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Theorem sigarcol 32035
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 1010 . . . . . . 7  |-  ( ph  ->  B  e.  CC )
42simp3d 1011 . . . . . . 7  |-  ( ph  ->  C  e.  CC )
52simp1d 1009 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
63, 4, 53jca 1177 . . . . . 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 32031 . . . . . . . . 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 32031 . . . . . . . . 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 2484 . . . . . . 7  |-  ( ph  ->  ( ( A  -  C ) G ( B  -  C ) )  =  ( ( C  -  B ) G ( A  -  B ) ) )
1514eqeq1d 2445 . . . . . 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 32032 . . . 4  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  (
( C  -  B
)  /  ( A  -  B ) )  e.  RR )
184, 3subcld 9936 . . . . . . . 8  |-  ( ph  ->  ( C  -  B
)  e.  CC )
1918adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  ( C  -  B )  e.  CC )
205, 3subcld 9936 . . . . . . . 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 )
249neqned 2646 . . . . . . . 8  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  A  =/=  B )
2522, 23, 24subne0d 9945 . . . . . . 7  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  ( A  -  B )  =/=  0 )
2619, 21, 25divcan1d 10328 . . . . . 6  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  (
( ( C  -  B )  /  ( A  -  B )
)  x.  ( A  -  B ) )  =  ( C  -  B ) )
2726oveq2d 6297 . . . . 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 9939 . . . . 5  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  ( B  +  ( C  -  B ) )  =  C )
3027, 29eqtr2d 2485 . . . 4  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  C  =  ( B  +  ( ( ( C  -  B )  / 
( A  -  B
) )  x.  ( A  -  B )
) ) )
31 oveq1 6288 . . . . . . 7  |-  ( t  =  ( ( C  -  B )  / 
( A  -  B
) )  ->  (
t  x.  ( A  -  B ) )  =  ( ( ( C  -  B )  /  ( A  -  B ) )  x.  ( A  -  B
) ) )
3231oveq2d 6297 . . . . . 6  |-  ( t  =  ( ( C  -  B )  / 
( A  -  B
) )  ->  ( B  +  ( t  x.  ( A  -  B
) ) )  =  ( B  +  ( ( ( C  -  B )  /  ( A  -  B )
)  x.  ( A  -  B ) ) ) )
3332eqeq2d 2457 . . . . 5  |-  ( t  =  ( ( C  -  B )  / 
( A  -  B
) )  ->  ( C  =  ( B  +  ( t  x.  ( A  -  B
) ) )  <->  C  =  ( B  +  (
( ( C  -  B )  /  ( A  -  B )
)  x.  ( A  -  B ) ) ) ) )
3433rspcev 3196 . . . 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 1018 . . . 4  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( ( A  -  C ) G ( B  -  C ) )  =  ( ( C  -  B ) G ( A  -  B ) ) )
38 simp3 999 . . . . . . . 8  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  C  =  ( B  +  (
t  x.  ( A  -  B ) ) ) )
3938oveq1d 6296 . . . . . . 7  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( C  -  B )  =  ( ( B  +  ( t  x.  ( A  -  B ) ) )  -  B ) )
4033ad2ant1 1018 . . . . . . . 8  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  B  e.  CC )
41 simp2 998 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  t  e.  RR )
4241recnd 9625 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  t  e.  CC )
4353ad2ant1 1018 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  A  e.  CC )
4443, 40subcld 9936 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( A  -  B )  e.  CC )
4542, 44mulcld 9619 . . . . . . . 8  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( t  x.  ( A  -  B
) )  e.  CC )
4640, 45pncan2d 9938 . . . . . . 7  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( ( B  +  ( t  x.  ( A  -  B
) ) )  -  B )  =  ( t  x.  ( A  -  B ) ) )
4739, 46eqtrd 2484 . . . . . 6  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( C  -  B )  =  ( t  x.  ( A  -  B ) ) )
4847oveq1d 6296 . . . . 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 9620 . . . . . 6  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( t  x.  ( A  -  B
) )  =  ( ( A  -  B
)  x.  t ) )
5049oveq1d 6296 . . . . 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 2484 . . . 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 9619 . . . . . 6  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( ( A  -  B )  x.  t )  e.  CC )
531sigarac 32023 . . . . . 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 32028 . . . . . . . 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 1229 . . . . . . 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 32029 . . . . . . . . 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 6296 . . . . . . 7  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( (
( A  -  B
) G ( A  -  B ) )  x.  t )  =  ( 0  x.  t
) )
6042mul02d 9781 . . . . . . 7  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( 0  x.  t )  =  0 )
6156, 59, 603eqtrd 2488 . . . . . 6  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( ( A  -  B ) G ( ( A  -  B )  x.  t ) )  =  0 )
6261negeqd 9819 . . . . 5  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  -u ( ( A  -  B ) G ( ( A  -  B )  x.  t ) )  = 
-u 0 )
63 neg0 9870 . . . . . 6  |-  -u 0  =  0
6463a1i 11 . . . . 5  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  -u 0  =  0 )
6554, 62, 643eqtrd 2488 . . . 4  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( (
( A  -  B
)  x.  t ) G ( A  -  B ) )  =  0 )
6637, 51, 653eqtrd 2488 . . 3  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( ( A  -  C ) G ( B  -  C ) )  =  0 )
6766rexlimdv3a 2937 . 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 974    = wceq 1383    e. wcel 1804   E.wrex 2794   ` cfv 5578  (class class class)co 6281    |-> cmpt2 6283   CCcc 9493   RRcr 9494   0cc0 9495    + caddc 9498    x. cmul 9500    - cmin 9810   -ucneg 9811    / cdiv 10213   *ccj 12911   Imcim 12913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-po 4790  df-so 4791  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-2 10601  df-cj 12914  df-re 12915  df-im 12916
This theorem is referenced by: (None)
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