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Theorem sigarcol 27957
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 968 . . . . . . 7  |-  ( ph  ->  B  e.  CC )
42simp3d 969 . . . . . . 7  |-  ( ph  ->  C  e.  CC )
52simp1d 967 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
63, 4, 53jca 1132 . . . . . 6  |-  ( ph  ->  ( B  e.  CC  /\  C  e.  CC  /\  A  e.  CC )
)
76adantr 451 . . . . 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 451 . . . . 5  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  -.  A  =  B )
101sigarperm 27953 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  C
) G ( B  -  C ) )  =  ( ( B  -  A ) G ( C  -  A
) ) )
112, 10syl 15 . . . . . . . 8  |-  ( ph  ->  ( ( A  -  C ) G ( B  -  C ) )  =  ( ( B  -  A ) G ( C  -  A ) ) )
121sigarperm 27953 . . . . . . . . 9  |-  ( ( B  e.  CC  /\  C  e.  CC  /\  A  e.  CC )  ->  (
( B  -  A
) G ( C  -  A ) )  =  ( ( C  -  B ) G ( A  -  B
) ) )
136, 12syl 15 . . . . . . . 8  |-  ( ph  ->  ( ( B  -  A ) G ( C  -  A ) )  =  ( ( C  -  B ) G ( A  -  B ) ) )
1411, 13eqtrd 2328 . . . . . . 7  |-  ( ph  ->  ( ( A  -  C ) G ( B  -  C ) )  =  ( ( C  -  B ) G ( A  -  B ) ) )
1514eqeq1d 2304 . . . . . 6  |-  ( ph  ->  ( ( ( A  -  C ) G ( B  -  C
) )  =  0  <-> 
( ( C  -  B ) G ( A  -  B ) )  =  0 ) )
1615biimpa 470 . . . . 5  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  (
( C  -  B
) G ( A  -  B ) )  =  0 )
171, 7, 9, 16sigardiv 27954 . . . 4  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  (
( C  -  B
)  /  ( A  -  B ) )  e.  RR )
184, 3subcld 9173 . . . . . . . 8  |-  ( ph  ->  ( C  -  B
)  e.  CC )
1918adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  ( C  -  B )  e.  CC )
205, 3subcld 9173 . . . . . . . 8  |-  ( ph  ->  ( A  -  B
)  e.  CC )
2120adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  ( A  -  B )  e.  CC )
225adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  A  e.  CC )
233adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  B  e.  CC )
249neneqad 2529 . . . . . . . 8  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  A  =/=  B )
2522, 23, 24subne0d 9182 . . . . . . 7  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  ( A  -  B )  =/=  0 )
2619, 21, 25divcan1d 9553 . . . . . 6  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  (
( ( C  -  B )  /  ( A  -  B )
)  x.  ( A  -  B ) )  =  ( C  -  B ) )
2726oveq2d 5890 . . . . 5  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  ( B  +  ( (
( C  -  B
)  /  ( A  -  B ) )  x.  ( A  -  B ) ) )  =  ( B  +  ( C  -  B
) ) )
284adantr 451 . . . . . 6  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  C  e.  CC )
2923, 28pncan3d 9176 . . . . 5  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  ( B  +  ( C  -  B ) )  =  C )
3027, 29eqtr2d 2329 . . . 4  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  C  =  ( B  +  ( ( ( C  -  B )  / 
( A  -  B
) )  x.  ( A  -  B )
) ) )
31 oveq1 5881 . . . . . . 7  |-  ( t  =  ( ( C  -  B )  / 
( A  -  B
) )  ->  (
t  x.  ( A  -  B ) )  =  ( ( ( C  -  B )  /  ( A  -  B ) )  x.  ( A  -  B
) ) )
3231oveq2d 5890 . . . . . 6  |-  ( t  =  ( ( C  -  B )  / 
( A  -  B
) )  ->  ( B  +  ( t  x.  ( A  -  B
) ) )  =  ( B  +  ( ( ( C  -  B )  /  ( A  -  B )
)  x.  ( A  -  B ) ) ) )
3332eqeq2d 2307 . . . . 5  |-  ( t  =  ( ( C  -  B )  / 
( A  -  B
) )  ->  ( C  =  ( B  +  ( t  x.  ( A  -  B
) ) )  <->  C  =  ( B  +  (
( ( C  -  B )  /  ( A  -  B )
)  x.  ( A  -  B ) ) ) ) )
3433rspcev 2897 . . . 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 642 . . 3  |-  ( (
ph  /\  ( ( A  -  C ) G ( B  -  C ) )  =  0 )  ->  E. t  e.  RR  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )
3635ex 423 . 2  |-  ( ph  ->  ( ( ( A  -  C ) G ( B  -  C
) )  =  0  ->  E. t  e.  RR  C  =  ( B  +  ( t  x.  ( A  -  B
) ) ) ) )
37143ad2ant1 976 . . . 4  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( ( A  -  C ) G ( B  -  C ) )  =  ( ( C  -  B ) G ( A  -  B ) ) )
38 simp3 957 . . . . . . . 8  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  C  =  ( B  +  (
t  x.  ( A  -  B ) ) ) )
3938oveq1d 5889 . . . . . . 7  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( C  -  B )  =  ( ( B  +  ( t  x.  ( A  -  B ) ) )  -  B ) )
4033ad2ant1 976 . . . . . . . 8  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  B  e.  CC )
41 simp2 956 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  t  e.  RR )
4241recnd 8877 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  t  e.  CC )
4353ad2ant1 976 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  A  e.  CC )
4443, 40subcld 9173 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( A  -  B )  e.  CC )
4542, 44mulcld 8871 . . . . . . . 8  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( t  x.  ( A  -  B
) )  e.  CC )
4640, 45pncan2d 9175 . . . . . . 7  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( ( B  +  ( t  x.  ( A  -  B
) ) )  -  B )  =  ( t  x.  ( A  -  B ) ) )
4739, 46eqtrd 2328 . . . . . 6  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( C  -  B )  =  ( t  x.  ( A  -  B ) ) )
4847oveq1d 5889 . . . . 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 8872 . . . . . 6  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( t  x.  ( A  -  B
) )  =  ( ( A  -  B
)  x.  t ) )
5049oveq1d 5889 . . . . 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 2328 . . . 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 8871 . . . . . 6  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( ( A  -  B )  x.  t )  e.  CC )
531sigarac 27945 . . . . . 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 642 . . . . 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 27950 . . . . . . . 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 1182 . . . . . . 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 27951 . . . . . . . . 9  |-  ( ( A  -  B )  e.  CC  ->  (
( A  -  B
) G ( A  -  B ) )  =  0 )
5844, 57syl 15 . . . . . . . 8  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( ( A  -  B ) G ( A  -  B ) )  =  0 )
5958oveq1d 5889 . . . . . . 7  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( (
( A  -  B
) G ( A  -  B ) )  x.  t )  =  ( 0  x.  t
) )
6042mul02d 9026 . . . . . . 7  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( 0  x.  t )  =  0 )
6156, 59, 603eqtrd 2332 . . . . . 6  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( ( A  -  B ) G ( ( A  -  B )  x.  t ) )  =  0 )
6261negeqd 9062 . . . . 5  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  -u ( ( A  -  B ) G ( ( A  -  B )  x.  t ) )  = 
-u 0 )
63 neg0 9109 . . . . . 6  |-  -u 0  =  0
6463a1i 10 . . . . 5  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  -u 0  =  0 )
6554, 62, 643eqtrd 2332 . . . 4  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( (
( A  -  B
)  x.  t ) G ( A  -  B ) )  =  0 )
6637, 51, 653eqtrd 2332 . . 3  |-  ( (
ph  /\  t  e.  RR  /\  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) )  ->  ( ( A  -  C ) G ( B  -  C ) )  =  0 )
6766rexlimdv3a 2682 . 2  |-  ( ph  ->  ( E. t  e.  RR  C  =  ( B  +  ( t  x.  ( A  -  B ) ) )  ->  ( ( A  -  C ) G ( B  -  C
) )  =  0 ) )
6836, 67impbid 183 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E.wrex 2557   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   CCcc 8751   RRcr 8752   0cc0 8753    + caddc 8756    x. cmul 8758    - cmin 9053   -ucneg 9054    / cdiv 9439   *ccj 11597   Imcim 11599
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-2 9820  df-cj 11600  df-re 11601  df-im 11602
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