  =  0 )
6766rexlimdv3a 2775 . 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 1649    e. wcel 1717   E.wrex 2650   ` cfv 5394  (class class class)co 6020    e. cmpt2 6022   CCcc 8921   RRcr 8922   0cc0 8923    + caddc 8926    x. cmul 8928    - cmin 9223   -ucneg 9224    / cdiv 9609   *ccj 11828   Imcim 11830
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-po 4444  df-so 4445  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-riota 6485  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-2 9990  df-cj 11831  df-re 11832  df-im 11833
  Copyright terms: Public domain W3C validator