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Theorem sigardiv 27827
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 970 . . . . . . 7  |-  ( ph  ->  B  e.  CC )
31simp1d 969 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
42, 3subcld 9411 . . . . . 6  |-  ( ph  ->  ( B  -  A
)  e.  CC )
51simp3d 971 . . . . . . 7  |-  ( ph  ->  C  e.  CC )
65, 3subcld 9411 . . . . . 6  |-  ( ph  ->  ( C  -  A
)  e.  CC )
7 sigardiv.b . . . . . . . 8  |-  ( ph  ->  -.  C  =  A )
87neneqad 2674 . . . . . . 7  |-  ( ph  ->  C  =/=  A )
95, 3, 8subne0d 9420 . . . . . 6  |-  ( ph  ->  ( C  -  A
)  =/=  0 )
104, 6, 9cjdivd 12028 . . . . 5  |-  ( ph  ->  ( * `  (
( B  -  A
)  /  ( C  -  A ) ) )  =  ( ( * `  ( B  -  A ) )  /  ( * `  ( C  -  A
) ) ) )
114cjcld 12001 . . . . . . 7  |-  ( ph  ->  ( * `  ( B  -  A )
)  e.  CC )
126cjcld 12001 . . . . . . 7  |-  ( ph  ->  ( * `  ( C  -  A )
)  e.  CC )
136, 9cjne0d 12008 . . . . . . 7  |-  ( ph  ->  ( * `  ( C  -  A )
)  =/=  0 )
1411, 12, 6, 13, 9divcan5rd 9817 . . . . . 6  |-  ( ph  ->  ( ( ( * `
 ( B  -  A ) )  x.  ( C  -  A
) )  /  (
( * `  ( C  -  A )
)  x.  ( C  -  A ) ) )  =  ( ( * `  ( B  -  A ) )  /  ( * `  ( C  -  A
) ) ) )
1511, 6mulcld 9108 . . . . . . . 8  |-  ( ph  ->  ( ( * `  ( B  -  A
) )  x.  ( C  -  A )
)  e.  CC )
16 sigar . . . . . . . . . . 11  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
1716sigarval 27816 . . . . . . . . . 10  |-  ( ( ( B  -  A
)  e.  CC  /\  ( C  -  A
)  e.  CC )  ->  ( ( B  -  A ) G ( C  -  A
) )  =  ( Im `  ( ( * `  ( B  -  A ) )  x.  ( C  -  A ) ) ) )
184, 6, 17syl2anc 643 . . . . . . . . 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 2470 . . . . . . . 8  |-  ( ph  ->  ( Im `  (
( * `  ( B  -  A )
)  x.  ( C  -  A ) ) )  =  0 )
2115, 20reim0bd 12005 . . . . . . 7  |-  ( ph  ->  ( ( * `  ( B  -  A
) )  x.  ( C  -  A )
)  e.  RR )
226, 12mulcomd 9109 . . . . . . . 8  |-  ( ph  ->  ( ( C  -  A )  x.  (
* `  ( C  -  A ) ) )  =  ( ( * `
 ( C  -  A ) )  x.  ( C  -  A
) ) )
236cjmulrcld 12011 . . . . . . . 8  |-  ( ph  ->  ( ( C  -  A )  x.  (
* `  ( C  -  A ) ) )  e.  RR )
2422, 23eqeltrrd 2511 . . . . . . 7  |-  ( ph  ->  ( ( * `  ( C  -  A
) )  x.  ( C  -  A )
)  e.  RR )
2512, 6, 13, 9mulne0d 9674 . . . . . . 7  |-  ( ph  ->  ( ( * `  ( C  -  A
) )  x.  ( C  -  A )
)  =/=  0 )
2621, 24, 25redivcld 9842 . . . . . 6  |-  ( ph  ->  ( ( ( * `
 ( B  -  A ) )  x.  ( C  -  A
) )  /  (
( * `  ( C  -  A )
)  x.  ( C  -  A ) ) )  e.  RR )
2714, 26eqeltrrd 2511 . . . . 5  |-  ( ph  ->  ( ( * `  ( B  -  A
) )  /  (
* `  ( C  -  A ) ) )  e.  RR )
2810, 27eqeltrd 2510 . . . 4  |-  ( ph  ->  ( * `  (
( B  -  A
)  /  ( C  -  A ) ) )  e.  RR )
2928cjred 12031 . . 3  |-  ( ph  ->  ( * `  (
* `  ( ( B  -  A )  /  ( C  -  A ) ) ) )  =  ( * `
 ( ( B  -  A )  / 
( C  -  A
) ) ) )
304, 6, 9divcld 9790 . . . 4  |-  ( ph  ->  ( ( B  -  A )  /  ( C  -  A )
)  e.  CC )
3130cjcjd 12004 . . 3  |-  ( ph  ->  ( * `  (
* `  ( ( B  -  A )  /  ( C  -  A ) ) ) )  =  ( ( B  -  A )  /  ( C  -  A ) ) )
3229, 31eqtr3d 2470 . 2  |-  ( ph  ->  ( * `  (
( B  -  A
)  /  ( C  -  A ) ) )  =  ( ( B  -  A )  /  ( C  -  A ) ) )
3332, 28eqeltrrd 2511 1  |-  ( ph  ->  ( ( B  -  A )  /  ( C  -  A )
)  e.  RR )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   CCcc 8988   RRcr 8989   0cc0 8990    x. cmul 8995    - cmin 9291    / cdiv 9677   *ccj 11901   Imcim 11903
This theorem is referenced by:  sigarcol  27830  sharhght  27831
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-2 10058  df-cj 11904  df-re 11905  df-im 11906
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