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Theorem sigardiv 32317
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 1007 . . . . . . 7  |-  ( ph  ->  B  e.  CC )
31simp1d 1006 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
42, 3subcld 9922 . . . . . 6  |-  ( ph  ->  ( B  -  A
)  e.  CC )
51simp3d 1008 . . . . . . 7  |-  ( ph  ->  C  e.  CC )
65, 3subcld 9922 . . . . . 6  |-  ( ph  ->  ( C  -  A
)  e.  CC )
7 sigardiv.b . . . . . . . 8  |-  ( ph  ->  -.  C  =  A )
87neqned 2657 . . . . . . 7  |-  ( ph  ->  C  =/=  A )
95, 3, 8subne0d 9931 . . . . . 6  |-  ( ph  ->  ( C  -  A
)  =/=  0 )
104, 6, 9cjdivd 13138 . . . . 5  |-  ( ph  ->  ( * `  (
( B  -  A
)  /  ( C  -  A ) ) )  =  ( ( * `  ( B  -  A ) )  /  ( * `  ( C  -  A
) ) ) )
114cjcld 13111 . . . . . . 7  |-  ( ph  ->  ( * `  ( B  -  A )
)  e.  CC )
126cjcld 13111 . . . . . . 7  |-  ( ph  ->  ( * `  ( C  -  A )
)  e.  CC )
136, 9cjne0d 13118 . . . . . . 7  |-  ( ph  ->  ( * `  ( C  -  A )
)  =/=  0 )
1411, 12, 6, 13, 9divcan5rd 10343 . . . . . 6  |-  ( ph  ->  ( ( ( * `
 ( B  -  A ) )  x.  ( C  -  A
) )  /  (
( * `  ( C  -  A )
)  x.  ( C  -  A ) ) )  =  ( ( * `  ( B  -  A ) )  /  ( * `  ( C  -  A
) ) ) )
1511, 6mulcld 9605 . . . . . . . 8  |-  ( ph  ->  ( ( * `  ( B  -  A
) )  x.  ( C  -  A )
)  e.  CC )
16 sigar . . . . . . . . . . 11  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
1716sigarval 32306 . . . . . . . . . 10  |-  ( ( ( B  -  A
)  e.  CC  /\  ( C  -  A
)  e.  CC )  ->  ( ( B  -  A ) G ( C  -  A
) )  =  ( Im `  ( ( * `  ( B  -  A ) )  x.  ( C  -  A ) ) ) )
184, 6, 17syl2anc 659 . . . . . . . . 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 2497 . . . . . . . 8  |-  ( ph  ->  ( Im `  (
( * `  ( B  -  A )
)  x.  ( C  -  A ) ) )  =  0 )
2115, 20reim0bd 13115 . . . . . . 7  |-  ( ph  ->  ( ( * `  ( B  -  A
) )  x.  ( C  -  A )
)  e.  RR )
226, 12mulcomd 9606 . . . . . . . 8  |-  ( ph  ->  ( ( C  -  A )  x.  (
* `  ( C  -  A ) ) )  =  ( ( * `
 ( C  -  A ) )  x.  ( C  -  A
) ) )
236cjmulrcld 13121 . . . . . . . 8  |-  ( ph  ->  ( ( C  -  A )  x.  (
* `  ( C  -  A ) ) )  e.  RR )
2422, 23eqeltrrd 2543 . . . . . . 7  |-  ( ph  ->  ( ( * `  ( C  -  A
) )  x.  ( C  -  A )
)  e.  RR )
2512, 6, 13, 9mulne0d 10197 . . . . . . 7  |-  ( ph  ->  ( ( * `  ( C  -  A
) )  x.  ( C  -  A )
)  =/=  0 )
2621, 24, 25redivcld 10368 . . . . . 6  |-  ( ph  ->  ( ( ( * `
 ( B  -  A ) )  x.  ( C  -  A
) )  /  (
( * `  ( C  -  A )
)  x.  ( C  -  A ) ) )  e.  RR )
2714, 26eqeltrrd 2543 . . . . 5  |-  ( ph  ->  ( ( * `  ( B  -  A
) )  /  (
* `  ( C  -  A ) ) )  e.  RR )
2810, 27eqeltrd 2542 . . . 4  |-  ( ph  ->  ( * `  (
( B  -  A
)  /  ( C  -  A ) ) )  e.  RR )
2928cjred 13141 . . 3  |-  ( ph  ->  ( * `  (
* `  ( ( B  -  A )  /  ( C  -  A ) ) ) )  =  ( * `
 ( ( B  -  A )  / 
( C  -  A
) ) ) )
304, 6, 9divcld 10316 . . . 4  |-  ( ph  ->  ( ( B  -  A )  /  ( C  -  A )
)  e.  CC )
3130cjcjd 13114 . . 3  |-  ( ph  ->  ( * `  (
* `  ( ( B  -  A )  /  ( C  -  A ) ) ) )  =  ( ( B  -  A )  /  ( C  -  A ) ) )
3229, 31eqtr3d 2497 . 2  |-  ( ph  ->  ( * `  (
( B  -  A
)  /  ( C  -  A ) ) )  =  ( ( B  -  A )  /  ( C  -  A ) ) )
3332, 28eqeltrrd 2543 1  |-  ( ph  ->  ( ( B  -  A )  /  ( C  -  A )
)  e.  RR )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ w3a 971    = wceq 1398    e. wcel 1823   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   CCcc 9479   RRcr 9480   0cc0 9481    x. cmul 9486    - cmin 9796    / cdiv 10202   *ccj 13011   Imcim 13013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-2 10590  df-cj 13014  df-re 13015  df-im 13016
This theorem is referenced by:  sigarcol  32320  sharhght  32321
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