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Theorem bj-bary1lem1 32923
Description: Existence and uniqueness (and actual computation) of barycentric coordinates in dimension 1 (complex line). (Contributed by BJ, 6-Jun-2019.)
Hypotheses
Ref Expression
bj-bary1.a  |-  ( ph  ->  A  e.  CC )
bj-bary1.b  |-  ( ph  ->  B  e.  CC )
bj-bary1.x  |-  ( ph  ->  X  e.  CC )
bj-bary1.neq  |-  ( ph  ->  A  =/=  B )
bj-bary1.s  |-  ( ph  ->  S  e.  CC )
bj-bary1.t  |-  ( ph  ->  T  e.  CC )
Assertion
Ref Expression
bj-bary1lem1  |-  ( ph  ->  ( ( X  =  ( ( S  x.  A )  +  ( T  x.  B ) )  /\  ( S  +  T )  =  1 )  ->  T  =  ( ( X  -  A )  / 
( B  -  A
) ) ) )

Proof of Theorem bj-bary1lem1
StepHypRef Expression
1 bj-bary1.s . . . . . . 7  |-  ( ph  ->  S  e.  CC )
2 bj-bary1.t . . . . . . 7  |-  ( ph  ->  T  e.  CC )
31, 2pncand 9830 . . . . . 6  |-  ( ph  ->  ( ( S  +  T )  -  T
)  =  S )
4 oveq1 6206 . . . . . 6  |-  ( ( S  +  T )  =  1  ->  (
( S  +  T
)  -  T )  =  ( 1  -  T ) )
5 pm5.31 588 . . . . . 6  |-  ( ( ( ( S  +  T )  -  T
)  =  S  /\  ( ( S  +  T )  =  1  ->  ( ( S  +  T )  -  T )  =  ( 1  -  T ) ) )  ->  (
( S  +  T
)  =  1  -> 
( ( ( S  +  T )  -  T )  =  ( 1  -  T )  /\  ( ( S  +  T )  -  T )  =  S ) ) )
63, 4, 5sylancl 662 . . . . 5  |-  ( ph  ->  ( ( S  +  T )  =  1  ->  ( ( ( S  +  T )  -  T )  =  ( 1  -  T
)  /\  ( ( S  +  T )  -  T )  =  S ) ) )
7 eqtr2 2481 . . . . . 6  |-  ( ( ( ( S  +  T )  -  T
)  =  ( 1  -  T )  /\  ( ( S  +  T )  -  T
)  =  S )  ->  ( 1  -  T )  =  S )
87eqcomd 2462 . . . . 5  |-  ( ( ( ( S  +  T )  -  T
)  =  ( 1  -  T )  /\  ( ( S  +  T )  -  T
)  =  S )  ->  S  =  ( 1  -  T ) )
96, 8syl6 33 . . . 4  |-  ( ph  ->  ( ( S  +  T )  =  1  ->  S  =  ( 1  -  T ) ) )
10 oveq1 6206 . . . . . . . 8  |-  ( S  =  ( 1  -  T )  ->  ( S  x.  A )  =  ( ( 1  -  T )  x.  A ) )
1110oveq1d 6214 . . . . . . 7  |-  ( S  =  ( 1  -  T )  ->  (
( S  x.  A
)  +  ( T  x.  B ) )  =  ( ( ( 1  -  T )  x.  A )  +  ( T  x.  B
) ) )
12 eqtr 2480 . . . . . . 7  |-  ( ( X  =  ( ( S  x.  A )  +  ( T  x.  B ) )  /\  ( ( S  x.  A )  +  ( T  x.  B ) )  =  ( ( ( 1  -  T
)  x.  A )  +  ( T  x.  B ) ) )  ->  X  =  ( ( ( 1  -  T )  x.  A
)  +  ( T  x.  B ) ) )
1311, 12sylan2 474 . . . . . 6  |-  ( ( X  =  ( ( S  x.  A )  +  ( T  x.  B ) )  /\  S  =  ( 1  -  T ) )  ->  X  =  ( ( ( 1  -  T )  x.  A
)  +  ( T  x.  B ) ) )
14 1cnd 9512 . . . . . . . . 9  |-  ( ph  ->  1  e.  CC )
15 bj-bary1.a . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
1614, 2, 15subdird 9911 . . . . . . . 8  |-  ( ph  ->  ( ( 1  -  T )  x.  A
)  =  ( ( 1  x.  A )  -  ( T  x.  A ) ) )
1715mulid2d 9514 . . . . . . . . 9  |-  ( ph  ->  ( 1  x.  A
)  =  A )
1817oveq1d 6214 . . . . . . . 8  |-  ( ph  ->  ( ( 1  x.  A )  -  ( T  x.  A )
)  =  ( A  -  ( T  x.  A ) ) )
1916, 18eqtrd 2495 . . . . . . 7  |-  ( ph  ->  ( ( 1  -  T )  x.  A
)  =  ( A  -  ( T  x.  A ) ) )
2019oveq1d 6214 . . . . . 6  |-  ( ph  ->  ( ( ( 1  -  T )  x.  A )  +  ( T  x.  B ) )  =  ( ( A  -  ( T  x.  A ) )  +  ( T  x.  B ) ) )
2113, 20sylan9eqr 2517 . . . . 5  |-  ( (
ph  /\  ( X  =  ( ( S  x.  A )  +  ( T  x.  B
) )  /\  S  =  ( 1  -  T ) ) )  ->  X  =  ( ( A  -  ( T  x.  A )
)  +  ( T  x.  B ) ) )
2221ex 434 . . . 4  |-  ( ph  ->  ( ( X  =  ( ( S  x.  A )  +  ( T  x.  B ) )  /\  S  =  ( 1  -  T
) )  ->  X  =  ( ( A  -  ( T  x.  A ) )  +  ( T  x.  B
) ) ) )
239, 22sylan2d 482 . . 3  |-  ( ph  ->  ( ( X  =  ( ( S  x.  A )  +  ( T  x.  B ) )  /\  ( S  +  T )  =  1 )  ->  X  =  ( ( A  -  ( T  x.  A ) )  +  ( T  x.  B
) ) ) )
242, 15mulcld 9516 . . . . . 6  |-  ( ph  ->  ( T  x.  A
)  e.  CC )
25 bj-bary1.b . . . . . . 7  |-  ( ph  ->  B  e.  CC )
262, 25mulcld 9516 . . . . . 6  |-  ( ph  ->  ( T  x.  B
)  e.  CC )
2715, 24, 26subadd23d 9851 . . . . 5  |-  ( ph  ->  ( ( A  -  ( T  x.  A
) )  +  ( T  x.  B ) )  =  ( A  +  ( ( T  x.  B )  -  ( T  x.  A
) ) ) )
282, 25, 15subdid 9910 . . . . . . 7  |-  ( ph  ->  ( T  x.  ( B  -  A )
)  =  ( ( T  x.  B )  -  ( T  x.  A ) ) )
2928eqcomd 2462 . . . . . 6  |-  ( ph  ->  ( ( T  x.  B )  -  ( T  x.  A )
)  =  ( T  x.  ( B  -  A ) ) )
3029oveq2d 6215 . . . . 5  |-  ( ph  ->  ( A  +  ( ( T  x.  B
)  -  ( T  x.  A ) ) )  =  ( A  +  ( T  x.  ( B  -  A
) ) ) )
3127, 30eqtrd 2495 . . . 4  |-  ( ph  ->  ( ( A  -  ( T  x.  A
) )  +  ( T  x.  B ) )  =  ( A  +  ( T  x.  ( B  -  A
) ) ) )
3231eqeq2d 2468 . . 3  |-  ( ph  ->  ( X  =  ( ( A  -  ( T  x.  A )
)  +  ( T  x.  B ) )  <-> 
X  =  ( A  +  ( T  x.  ( B  -  A
) ) ) ) )
3323, 32sylibd 214 . 2  |-  ( ph  ->  ( ( X  =  ( ( S  x.  A )  +  ( T  x.  B ) )  /\  ( S  +  T )  =  1 )  ->  X  =  ( A  +  ( T  x.  ( B  -  A )
) ) ) )
34 oveq1 6206 . . 3  |-  ( X  =  ( A  +  ( T  x.  ( B  -  A )
) )  ->  ( X  -  A )  =  ( ( A  +  ( T  x.  ( B  -  A
) ) )  -  A ) )
3525, 15subcld 9829 . . . . . 6  |-  ( ph  ->  ( B  -  A
)  e.  CC )
362, 35mulcld 9516 . . . . 5  |-  ( ph  ->  ( T  x.  ( B  -  A )
)  e.  CC )
3715, 36pncan2d 9831 . . . 4  |-  ( ph  ->  ( ( A  +  ( T  x.  ( B  -  A )
) )  -  A
)  =  ( T  x.  ( B  -  A ) ) )
3837eqeq2d 2468 . . 3  |-  ( ph  ->  ( ( X  -  A )  =  ( ( A  +  ( T  x.  ( B  -  A ) ) )  -  A )  <-> 
( X  -  A
)  =  ( T  x.  ( B  -  A ) ) ) )
3934, 38syl5ib 219 . 2  |-  ( ph  ->  ( X  =  ( A  +  ( T  x.  ( B  -  A ) ) )  ->  ( X  -  A )  =  ( T  x.  ( B  -  A ) ) ) )
40 eqcom 2463 . . 3  |-  ( ( X  -  A )  =  ( T  x.  ( B  -  A
) )  <->  ( T  x.  ( B  -  A
) )  =  ( X  -  A ) )
412, 35mulcomd 9517 . . . . 5  |-  ( ph  ->  ( T  x.  ( B  -  A )
)  =  ( ( B  -  A )  x.  T ) )
4241eqeq1d 2456 . . . 4  |-  ( ph  ->  ( ( T  x.  ( B  -  A
) )  =  ( X  -  A )  <-> 
( ( B  -  A )  x.  T
)  =  ( X  -  A ) ) )
43 bj-bary1.x . . . . . . 7  |-  ( ph  ->  X  e.  CC )
4443, 15subcld 9829 . . . . . 6  |-  ( ph  ->  ( X  -  A
)  e.  CC )
45 bj-bary1.neq . . . . . . . 8  |-  ( ph  ->  A  =/=  B )
4645necomd 2722 . . . . . . 7  |-  ( ph  ->  B  =/=  A )
4725, 15, 46subne0d 9838 . . . . . 6  |-  ( ph  ->  ( B  -  A
)  =/=  0 )
4835, 2, 44, 47bj-rdiv 32918 . . . . 5  |-  ( ph  ->  ( ( ( B  -  A )  x.  T )  =  ( X  -  A )  <-> 
T  =  ( ( X  -  A )  /  ( B  -  A ) ) ) )
4948biimpd 207 . . . 4  |-  ( ph  ->  ( ( ( B  -  A )  x.  T )  =  ( X  -  A )  ->  T  =  ( ( X  -  A
)  /  ( B  -  A ) ) ) )
5042, 49sylbid 215 . . 3  |-  ( ph  ->  ( ( T  x.  ( B  -  A
) )  =  ( X  -  A )  ->  T  =  ( ( X  -  A
)  /  ( B  -  A ) ) ) )
5140, 50syl5bi 217 . 2  |-  ( ph  ->  ( ( X  -  A )  =  ( T  x.  ( B  -  A ) )  ->  T  =  ( ( X  -  A
)  /  ( B  -  A ) ) ) )
5233, 39, 513syld 55 1  |-  ( ph  ->  ( ( X  =  ( ( S  x.  A )  +  ( T  x.  B ) )  /\  ( S  +  T )  =  1 )  ->  T  =  ( ( X  -  A )  / 
( B  -  A
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2647  (class class class)co 6199   CCcc 9390   1c1 9393    + caddc 9395    x. cmul 9397    - cmin 9705    / cdiv 10103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-po 4748  df-so 4749  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104
This theorem is referenced by:  bj-bary1  32924
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