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Theorem sigardiv 38470
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 1021 . . . . . . 7  |-  ( ph  ->  B  e.  CC )
31simp1d 1020 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
42, 3subcld 9986 . . . . . 6  |-  ( ph  ->  ( B  -  A
)  e.  CC )
51simp3d 1022 . . . . . . 7  |-  ( ph  ->  C  e.  CC )
65, 3subcld 9986 . . . . . 6  |-  ( ph  ->  ( C  -  A
)  e.  CC )
7 sigardiv.b . . . . . . . 8  |-  ( ph  ->  -.  C  =  A )
87neqned 2631 . . . . . . 7  |-  ( ph  ->  C  =/=  A )
95, 3, 8subne0d 9995 . . . . . 6  |-  ( ph  ->  ( C  -  A
)  =/=  0 )
104, 6, 9cjdivd 13286 . . . . 5  |-  ( ph  ->  ( * `  (
( B  -  A
)  /  ( C  -  A ) ) )  =  ( ( * `  ( B  -  A ) )  /  ( * `  ( C  -  A
) ) ) )
114cjcld 13259 . . . . . . 7  |-  ( ph  ->  ( * `  ( B  -  A )
)  e.  CC )
126cjcld 13259 . . . . . . 7  |-  ( ph  ->  ( * `  ( C  -  A )
)  e.  CC )
136, 9cjne0d 13266 . . . . . . 7  |-  ( ph  ->  ( * `  ( C  -  A )
)  =/=  0 )
1411, 12, 6, 13, 9divcan5rd 10410 . . . . . 6  |-  ( ph  ->  ( ( ( * `
 ( B  -  A ) )  x.  ( C  -  A
) )  /  (
( * `  ( C  -  A )
)  x.  ( C  -  A ) ) )  =  ( ( * `  ( B  -  A ) )  /  ( * `  ( C  -  A
) ) ) )
1511, 6mulcld 9663 . . . . . . . 8  |-  ( ph  ->  ( ( * `  ( B  -  A
) )  x.  ( C  -  A )
)  e.  CC )
16 sigar . . . . . . . . . . 11  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
1716sigarval 38459 . . . . . . . . . 10  |-  ( ( ( B  -  A
)  e.  CC  /\  ( C  -  A
)  e.  CC )  ->  ( ( B  -  A ) G ( C  -  A
) )  =  ( Im `  ( ( * `  ( B  -  A ) )  x.  ( C  -  A ) ) ) )
184, 6, 17syl2anc 667 . . . . . . . . 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 2487 . . . . . . . 8  |-  ( ph  ->  ( Im `  (
( * `  ( B  -  A )
)  x.  ( C  -  A ) ) )  =  0 )
2115, 20reim0bd 13263 . . . . . . 7  |-  ( ph  ->  ( ( * `  ( B  -  A
) )  x.  ( C  -  A )
)  e.  RR )
226, 12mulcomd 9664 . . . . . . . 8  |-  ( ph  ->  ( ( C  -  A )  x.  (
* `  ( C  -  A ) ) )  =  ( ( * `
 ( C  -  A ) )  x.  ( C  -  A
) ) )
236cjmulrcld 13269 . . . . . . . 8  |-  ( ph  ->  ( ( C  -  A )  x.  (
* `  ( C  -  A ) ) )  e.  RR )
2422, 23eqeltrrd 2530 . . . . . . 7  |-  ( ph  ->  ( ( * `  ( C  -  A
) )  x.  ( C  -  A )
)  e.  RR )
2512, 6, 13, 9mulne0d 10264 . . . . . . 7  |-  ( ph  ->  ( ( * `  ( C  -  A
) )  x.  ( C  -  A )
)  =/=  0 )
2621, 24, 25redivcld 10435 . . . . . 6  |-  ( ph  ->  ( ( ( * `
 ( B  -  A ) )  x.  ( C  -  A
) )  /  (
( * `  ( C  -  A )
)  x.  ( C  -  A ) ) )  e.  RR )
2714, 26eqeltrrd 2530 . . . . 5  |-  ( ph  ->  ( ( * `  ( B  -  A
) )  /  (
* `  ( C  -  A ) ) )  e.  RR )
2810, 27eqeltrd 2529 . . . 4  |-  ( ph  ->  ( * `  (
( B  -  A
)  /  ( C  -  A ) ) )  e.  RR )
2928cjred 13289 . . 3  |-  ( ph  ->  ( * `  (
* `  ( ( B  -  A )  /  ( C  -  A ) ) ) )  =  ( * `
 ( ( B  -  A )  / 
( C  -  A
) ) ) )
304, 6, 9divcld 10383 . . . 4  |-  ( ph  ->  ( ( B  -  A )  /  ( C  -  A )
)  e.  CC )
3130cjcjd 13262 . . 3  |-  ( ph  ->  ( * `  (
* `  ( ( B  -  A )  /  ( C  -  A ) ) ) )  =  ( ( B  -  A )  /  ( C  -  A ) ) )
3229, 31eqtr3d 2487 . 2  |-  ( ph  ->  ( * `  (
( B  -  A
)  /  ( C  -  A ) ) )  =  ( ( B  -  A )  /  ( C  -  A ) ) )
3332, 28eqeltrrd 2530 1  |-  ( ph  ->  ( ( B  -  A )  /  ( C  -  A )
)  e.  RR )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ w3a 985    = wceq 1444    e. wcel 1887   ` cfv 5582  (class class class)co 6290    |-> cmpt2 6292   CCcc 9537   RRcr 9538   0cc0 9539    x. cmul 9544    - cmin 9860    / cdiv 10269   *ccj 13159   Imcim 13161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-po 4755  df-so 4756  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-2 10668  df-cj 13162  df-re 13163  df-im 13164
This theorem is referenced by:  sigarcol  38473  sharhght  38474
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