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Theorem bj-bary1lem1 33770
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 9931 . . . . . 6  |-  ( ph  ->  ( ( S  +  T )  -  T
)  =  S )
4 oveq1 6291 . . . . . 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 2494 . . . . . 6  |-  ( ( ( ( S  +  T )  -  T
)  =  ( 1  -  T )  /\  ( ( S  +  T )  -  T
)  =  S )  ->  ( 1  -  T )  =  S )
87eqcomd 2475 . . . . 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 6291 . . . . . . . 8  |-  ( S  =  ( 1  -  T )  ->  ( S  x.  A )  =  ( ( 1  -  T )  x.  A ) )
1110oveq1d 6299 . . . . . . 7  |-  ( S  =  ( 1  -  T )  ->  (
( S  x.  A
)  +  ( T  x.  B ) )  =  ( ( ( 1  -  T )  x.  A )  +  ( T  x.  B
) ) )
12 eqtr 2493 . . . . . . 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 9612 . . . . . . . . 9  |-  ( ph  ->  1  e.  CC )
15 bj-bary1.a . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
1614, 2, 15subdird 10013 . . . . . . . 8  |-  ( ph  ->  ( ( 1  -  T )  x.  A
)  =  ( ( 1  x.  A )  -  ( T  x.  A ) ) )
1715mulid2d 9614 . . . . . . . . 9  |-  ( ph  ->  ( 1  x.  A
)  =  A )
1817oveq1d 6299 . . . . . . . 8  |-  ( ph  ->  ( ( 1  x.  A )  -  ( T  x.  A )
)  =  ( A  -  ( T  x.  A ) ) )
1916, 18eqtrd 2508 . . . . . . 7  |-  ( ph  ->  ( ( 1  -  T )  x.  A
)  =  ( A  -  ( T  x.  A ) ) )
2019oveq1d 6299 . . . . . 6  |-  ( ph  ->  ( ( ( 1  -  T )  x.  A )  +  ( T  x.  B ) )  =  ( ( A  -  ( T  x.  A ) )  +  ( T  x.  B ) ) )
2113, 20sylan9eqr 2530 . . . . 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 9616 . . . . . 6  |-  ( ph  ->  ( T  x.  A
)  e.  CC )
25 bj-bary1.b . . . . . . 7  |-  ( ph  ->  B  e.  CC )
262, 25mulcld 9616 . . . . . 6  |-  ( ph  ->  ( T  x.  B
)  e.  CC )
2715, 24, 26subadd23d 9952 . . . . 5  |-  ( ph  ->  ( ( A  -  ( T  x.  A
) )  +  ( T  x.  B ) )  =  ( A  +  ( ( T  x.  B )  -  ( T  x.  A
) ) ) )
282, 25, 15subdid 10012 . . . . . . 7  |-  ( ph  ->  ( T  x.  ( B  -  A )
)  =  ( ( T  x.  B )  -  ( T  x.  A ) ) )
2928eqcomd 2475 . . . . . 6  |-  ( ph  ->  ( ( T  x.  B )  -  ( T  x.  A )
)  =  ( T  x.  ( B  -  A ) ) )
3029oveq2d 6300 . . . . 5  |-  ( ph  ->  ( A  +  ( ( T  x.  B
)  -  ( T  x.  A ) ) )  =  ( A  +  ( T  x.  ( B  -  A
) ) ) )
3127, 30eqtrd 2508 . . . 4  |-  ( ph  ->  ( ( A  -  ( T  x.  A
) )  +  ( T  x.  B ) )  =  ( A  +  ( T  x.  ( B  -  A
) ) ) )
3231eqeq2d 2481 . . 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 6291 . . 3  |-  ( X  =  ( A  +  ( T  x.  ( B  -  A )
) )  ->  ( X  -  A )  =  ( ( A  +  ( T  x.  ( B  -  A
) ) )  -  A ) )
3525, 15subcld 9930 . . . . . 6  |-  ( ph  ->  ( B  -  A
)  e.  CC )
362, 35mulcld 9616 . . . . 5  |-  ( ph  ->  ( T  x.  ( B  -  A )
)  e.  CC )
3715, 36pncan2d 9932 . . . 4  |-  ( ph  ->  ( ( A  +  ( T  x.  ( B  -  A )
) )  -  A
)  =  ( T  x.  ( B  -  A ) ) )
3837eqeq2d 2481 . . 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 2476 . . 3  |-  ( ( X  -  A )  =  ( T  x.  ( B  -  A
) )  <->  ( T  x.  ( B  -  A
) )  =  ( X  -  A ) )
412, 35mulcomd 9617 . . . . 5  |-  ( ph  ->  ( T  x.  ( B  -  A )
)  =  ( ( B  -  A )  x.  T ) )
4241eqeq1d 2469 . . . 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 9930 . . . . . 6  |-  ( ph  ->  ( X  -  A
)  e.  CC )
45 bj-bary1.neq . . . . . . . 8  |-  ( ph  ->  A  =/=  B )
4645necomd 2738 . . . . . . 7  |-  ( ph  ->  B  =/=  A )
4725, 15, 46subne0d 9939 . . . . . 6  |-  ( ph  ->  ( B  -  A
)  =/=  0 )
4835, 2, 44, 47bj-rdiv 33765 . . . . 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 1379    e. wcel 1767    =/= wne 2662  (class class class)co 6284   CCcc 9490   1c1 9493    + caddc 9495    x. cmul 9497    - cmin 9805    / cdiv 10206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207
This theorem is referenced by:  bj-bary1  33771
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