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Theorem sigarcol 30047
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 1001 . . . . . . 7  |-  ( ph  ->  B  e.  CC )
42simp3d 1002 . . . . . . 7  |-  ( ph  ->  C  e.  CC )
52simp1d 1000 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
63, 4, 53jca 1168 . . . . . 6  |-  ( ph  ->  ( B  e.  CC  /\  C  e.  CC  /\  A  e.  CC )
)
76adantr 465 . . . . 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 465 . . . . 5  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  -.  A  =  B )
101sigarperm 30043 . . . . . . . . 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 30043 . . . . . . . . 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 2495 . . . . . . 7  |-  ( ph  ->  ( ( A  -  C ) G ( B  -  C ) )  =  ( ( C  -  B ) G ( A  -  B ) ) )
1514eqeq1d 2456 . . . . . 6  |-  ( ph  ->  ( ( ( A  -  C ) G ( B  -  C
) )  =  0  <-> 
( ( C  -  B ) G ( A  -  B ) )  =  0 ) )
1615biimpa 484 . . . . 5  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  (
( C  -  B
) G ( A  -  B ) )  =  0 )
171, 7, 9, 16sigardiv 30044 . . . 4  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  (
( C  -  B
)  /  ( A  -  B ) )  e.  RR )
184, 3subcld 9829 . . . . . . . 8  |-  ( ph  ->  ( C  -  B
)  e.  CC )
1918adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  ( C  -  B )  e.  CC )
205, 3subcld 9829 . . . . . . . 8  |-  ( ph  ->  ( A  -  B
)  e.  CC )
2120adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  ( A  -  B )  e.  CC )
225adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  A  e.  CC )
233adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  B  e.  CC )
249neneqad 2655 . . . . . . . 8  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  A  =/=  B )
2522, 23, 24subne0d 9838 . . . . . . 7  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  ( A  -  B )  =/=  0 )
2619, 21, 25divcan1d 10218 . . . . . 6  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  (
( ( C  -  B )  /  ( A  -  B )
)  x.  ( A  -  B ) )  =  ( C  -  B ) )
2726oveq2d 6215 . . . . 5  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  ( B  +  ( (
( C  -  B
)  /  ( A  -  B ) )  x.  ( A  -  B ) ) )  =  ( B  +  ( C  -  B
) ) )
284adantr 465 . . . . . 6  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  C  e.  CC )
2923, 28pncan3d 9832 . . . . 5  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  ( B  +  ( C  -  B ) )  =  C )
3027, 29eqtr2d 2496 . . . 4  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  C  =  ( B  +  ( ( ( C  -  B )  / 
( A  -  B
) )  x.  ( A  -  B )
) ) )
31 oveq1 6206 . . . . . . 7  |-  ( t  =  ( ( C  -  B )  / 
( A  -  B
) )  ->  (
t  x.  ( A  -  B ) )  =  ( ( ( C  -  B )  /  ( A  -  B ) )  x.  ( A  -  B
) ) )
3231oveq2d 6215 . . . . . 6  |-  ( t  =  ( ( C  -  B )  / 
( A  -  B
) )  ->  ( B  +  ( t  x.  ( A  -  B
) ) )  =  ( B  +  ( ( ( C  -  B )  /  ( A  -  B )
)  x.  ( A  -  B ) ) ) )
3332eqeq2d 2468 . . . . 5  |-  ( t  =  ( ( C  -  B )  / 
( A  -  B
) )  ->  ( C  =  ( B  +  ( t  x.  ( A  -  B
) ) )  <->  C  =  ( B  +  (
( ( C  -  B )  /  ( A  -  B )
)  x.  ( A  -  B ) ) ) ) )
3433rspcev 3177 . . . 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 661 . . 3  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  E. t  e.  RR  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )
3635ex 434 . 2  |-  ( ph  ->  ( ( ( A  -  C ) G ( B  -  C
) )  =  0  ->  E. t  e.  RR  C  =  ( B  +  ( t  x.  ( A  -  B
) ) ) ) )
37143ad2ant1 1009 . . . 4  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( ( A  -  C ) G ( B  -  C ) )  =  ( ( C  -  B ) G ( A  -  B ) ) )
38 simp3 990 . . . . . . . 8  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  C  =  ( B  +  (
t  x.  ( A  -  B ) ) ) )
3938oveq1d 6214 . . . . . . 7  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( C  -  B )  =  ( ( B  +  ( t  x.  ( A  -  B ) ) )  -  B ) )
4033ad2ant1 1009 . . . . . . . 8  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  B  e.  CC )
41 simp2 989 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  t  e.  RR )
4241recnd 9522 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  t  e.  CC )
4353ad2ant1 1009 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  A  e.  CC )
4443, 40subcld 9829 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( A  -  B )  e.  CC )
4542, 44mulcld 9516 . . . . . . . 8  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( t  x.  ( A  -  B
) )  e.  CC )
4640, 45pncan2d 9831 . . . . . . 7  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( ( B  +  ( t  x.  ( A  -  B
) ) )  -  B )  =  ( t  x.  ( A  -  B ) ) )
4739, 46eqtrd 2495 . . . . . 6  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( C  -  B )  =  ( t  x.  ( A  -  B ) ) )
4847oveq1d 6214 . . . . 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 9517 . . . . . 6  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( t  x.  ( A  -  B
) )  =  ( ( A  -  B
)  x.  t ) )
5049oveq1d 6214 . . . . 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 2495 . . . 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 9516 . . . . . 6  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( ( A  -  B )  x.  t )  e.  CC )
531sigarac 30035 . . . . . 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 661 . . . . 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 30040 . . . . . . . 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 1219 . . . . . . 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 30041 . . . . . . . . 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 6214 . . . . . . 7  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( (
( A  -  B
) G ( A  -  B ) )  x.  t )  =  ( 0  x.  t
) )
6042mul02d 9677 . . . . . . 7  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( 0  x.  t )  =  0 )
6156, 59, 603eqtrd 2499 . . . . . 6  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( ( A  -  B ) G ( ( A  -  B )  x.  t ) )  =  0 )
6261negeqd 9714 . . . . 5  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  -u ( ( A  -  B ) G ( ( A  -  B )  x.  t ) )  = 
-u 0 )
63 neg0 9765 . . . . . 6  |-  -u 0  =  0
6463a1i 11 . . . . 5  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  -u 0  =  0 )
6554, 62, 643eqtrd 2499 . . . 4  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( (
( A  -  B
)  x.  t ) G ( A  -  B ) )  =  0 )
6637, 51, 653eqtrd 2499 . . 3  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( ( A  -  C ) G ( B  -  C ) )  =  0 )
6766rexlimdv3a 2947 . 2  |-  ( ph  ->  ( E. t  e.  RR  C  =  ( B  +  ( t  x.  ( A  -  B ) ) )  ->  ( ( A  -  C ) G ( B  -  C
) )  =  0 ) )
6836, 67impbid 191 1  |-  ( ph  ->  ( ( ( A  -  C ) G ( B  -  C
) )  =  0  <->  E. t  e.  RR  C  =  ( B  +  ( t  x.  ( A  -  B
) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   E.wrex 2799   ` cfv 5525  (class class class)co 6199    |-> cmpt2 6201   CCcc 9390   RRcr 9391   0cc0 9392    + caddc 9395    x. cmul 9397    - cmin 9705   -ucneg 9706    / cdiv 10103   *ccj 12702   Imcim 12704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-po 4748  df-so 4749  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-2 10490  df-cj 12705  df-re 12706  df-im 12707
This theorem is referenced by: (None)
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