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Theorem bj-bary1lem1 32442
Description: A lemma for 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.s  |-  ( ph  ->  S  e.  CC )
bj-bary1.t  |-  ( ph  ->  T  e.  CC )
bj-bary1.x  |-  ( ph  ->  X  e.  CC )
bj-bary1.neq  |-  ( ph  ->  A  =/=  B )
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 9712 . . . . . 6  |-  ( ph  ->  ( ( S  +  T )  -  T
)  =  S )
4 oveq1 6093 . . . . . 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 2456 . . . . . 6  |-  ( ( ( ( S  +  T )  -  T
)  =  ( 1  -  T )  /\  ( ( S  +  T )  -  T
)  =  S )  ->  ( 1  -  T )  =  S )
87eqcomd 2443 . . . . 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 6093 . . . . . . . 8  |-  ( S  =  ( 1  -  T )  ->  ( S  x.  A )  =  ( ( 1  -  T )  x.  A ) )
1110oveq1d 6101 . . . . . . 7  |-  ( S  =  ( 1  -  T )  ->  (
( S  x.  A
)  +  ( T  x.  B ) )  =  ( ( ( 1  -  T )  x.  A )  +  ( T  x.  B
) ) )
12 eqtr 2455 . . . . . . 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 9394 . . . . . . . . 9  |-  ( ph  ->  1  e.  CC )
15 bj-bary1.a . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
1614, 2, 15subdird 9793 . . . . . . . 8  |-  ( ph  ->  ( ( 1  -  T )  x.  A
)  =  ( ( 1  x.  A )  -  ( T  x.  A ) ) )
1715mulid2d 9396 . . . . . . . . 9  |-  ( ph  ->  ( 1  x.  A
)  =  A )
1817oveq1d 6101 . . . . . . . 8  |-  ( ph  ->  ( ( 1  x.  A )  -  ( T  x.  A )
)  =  ( A  -  ( T  x.  A ) ) )
1916, 18eqtrd 2470 . . . . . . 7  |-  ( ph  ->  ( ( 1  -  T )  x.  A
)  =  ( A  -  ( T  x.  A ) ) )
2019oveq1d 6101 . . . . . 6  |-  ( ph  ->  ( ( ( 1  -  T )  x.  A )  +  ( T  x.  B ) )  =  ( ( A  -  ( T  x.  A ) )  +  ( T  x.  B ) ) )
2113, 20sylan9eqr 2492 . . . . 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 9398 . . . . . 6  |-  ( ph  ->  ( T  x.  A
)  e.  CC )
25 bj-bary1.b . . . . . . 7  |-  ( ph  ->  B  e.  CC )
262, 25mulcld 9398 . . . . . 6  |-  ( ph  ->  ( T  x.  B
)  e.  CC )
2715, 24, 26subadd23d 9733 . . . . 5  |-  ( ph  ->  ( ( A  -  ( T  x.  A
) )  +  ( T  x.  B ) )  =  ( A  +  ( ( T  x.  B )  -  ( T  x.  A
) ) ) )
282, 25, 15subdid 9792 . . . . . . 7  |-  ( ph  ->  ( T  x.  ( B  -  A )
)  =  ( ( T  x.  B )  -  ( T  x.  A ) ) )
2928eqcomd 2443 . . . . . 6  |-  ( ph  ->  ( ( T  x.  B )  -  ( T  x.  A )
)  =  ( T  x.  ( B  -  A ) ) )
3029oveq2d 6102 . . . . 5  |-  ( ph  ->  ( A  +  ( ( T  x.  B
)  -  ( T  x.  A ) ) )  =  ( A  +  ( T  x.  ( B  -  A
) ) ) )
3127, 30eqtrd 2470 . . . 4  |-  ( ph  ->  ( ( A  -  ( T  x.  A
) )  +  ( T  x.  B ) )  =  ( A  +  ( T  x.  ( B  -  A
) ) ) )
3231eqeq2d 2449 . . 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 6093 . . 3  |-  ( X  =  ( A  +  ( T  x.  ( B  -  A )
) )  ->  ( X  -  A )  =  ( ( A  +  ( T  x.  ( B  -  A
) ) )  -  A ) )
3525, 15subcld 9711 . . . . . 6  |-  ( ph  ->  ( B  -  A
)  e.  CC )
362, 35mulcld 9398 . . . . 5  |-  ( ph  ->  ( T  x.  ( B  -  A )
)  e.  CC )
3715, 36pncan2d 9713 . . . 4  |-  ( ph  ->  ( ( A  +  ( T  x.  ( B  -  A )
) )  -  A
)  =  ( T  x.  ( B  -  A ) ) )
3837eqeq2d 2449 . . 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 2440 . . 3  |-  ( ( X  -  A )  =  ( T  x.  ( B  -  A
) )  <->  ( T  x.  ( B  -  A
) )  =  ( X  -  A ) )
412, 35mulcomd 9399 . . . . 5  |-  ( ph  ->  ( T  x.  ( B  -  A )
)  =  ( ( B  -  A )  x.  T ) )
4241eqeq1d 2446 . . . 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 9711 . . . . . 6  |-  ( ph  ->  ( X  -  A
)  e.  CC )
45 bj-bary1.neq . . . . . . . 8  |-  ( ph  ->  A  =/=  B )
4645necomd 2690 . . . . . . 7  |-  ( ph  ->  B  =/=  A )
4725, 15, 46subne0d 9720 . . . . . 6  |-  ( ph  ->  ( B  -  A
)  =/=  0 )
4835, 2, 44, 47bj-rdiv 32437 . . . . 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 1369    e. wcel 1756    =/= wne 2601  (class class class)co 6086   CCcc 9272   1c1 9275    + caddc 9277    x. cmul 9279    - cmin 9587    / cdiv 9985
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 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-po 4636  df-so 4637  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986
This theorem is referenced by:  bj-bary1  32443
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