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Theorem sigardiv 38616
Description: If signed area between vectors  B  -  A and  C  -  A is zero, then those vectors lie on the same line. (Contributed by Saveliy Skresanov, 22-Sep-2017.)
Hypotheses
Ref Expression
sigar  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
sigardiv.a  |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
)
sigardiv.b  |-  ( ph  ->  -.  C  =  A )
sigardiv.c  |-  ( ph  ->  ( ( B  -  A ) G ( C  -  A ) )  =  0 )
Assertion
Ref Expression
sigardiv  |-  ( ph  ->  ( ( B  -  A )  /  ( C  -  A )
)  e.  RR )
Distinct variable groups:    x, y, A    x, B, y    x, C, y
Allowed substitution hints:    ph( x, y)    G( x, y)

Proof of Theorem sigardiv
StepHypRef Expression
1 sigardiv.a . . . . . . . 8  |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
)
21simp2d 1043 . . . . . . 7  |-  ( ph  ->  B  e.  CC )
31simp1d 1042 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
42, 3subcld 10005 . . . . . 6  |-  ( ph  ->  ( B  -  A
)  e.  CC )
51simp3d 1044 . . . . . . 7  |-  ( ph  ->  C  e.  CC )
65, 3subcld 10005 . . . . . 6  |-  ( ph  ->  ( C  -  A
)  e.  CC )
7 sigardiv.b . . . . . . . 8  |-  ( ph  ->  -.  C  =  A )
87neqned 2650 . . . . . . 7  |-  ( ph  ->  C  =/=  A )
95, 3, 8subne0d 10014 . . . . . 6  |-  ( ph  ->  ( C  -  A
)  =/=  0 )
104, 6, 9cjdivd 13363 . . . . 5  |-  ( ph  ->  ( * `  (
( B  -  A
)  /  ( C  -  A ) ) )  =  ( ( * `  ( B  -  A ) )  /  ( * `  ( C  -  A
) ) ) )
114cjcld 13336 . . . . . . 7  |-  ( ph  ->  ( * `  ( B  -  A )
)  e.  CC )
126cjcld 13336 . . . . . . 7  |-  ( ph  ->  ( * `  ( C  -  A )
)  e.  CC )
136, 9cjne0d 13343 . . . . . . 7  |-  ( ph  ->  ( * `  ( C  -  A )
)  =/=  0 )
1411, 12, 6, 13, 9divcan5rd 10432 . . . . . 6  |-  ( ph  ->  ( ( ( * `
 ( B  -  A ) )  x.  ( C  -  A
) )  /  (
( * `  ( C  -  A )
)  x.  ( C  -  A ) ) )  =  ( ( * `  ( B  -  A ) )  /  ( * `  ( C  -  A
) ) ) )
1511, 6mulcld 9681 . . . . . . . 8  |-  ( ph  ->  ( ( * `  ( B  -  A
) )  x.  ( C  -  A )
)  e.  CC )
16 sigar . . . . . . . . . . 11  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
1716sigarval 38605 . . . . . . . . . 10  |-  ( ( ( B  -  A
)  e.  CC  /\  ( C  -  A
)  e.  CC )  ->  ( ( B  -  A ) G ( C  -  A
) )  =  ( Im `  ( ( * `  ( B  -  A ) )  x.  ( C  -  A ) ) ) )
184, 6, 17syl2anc 673 . . . . . . . . 9  |-  ( ph  ->  ( ( B  -  A ) G ( C  -  A ) )  =  ( Im
`  ( ( * `
 ( B  -  A ) )  x.  ( C  -  A
) ) ) )
19 sigardiv.c . . . . . . . . 9  |-  ( ph  ->  ( ( B  -  A ) G ( C  -  A ) )  =  0 )
2018, 19eqtr3d 2507 . . . . . . . 8  |-  ( ph  ->  ( Im `  (
( * `  ( B  -  A )
)  x.  ( C  -  A ) ) )  =  0 )
2115, 20reim0bd 13340 . . . . . . 7  |-  ( ph  ->  ( ( * `  ( B  -  A
) )  x.  ( C  -  A )
)  e.  RR )
226, 12mulcomd 9682 . . . . . . . 8  |-  ( ph  ->  ( ( C  -  A )  x.  (
* `  ( C  -  A ) ) )  =  ( ( * `
 ( C  -  A ) )  x.  ( C  -  A
) ) )
236cjmulrcld 13346 . . . . . . . 8  |-  ( ph  ->  ( ( C  -  A )  x.  (
* `  ( C  -  A ) ) )  e.  RR )
2422, 23eqeltrrd 2550 . . . . . . 7  |-  ( ph  ->  ( ( * `  ( C  -  A
) )  x.  ( C  -  A )
)  e.  RR )
2512, 6, 13, 9mulne0d 10286 . . . . . . 7  |-  ( ph  ->  ( ( * `  ( C  -  A
) )  x.  ( C  -  A )
)  =/=  0 )
2621, 24, 25redivcld 10457 . . . . . 6  |-  ( ph  ->  ( ( ( * `
 ( B  -  A ) )  x.  ( C  -  A
) )  /  (
( * `  ( C  -  A )
)  x.  ( C  -  A ) ) )  e.  RR )
2714, 26eqeltrrd 2550 . . . . 5  |-  ( ph  ->  ( ( * `  ( B  -  A
) )  /  (
* `  ( C  -  A ) ) )  e.  RR )
2810, 27eqeltrd 2549 . . . 4  |-  ( ph  ->  ( * `  (
( B  -  A
)  /  ( C  -  A ) ) )  e.  RR )
2928cjred 13366 . . 3  |-  ( ph  ->  ( * `  (
* `  ( ( B  -  A )  /  ( C  -  A ) ) ) )  =  ( * `
 ( ( B  -  A )  / 
( C  -  A
) ) ) )
304, 6, 9divcld 10405 . . . 4  |-  ( ph  ->  ( ( B  -  A )  /  ( C  -  A )
)  e.  CC )
3130cjcjd 13339 . . 3  |-  ( ph  ->  ( * `  (
* `  ( ( B  -  A )  /  ( C  -  A ) ) ) )  =  ( ( B  -  A )  /  ( C  -  A ) ) )
3229, 31eqtr3d 2507 . 2  |-  ( ph  ->  ( * `  (
( B  -  A
)  /  ( C  -  A ) ) )  =  ( ( B  -  A )  /  ( C  -  A ) ) )
3332, 28eqeltrrd 2550 1  |-  ( ph  ->  ( ( B  -  A )  /  ( C  -  A )
)  e.  RR )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ w3a 1007    = wceq 1452    e. wcel 1904   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   CCcc 9555   RRcr 9556   0cc0 9557    x. cmul 9562    - cmin 9880    / cdiv 10291   *ccj 13236   Imcim 13238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-po 4760  df-so 4761  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-2 10690  df-cj 13239  df-re 13240  df-im 13241
This theorem is referenced by:  sigarcol  38619  sharhght  38620
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