Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-bary1 Structured version   Unicode version

Theorem bj-bary1 31488
Description: Barycentric coordinates in one dimension. (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-bary1  |-  ( ph  ->  ( ( X  =  ( ( S  x.  A )  +  ( T  x.  B ) )  /\  ( S  +  T )  =  1 )  <->  ( S  =  ( ( B  -  X )  / 
( B  -  A
) )  /\  T  =  ( ( X  -  A )  / 
( B  -  A
) ) ) ) )

Proof of Theorem bj-bary1
StepHypRef Expression
1 bj-bary1.s . . . . . . . . 9  |-  ( ph  ->  S  e.  CC )
2 bj-bary1.a . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
31, 2mulcld 9652 . . . . . . . 8  |-  ( ph  ->  ( S  x.  A
)  e.  CC )
4 bj-bary1.t . . . . . . . . 9  |-  ( ph  ->  T  e.  CC )
5 bj-bary1.b . . . . . . . . 9  |-  ( ph  ->  B  e.  CC )
64, 5mulcld 9652 . . . . . . . 8  |-  ( ph  ->  ( T  x.  B
)  e.  CC )
73, 6addcomd 9824 . . . . . . 7  |-  ( ph  ->  ( ( S  x.  A )  +  ( T  x.  B ) )  =  ( ( T  x.  B )  +  ( S  x.  A ) ) )
87eqeq2d 2434 . . . . . 6  |-  ( ph  ->  ( X  =  ( ( S  x.  A
)  +  ( T  x.  B ) )  <-> 
X  =  ( ( T  x.  B )  +  ( S  x.  A ) ) ) )
98biimpd 210 . . . . 5  |-  ( ph  ->  ( X  =  ( ( S  x.  A
)  +  ( T  x.  B ) )  ->  X  =  ( ( T  x.  B
)  +  ( S  x.  A ) ) ) )
101, 4addcomd 9824 . . . . . . 7  |-  ( ph  ->  ( S  +  T
)  =  ( T  +  S ) )
1110eqeq1d 2422 . . . . . 6  |-  ( ph  ->  ( ( S  +  T )  =  1  <-> 
( T  +  S
)  =  1 ) )
1211biimpd 210 . . . . 5  |-  ( ph  ->  ( ( S  +  T )  =  1  ->  ( T  +  S )  =  1 ) )
13 bj-bary1.x . . . . . 6  |-  ( ph  ->  X  e.  CC )
14 bj-bary1.neq . . . . . . 7  |-  ( ph  ->  A  =/=  B )
1514necomd 2693 . . . . . 6  |-  ( ph  ->  B  =/=  A )
165, 2, 13, 15, 4, 1bj-bary1lem1 31487 . . . . 5  |-  ( ph  ->  ( ( X  =  ( ( T  x.  B )  +  ( S  x.  A ) )  /\  ( T  +  S )  =  1 )  ->  S  =  ( ( X  -  B )  / 
( A  -  B
) ) ) )
179, 12, 16syl2and 485 . . . 4  |-  ( ph  ->  ( ( X  =  ( ( S  x.  A )  +  ( T  x.  B ) )  /\  ( S  +  T )  =  1 )  ->  S  =  ( ( X  -  B )  / 
( A  -  B
) ) ) )
1813, 5, 2, 5, 14div2subd 10422 . . . . 5  |-  ( ph  ->  ( ( X  -  B )  /  ( A  -  B )
)  =  ( ( B  -  X )  /  ( B  -  A ) ) )
1918eqeq2d 2434 . . . 4  |-  ( ph  ->  ( S  =  ( ( X  -  B
)  /  ( A  -  B ) )  <-> 
S  =  ( ( B  -  X )  /  ( B  -  A ) ) ) )
2017, 19sylibd 217 . . 3  |-  ( ph  ->  ( ( X  =  ( ( S  x.  A )  +  ( T  x.  B ) )  /\  ( S  +  T )  =  1 )  ->  S  =  ( ( B  -  X )  / 
( B  -  A
) ) ) )
212, 5, 13, 14, 1, 4bj-bary1lem1 31487 . . 3  |-  ( ph  ->  ( ( X  =  ( ( S  x.  A )  +  ( T  x.  B ) )  /\  ( S  +  T )  =  1 )  ->  T  =  ( ( X  -  A )  / 
( B  -  A
) ) ) )
2220, 21jcad 535 . 2  |-  ( ph  ->  ( ( X  =  ( ( S  x.  A )  +  ( T  x.  B ) )  /\  ( S  +  T )  =  1 )  ->  ( S  =  ( ( B  -  X )  /  ( B  -  A ) )  /\  T  =  ( ( X  -  A )  /  ( B  -  A ) ) ) ) )
232, 5, 13, 14bj-bary1lem 31486 . . . 4  |-  ( ph  ->  X  =  ( ( ( ( B  -  X )  /  ( B  -  A )
)  x.  A )  +  ( ( ( X  -  A )  /  ( B  -  A ) )  x.  B ) ) )
24 oveq1 6303 . . . . . 6  |-  ( S  =  ( ( B  -  X )  / 
( B  -  A
) )  ->  ( S  x.  A )  =  ( ( ( B  -  X )  /  ( B  -  A ) )  x.  A ) )
25 oveq1 6303 . . . . . 6  |-  ( T  =  ( ( X  -  A )  / 
( B  -  A
) )  ->  ( T  x.  B )  =  ( ( ( X  -  A )  /  ( B  -  A ) )  x.  B ) )
2624, 25oveqan12d 6315 . . . . 5  |-  ( ( S  =  ( ( B  -  X )  /  ( B  -  A ) )  /\  T  =  ( ( X  -  A )  /  ( B  -  A ) ) )  ->  ( ( S  x.  A )  +  ( T  x.  B
) )  =  ( ( ( ( B  -  X )  / 
( B  -  A
) )  x.  A
)  +  ( ( ( X  -  A
)  /  ( B  -  A ) )  x.  B ) ) )
2726a1i 11 . . . 4  |-  ( ph  ->  ( ( S  =  ( ( B  -  X )  /  ( B  -  A )
)  /\  T  =  ( ( X  -  A )  /  ( B  -  A )
) )  ->  (
( S  x.  A
)  +  ( T  x.  B ) )  =  ( ( ( ( B  -  X
)  /  ( B  -  A ) )  x.  A )  +  ( ( ( X  -  A )  / 
( B  -  A
) )  x.  B
) ) ) )
28 eqtr3 2448 . . . 4  |-  ( ( X  =  ( ( ( ( B  -  X )  /  ( B  -  A )
)  x.  A )  +  ( ( ( X  -  A )  /  ( B  -  A ) )  x.  B ) )  /\  ( ( S  x.  A )  +  ( T  x.  B ) )  =  ( ( ( ( B  -  X )  /  ( B  -  A )
)  x.  A )  +  ( ( ( X  -  A )  /  ( B  -  A ) )  x.  B ) ) )  ->  X  =  ( ( S  x.  A
)  +  ( T  x.  B ) ) )
2923, 27, 28syl6an 547 . . 3  |-  ( ph  ->  ( ( S  =  ( ( B  -  X )  /  ( B  -  A )
)  /\  T  =  ( ( X  -  A )  /  ( B  -  A )
) )  ->  X  =  ( ( S  x.  A )  +  ( T  x.  B
) ) ) )
30 oveq12 6305 . . . 4  |-  ( ( S  =  ( ( B  -  X )  /  ( B  -  A ) )  /\  T  =  ( ( X  -  A )  /  ( B  -  A ) ) )  ->  ( S  +  T )  =  ( ( ( B  -  X )  /  ( B  -  A )
)  +  ( ( X  -  A )  /  ( B  -  A ) ) ) )
315, 13subcld 9975 . . . . . . 7  |-  ( ph  ->  ( B  -  X
)  e.  CC )
3213, 2subcld 9975 . . . . . . 7  |-  ( ph  ->  ( X  -  A
)  e.  CC )
335, 2subcld 9975 . . . . . . 7  |-  ( ph  ->  ( B  -  A
)  e.  CC )
345, 2, 15subne0d 9984 . . . . . . 7  |-  ( ph  ->  ( B  -  A
)  =/=  0 )
3531, 32, 33, 34divdird 10410 . . . . . 6  |-  ( ph  ->  ( ( ( B  -  X )  +  ( X  -  A
) )  /  ( B  -  A )
)  =  ( ( ( B  -  X
)  /  ( B  -  A ) )  +  ( ( X  -  A )  / 
( B  -  A
) ) ) )
365, 13, 2npncand 9999 . . . . . . 7  |-  ( ph  ->  ( ( B  -  X )  +  ( X  -  A ) )  =  ( B  -  A ) )
3733, 34, 36diveq1bd 10420 . . . . . 6  |-  ( ph  ->  ( ( ( B  -  X )  +  ( X  -  A
) )  /  ( B  -  A )
)  =  1 )
3835, 37eqtr3d 2463 . . . . 5  |-  ( ph  ->  ( ( ( B  -  X )  / 
( B  -  A
) )  +  ( ( X  -  A
)  /  ( B  -  A ) ) )  =  1 )
3938eqeq2d 2434 . . . 4  |-  ( ph  ->  ( ( S  +  T )  =  ( ( ( B  -  X )  /  ( B  -  A )
)  +  ( ( X  -  A )  /  ( B  -  A ) ) )  <-> 
( S  +  T
)  =  1 ) )
4030, 39syl5ib 222 . . 3  |-  ( ph  ->  ( ( S  =  ( ( B  -  X )  /  ( B  -  A )
)  /\  T  =  ( ( X  -  A )  /  ( B  -  A )
) )  ->  ( S  +  T )  =  1 ) )
4129, 40jcad 535 . 2  |-  ( ph  ->  ( ( S  =  ( ( B  -  X )  /  ( B  -  A )
)  /\  T  =  ( ( X  -  A )  /  ( B  -  A )
) )  ->  ( X  =  ( ( S  x.  A )  +  ( T  x.  B ) )  /\  ( S  +  T
)  =  1 ) ) )
4222, 41impbid 193 1  |-  ( ph  ->  ( ( X  =  ( ( S  x.  A )  +  ( T  x.  B ) )  /\  ( S  +  T )  =  1 )  <->  ( S  =  ( ( B  -  X )  / 
( B  -  A
) )  /\  T  =  ( ( X  -  A )  / 
( B  -  A
) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ 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: (None)
  Copyright terms: Public domain W3C validator