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Theorem bj-bary1lem1 31487
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 9976 . . . . . 6  |-  ( ph  ->  ( ( S  +  T )  -  T
)  =  S )
4 oveq1 6303 . . . . . 6  |-  ( ( S  +  T )  =  1  ->  (
( S  +  T
)  -  T )  =  ( 1  -  T ) )
5 pm5.31 590 . . . . . 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 666 . . . . 5  |-  ( ph  ->  ( ( S  +  T )  =  1  ->  ( ( ( S  +  T )  -  T )  =  ( 1  -  T
)  /\  ( ( S  +  T )  -  T )  =  S ) ) )
7 eqtr2 2447 . . . . . 6  |-  ( ( ( ( S  +  T )  -  T
)  =  ( 1  -  T )  /\  ( ( S  +  T )  -  T
)  =  S )  ->  ( 1  -  T )  =  S )
87eqcomd 2428 . . . . 5  |-  ( ( ( ( S  +  T )  -  T
)  =  ( 1  -  T )  /\  ( ( S  +  T )  -  T
)  =  S )  ->  S  =  ( 1  -  T ) )
96, 8syl6 34 . . . 4  |-  ( ph  ->  ( ( S  +  T )  =  1  ->  S  =  ( 1  -  T ) ) )
10 oveq1 6303 . . . . . . . 8  |-  ( S  =  ( 1  -  T )  ->  ( S  x.  A )  =  ( ( 1  -  T )  x.  A ) )
1110oveq1d 6311 . . . . . . 7  |-  ( S  =  ( 1  -  T )  ->  (
( S  x.  A
)  +  ( T  x.  B ) )  =  ( ( ( 1  -  T )  x.  A )  +  ( T  x.  B
) ) )
12 eqtr 2446 . . . . . . 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 476 . . . . . 6  |-  ( ( X  =  ( ( S  x.  A )  +  ( T  x.  B ) )  /\  S  =  ( 1  -  T ) )  ->  X  =  ( ( ( 1  -  T )  x.  A
)  +  ( T  x.  B ) ) )
14 1cnd 9648 . . . . . . . . 9  |-  ( ph  ->  1  e.  CC )
15 bj-bary1.a . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
1614, 2, 15subdird 10064 . . . . . . . 8  |-  ( ph  ->  ( ( 1  -  T )  x.  A
)  =  ( ( 1  x.  A )  -  ( T  x.  A ) ) )
1715mulid2d 9650 . . . . . . . . 9  |-  ( ph  ->  ( 1  x.  A
)  =  A )
1817oveq1d 6311 . . . . . . . 8  |-  ( ph  ->  ( ( 1  x.  A )  -  ( T  x.  A )
)  =  ( A  -  ( T  x.  A ) ) )
1916, 18eqtrd 2461 . . . . . . 7  |-  ( ph  ->  ( ( 1  -  T )  x.  A
)  =  ( A  -  ( T  x.  A ) ) )
2019oveq1d 6311 . . . . . 6  |-  ( ph  ->  ( ( ( 1  -  T )  x.  A )  +  ( T  x.  B ) )  =  ( ( A  -  ( T  x.  A ) )  +  ( T  x.  B ) ) )
2113, 20sylan9eqr 2483 . . . . 5  |-  ( (
ph  /\  ( X  =  ( ( S  x.  A )  +  ( T  x.  B
) )  /\  S  =  ( 1  -  T ) ) )  ->  X  =  ( ( A  -  ( T  x.  A )
)  +  ( T  x.  B ) ) )
2221ex 435 . . . 4  |-  ( ph  ->  ( ( X  =  ( ( S  x.  A )  +  ( T  x.  B ) )  /\  S  =  ( 1  -  T
) )  ->  X  =  ( ( A  -  ( T  x.  A ) )  +  ( T  x.  B
) ) ) )
239, 22sylan2d 484 . . 3  |-  ( ph  ->  ( ( X  =  ( ( S  x.  A )  +  ( T  x.  B ) )  /\  ( S  +  T )  =  1 )  ->  X  =  ( ( A  -  ( T  x.  A ) )  +  ( T  x.  B
) ) ) )
242, 15mulcld 9652 . . . . . 6  |-  ( ph  ->  ( T  x.  A
)  e.  CC )
25 bj-bary1.b . . . . . . 7  |-  ( ph  ->  B  e.  CC )
262, 25mulcld 9652 . . . . . 6  |-  ( ph  ->  ( T  x.  B
)  e.  CC )
2715, 24, 26subadd23d 9997 . . . . 5  |-  ( ph  ->  ( ( A  -  ( T  x.  A
) )  +  ( T  x.  B ) )  =  ( A  +  ( ( T  x.  B )  -  ( T  x.  A
) ) ) )
282, 25, 15subdid 10063 . . . . . . 7  |-  ( ph  ->  ( T  x.  ( B  -  A )
)  =  ( ( T  x.  B )  -  ( T  x.  A ) ) )
2928eqcomd 2428 . . . . . 6  |-  ( ph  ->  ( ( T  x.  B )  -  ( T  x.  A )
)  =  ( T  x.  ( B  -  A ) ) )
3029oveq2d 6312 . . . . 5  |-  ( ph  ->  ( A  +  ( ( T  x.  B
)  -  ( T  x.  A ) ) )  =  ( A  +  ( T  x.  ( B  -  A
) ) ) )
3127, 30eqtrd 2461 . . . 4  |-  ( ph  ->  ( ( A  -  ( T  x.  A
) )  +  ( T  x.  B ) )  =  ( A  +  ( T  x.  ( B  -  A
) ) ) )
3231eqeq2d 2434 . . 3  |-  ( ph  ->  ( X  =  ( ( A  -  ( T  x.  A )
)  +  ( T  x.  B ) )  <-> 
X  =  ( A  +  ( T  x.  ( B  -  A
) ) ) ) )
3323, 32sylibd 217 . 2  |-  ( ph  ->  ( ( X  =  ( ( S  x.  A )  +  ( T  x.  B ) )  /\  ( S  +  T )  =  1 )  ->  X  =  ( A  +  ( T  x.  ( B  -  A )
) ) ) )
34 oveq1 6303 . . 3  |-  ( X  =  ( A  +  ( T  x.  ( B  -  A )
) )  ->  ( X  -  A )  =  ( ( A  +  ( T  x.  ( B  -  A
) ) )  -  A ) )
3525, 15subcld 9975 . . . . . 6  |-  ( ph  ->  ( B  -  A
)  e.  CC )
362, 35mulcld 9652 . . . . 5  |-  ( ph  ->  ( T  x.  ( B  -  A )
)  e.  CC )
3715, 36pncan2d 9977 . . . 4  |-  ( ph  ->  ( ( A  +  ( T  x.  ( B  -  A )
) )  -  A
)  =  ( T  x.  ( B  -  A ) ) )
3837eqeq2d 2434 . . 3  |-  ( ph  ->  ( ( X  -  A )  =  ( ( A  +  ( T  x.  ( B  -  A ) ) )  -  A )  <-> 
( X  -  A
)  =  ( T  x.  ( B  -  A ) ) ) )
3934, 38syl5ib 222 . 2  |-  ( ph  ->  ( X  =  ( A  +  ( T  x.  ( B  -  A ) ) )  ->  ( X  -  A )  =  ( T  x.  ( B  -  A ) ) ) )
40 eqcom 2429 . . 3  |-  ( ( X  -  A )  =  ( T  x.  ( B  -  A
) )  <->  ( T  x.  ( B  -  A
) )  =  ( X  -  A ) )
412, 35mulcomd 9653 . . . . 5  |-  ( ph  ->  ( T  x.  ( B  -  A )
)  =  ( ( B  -  A )  x.  T ) )
4241eqeq1d 2422 . . . 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 9975 . . . . . 6  |-  ( ph  ->  ( X  -  A
)  e.  CC )
45 bj-bary1.neq . . . . . . . 8  |-  ( ph  ->  A  =/=  B )
4645necomd 2693 . . . . . . 7  |-  ( ph  ->  B  =/=  A )
4725, 15, 46subne0d 9984 . . . . . 6  |-  ( ph  ->  ( B  -  A
)  =/=  0 )
4835, 2, 44, 47bj-rdiv 31482 . . . . 5  |-  ( ph  ->  ( ( ( B  -  A )  x.  T )  =  ( X  -  A )  <-> 
T  =  ( ( X  -  A )  /  ( B  -  A ) ) ) )
4948biimpd 210 . . . 4  |-  ( ph  ->  ( ( ( B  -  A )  x.  T )  =  ( X  -  A )  ->  T  =  ( ( X  -  A
)  /  ( B  -  A ) ) ) )
5042, 49sylbid 218 . . 3  |-  ( ph  ->  ( ( T  x.  ( B  -  A
) )  =  ( X  -  A )  ->  T  =  ( ( X  -  A
)  /  ( B  -  A ) ) ) )
5140, 50syl5bi 220 . 2  |-  ( ph  ->  ( ( X  -  A )  =  ( T  x.  ( B  -  A ) )  ->  T  =  ( ( X  -  A
)  /  ( B  -  A ) ) ) )
5233, 39, 513syld 57 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 370    = wceq 1437    e. wcel 1867    =/= wne 2616  (class class class)co 6296   CCcc 9526   1c1 9529    + caddc 9531    x. cmul 9533    - cmin 9849    / cdiv 10258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-po 4766  df-so 4767  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259
This theorem is referenced by:  bj-bary1  31488
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