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Theorem bj-bary1 32606
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 9411 . . . . . . . 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 9411 . . . . . . . 8  |-  ( ph  ->  ( T  x.  B
)  e.  CC )
73, 6addcomd 9576 . . . . . . 7  |-  ( ph  ->  ( ( S  x.  A )  +  ( T  x.  B ) )  =  ( ( T  x.  B )  +  ( S  x.  A ) ) )
87eqeq2d 2454 . . . . . 6  |-  ( ph  ->  ( X  =  ( ( S  x.  A
)  +  ( T  x.  B ) )  <-> 
X  =  ( ( T  x.  B )  +  ( S  x.  A ) ) ) )
98biimpd 207 . . . . 5  |-  ( ph  ->  ( X  =  ( ( S  x.  A
)  +  ( T  x.  B ) )  ->  X  =  ( ( T  x.  B
)  +  ( S  x.  A ) ) ) )
101, 4addcomd 9576 . . . . . . 7  |-  ( ph  ->  ( S  +  T
)  =  ( T  +  S ) )
1110eqeq1d 2451 . . . . . 6  |-  ( ph  ->  ( ( S  +  T )  =  1  <-> 
( T  +  S
)  =  1 ) )
1211biimpd 207 . . . . 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 2700 . . . . . 6  |-  ( ph  ->  B  =/=  A )
165, 2, 13, 15, 4, 1bj-bary1lem1 32605 . . . . 5  |-  ( ph  ->  ( ( X  =  ( ( T  x.  B )  +  ( S  x.  A ) )  /\  ( T  +  S )  =  1 )  ->  S  =  ( ( X  -  B )  / 
( A  -  B
) ) ) )
179, 12, 16syl2and 483 . . . 4  |-  ( ph  ->  ( ( X  =  ( ( S  x.  A )  +  ( T  x.  B ) )  /\  ( S  +  T )  =  1 )  ->  S  =  ( ( X  -  B )  / 
( A  -  B
) ) ) )
1813, 5, 2, 5, 14div2subd 10162 . . . . 5  |-  ( ph  ->  ( ( X  -  B )  /  ( A  -  B )
)  =  ( ( B  -  X )  /  ( B  -  A ) ) )
1918eqeq2d 2454 . . . 4  |-  ( ph  ->  ( S  =  ( ( X  -  B
)  /  ( A  -  B ) )  <-> 
S  =  ( ( B  -  X )  /  ( B  -  A ) ) ) )
2017, 19sylibd 214 . . 3  |-  ( ph  ->  ( ( X  =  ( ( S  x.  A )  +  ( T  x.  B ) )  /\  ( S  +  T )  =  1 )  ->  S  =  ( ( B  -  X )  / 
( B  -  A
) ) ) )
212, 5, 13, 14, 1, 4bj-bary1lem1 32605 . . 3  |-  ( ph  ->  ( ( X  =  ( ( S  x.  A )  +  ( T  x.  B ) )  /\  ( S  +  T )  =  1 )  ->  T  =  ( ( X  -  A )  / 
( B  -  A
) ) ) )
2220, 21jcad 533 . 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 32604 . . . 4  |-  ( ph  ->  X  =  ( ( ( ( B  -  X )  /  ( B  -  A )
)  x.  A )  +  ( ( ( X  -  A )  /  ( B  -  A ) )  x.  B ) ) )
24 oveq1 6103 . . . . . 6  |-  ( S  =  ( ( B  -  X )  / 
( B  -  A
) )  ->  ( S  x.  A )  =  ( ( ( B  -  X )  /  ( B  -  A ) )  x.  A ) )
25 oveq1 6103 . . . . . 6  |-  ( T  =  ( ( X  -  A )  / 
( B  -  A
) )  ->  ( T  x.  B )  =  ( ( ( X  -  A )  /  ( B  -  A ) )  x.  B ) )
2624, 25oveqan12d 6115 . . . . 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 2462 . . . 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 545 . . 3  |-  ( ph  ->  ( ( S  =  ( ( B  -  X )  /  ( B  -  A )
)  /\  T  =  ( ( X  -  A )  /  ( B  -  A )
) )  ->  X  =  ( ( S  x.  A )  +  ( T  x.  B
) ) ) )
30 oveq12 6105 . . . 4  |-  ( ( S  =  ( ( B  -  X )  /  ( B  -  A ) )  /\  T  =  ( ( X  -  A )  /  ( B  -  A ) ) )  ->  ( S  +  T )  =  ( ( ( B  -  X )  /  ( B  -  A )
)  +  ( ( X  -  A )  /  ( B  -  A ) ) ) )
315, 13subcld 9724 . . . . . . 7  |-  ( ph  ->  ( B  -  X
)  e.  CC )
3213, 2subcld 9724 . . . . . . 7  |-  ( ph  ->  ( X  -  A
)  e.  CC )
335, 2subcld 9724 . . . . . . 7  |-  ( ph  ->  ( B  -  A
)  e.  CC )
345, 2, 15subne0d 9733 . . . . . . 7  |-  ( ph  ->  ( B  -  A
)  =/=  0 )
3531, 32, 33, 34divdird 10150 . . . . . 6  |-  ( ph  ->  ( ( ( B  -  X )  +  ( X  -  A
) )  /  ( B  -  A )
)  =  ( ( ( B  -  X
)  /  ( B  -  A ) )  +  ( ( X  -  A )  / 
( B  -  A
) ) ) )
365, 13, 2npncand 9748 . . . . . . 7  |-  ( ph  ->  ( ( B  -  X )  +  ( X  -  A ) )  =  ( B  -  A ) )
3733, 34, 36diveq1bd 10160 . . . . . 6  |-  ( ph  ->  ( ( ( B  -  X )  +  ( X  -  A
) )  /  ( B  -  A )
)  =  1 )
3835, 37eqtr3d 2477 . . . . 5  |-  ( ph  ->  ( ( ( B  -  X )  / 
( B  -  A
) )  +  ( ( X  -  A
)  /  ( B  -  A ) ) )  =  1 )
3938eqeq2d 2454 . . . 4  |-  ( ph  ->  ( ( S  +  T )  =  ( ( ( B  -  X )  /  ( B  -  A )
)  +  ( ( X  -  A )  /  ( B  -  A ) ) )  <-> 
( S  +  T
)  =  1 ) )
4030, 39syl5ib 219 . . 3  |-  ( ph  ->  ( ( S  =  ( ( B  -  X )  /  ( B  -  A )
)  /\  T  =  ( ( X  -  A )  /  ( B  -  A )
) )  ->  ( S  +  T )  =  1 ) )
4129, 40jcad 533 . 2  |-  ( ph  ->  ( ( S  =  ( ( B  -  X )  /  ( B  -  A )
)  /\  T  =  ( ( X  -  A )  /  ( B  -  A )
) )  ->  ( X  =  ( ( S  x.  A )  +  ( T  x.  B ) )  /\  ( S  +  T
)  =  1 ) ) )
4222, 41impbid 191 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 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2611  (class class class)co 6096   CCcc 9285   1c1 9288    + caddc 9290    x. cmul 9292    - cmin 9600    / cdiv 9998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  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 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999
This theorem is referenced by: (None)
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