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Theorem sigardiv 27954
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 968 . . . . . . 7  |-  ( ph  ->  B  e.  CC )
31simp1d 967 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
42, 3subcld 9173 . . . . . 6  |-  ( ph  ->  ( B  -  A
)  e.  CC )
51simp3d 969 . . . . . . 7  |-  ( ph  ->  C  e.  CC )
65, 3subcld 9173 . . . . . 6  |-  ( ph  ->  ( C  -  A
)  e.  CC )
7 sigardiv.b . . . . . . . 8  |-  ( ph  ->  -.  C  =  A )
87neneqad 2529 . . . . . . 7  |-  ( ph  ->  C  =/=  A )
95, 3, 8subne0d 9182 . . . . . 6  |-  ( ph  ->  ( C  -  A
)  =/=  0 )
104, 6, 9cjdivd 11724 . . . . 5  |-  ( ph  ->  ( * `  (
( B  -  A
)  /  ( C  -  A ) ) )  =  ( ( * `  ( B  -  A ) )  /  ( * `  ( C  -  A
) ) ) )
114cjcld 11697 . . . . . . 7  |-  ( ph  ->  ( * `  ( B  -  A )
)  e.  CC )
126cjcld 11697 . . . . . . 7  |-  ( ph  ->  ( * `  ( C  -  A )
)  e.  CC )
136, 9cjne0d 11704 . . . . . . 7  |-  ( ph  ->  ( * `  ( C  -  A )
)  =/=  0 )
1411, 12, 6, 13, 9divcan5rd 9579 . . . . . 6  |-  ( ph  ->  ( ( ( * `
 ( B  -  A ) )  x.  ( C  -  A
) )  /  (
( * `  ( C  -  A )
)  x.  ( C  -  A ) ) )  =  ( ( * `  ( B  -  A ) )  /  ( * `  ( C  -  A
) ) ) )
1511, 6mulcld 8871 . . . . . . . 8  |-  ( ph  ->  ( ( * `  ( B  -  A
) )  x.  ( C  -  A )
)  e.  CC )
16 sigar . . . . . . . . . . 11  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
1716sigarval 27943 . . . . . . . . . 10  |-  ( ( ( B  -  A
)  e.  CC  /\  ( C  -  A
)  e.  CC )  ->  ( ( B  -  A ) G ( C  -  A
) )  =  ( Im `  ( ( * `  ( B  -  A ) )  x.  ( C  -  A ) ) ) )
184, 6, 17syl2anc 642 . . . . . . . . 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 2330 . . . . . . . 8  |-  ( ph  ->  ( Im `  (
( * `  ( B  -  A )
)  x.  ( C  -  A ) ) )  =  0 )
2115, 20reim0bd 11701 . . . . . . 7  |-  ( ph  ->  ( ( * `  ( B  -  A
) )  x.  ( C  -  A )
)  e.  RR )
226, 12mulcomd 8872 . . . . . . . 8  |-  ( ph  ->  ( ( C  -  A )  x.  (
* `  ( C  -  A ) ) )  =  ( ( * `
 ( C  -  A ) )  x.  ( C  -  A
) ) )
236cjmulrcld 11707 . . . . . . . 8  |-  ( ph  ->  ( ( C  -  A )  x.  (
* `  ( C  -  A ) ) )  e.  RR )
2422, 23eqeltrrd 2371 . . . . . . 7  |-  ( ph  ->  ( ( * `  ( C  -  A
) )  x.  ( C  -  A )
)  e.  RR )
2512, 6, 13, 9mulne0d 9436 . . . . . . 7  |-  ( ph  ->  ( ( * `  ( C  -  A
) )  x.  ( C  -  A )
)  =/=  0 )
2621, 24, 25redivcld 9604 . . . . . 6  |-  ( ph  ->  ( ( ( * `
 ( B  -  A ) )  x.  ( C  -  A
) )  /  (
( * `  ( C  -  A )
)  x.  ( C  -  A ) ) )  e.  RR )
2714, 26eqeltrrd 2371 . . . . 5  |-  ( ph  ->  ( ( * `  ( B  -  A
) )  /  (
* `  ( C  -  A ) ) )  e.  RR )
2810, 27eqeltrd 2370 . . . 4  |-  ( ph  ->  ( * `  (
( B  -  A
)  /  ( C  -  A ) ) )  e.  RR )
2928cjred 11727 . . 3  |-  ( ph  ->  ( * `  (
* `  ( ( B  -  A )  /  ( C  -  A ) ) ) )  =  ( * `
 ( ( B  -  A )  / 
( C  -  A
) ) ) )
304, 6, 9divcld 9552 . . . 4  |-  ( ph  ->  ( ( B  -  A )  /  ( C  -  A )
)  e.  CC )
3130cjcjd 11700 . . 3  |-  ( ph  ->  ( * `  (
* `  ( ( B  -  A )  /  ( C  -  A ) ) ) )  =  ( ( B  -  A )  /  ( C  -  A ) ) )
3229, 31eqtr3d 2330 . 2  |-  ( ph  ->  ( * `  (
( B  -  A
)  /  ( C  -  A ) ) )  =  ( ( B  -  A )  /  ( C  -  A ) ) )
3332, 28eqeltrrd 2371 1  |-  ( ph  ->  ( ( B  -  A )  /  ( C  -  A )
)  e.  RR )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   CCcc 8751   RRcr 8752   0cc0 8753    x. cmul 8758    - cmin 9053    / cdiv 9439   *ccj 11597   Imcim 11599
This theorem is referenced by:  sigarcol  27957  sharhght  27958
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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