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Theorem bj-bary1lem 32611
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.x  |-  ( ph  ->  X  e.  CC )
bj-bary1.neq  |-  ( ph  ->  A  =/=  B )
Assertion
Ref Expression
bj-bary1lem  |-  ( ph  ->  X  =  ( ( ( ( B  -  X )  /  ( B  -  A )
)  x.  A )  +  ( ( ( X  -  A )  /  ( B  -  A ) )  x.  B ) ) )

Proof of Theorem bj-bary1lem
StepHypRef Expression
1 bj-bary1.b . . . . . . . . . 10  |-  ( ph  ->  B  e.  CC )
2 bj-bary1.a . . . . . . . . . 10  |-  ( ph  ->  A  e.  CC )
31, 2mulcld 9418 . . . . . . . . 9  |-  ( ph  ->  ( B  x.  A
)  e.  CC )
4 bj-bary1.x . . . . . . . . . 10  |-  ( ph  ->  X  e.  CC )
54, 2mulcld 9418 . . . . . . . . 9  |-  ( ph  ->  ( X  x.  A
)  e.  CC )
63, 5subcld 9731 . . . . . . . 8  |-  ( ph  ->  ( ( B  x.  A )  -  ( X  x.  A )
)  e.  CC )
74, 1mulcld 9418 . . . . . . . 8  |-  ( ph  ->  ( X  x.  B
)  e.  CC )
82, 1mulcld 9418 . . . . . . . 8  |-  ( ph  ->  ( A  x.  B
)  e.  CC )
96, 7, 8addsub12d 9754 . . . . . . 7  |-  ( ph  ->  ( ( ( B  x.  A )  -  ( X  x.  A
) )  +  ( ( X  x.  B
)  -  ( A  x.  B ) ) )  =  ( ( X  x.  B )  +  ( ( ( B  x.  A )  -  ( X  x.  A ) )  -  ( A  x.  B
) ) ) )
103, 5, 8sub32d 9763 . . . . . . . . 9  |-  ( ph  ->  ( ( ( B  x.  A )  -  ( X  x.  A
) )  -  ( A  x.  B )
)  =  ( ( ( B  x.  A
)  -  ( A  x.  B ) )  -  ( X  x.  A ) ) )
111, 2mulcomd 9419 . . . . . . . . . . 11  |-  ( ph  ->  ( B  x.  A
)  =  ( A  x.  B ) )
123, 11subeq0bd 9786 . . . . . . . . . 10  |-  ( ph  ->  ( ( B  x.  A )  -  ( A  x.  B )
)  =  0 )
1312oveq1d 6118 . . . . . . . . 9  |-  ( ph  ->  ( ( ( B  x.  A )  -  ( A  x.  B
) )  -  ( X  x.  A )
)  =  ( 0  -  ( X  x.  A ) ) )
1410, 13eqtrd 2475 . . . . . . . 8  |-  ( ph  ->  ( ( ( B  x.  A )  -  ( X  x.  A
) )  -  ( A  x.  B )
)  =  ( 0  -  ( X  x.  A ) ) )
1514oveq2d 6119 . . . . . . 7  |-  ( ph  ->  ( ( X  x.  B )  +  ( ( ( B  x.  A )  -  ( X  x.  A )
)  -  ( A  x.  B ) ) )  =  ( ( X  x.  B )  +  ( 0  -  ( X  x.  A
) ) ) )
169, 15eqtrd 2475 . . . . . 6  |-  ( ph  ->  ( ( ( B  x.  A )  -  ( X  x.  A
) )  +  ( ( X  x.  B
)  -  ( A  x.  B ) ) )  =  ( ( X  x.  B )  +  ( 0  -  ( X  x.  A
) ) ) )
17 0cnd 9391 . . . . . . 7  |-  ( ph  ->  0  e.  CC )
187, 17, 5addsubassd 9751 . . . . . 6  |-  ( ph  ->  ( ( ( X  x.  B )  +  0 )  -  ( X  x.  A )
)  =  ( ( X  x.  B )  +  ( 0  -  ( X  x.  A
) ) ) )
197addid1d 9581 . . . . . . 7  |-  ( ph  ->  ( ( X  x.  B )  +  0 )  =  ( X  x.  B ) )
2019oveq1d 6118 . . . . . 6  |-  ( ph  ->  ( ( ( X  x.  B )  +  0 )  -  ( X  x.  A )
)  =  ( ( X  x.  B )  -  ( X  x.  A ) ) )
2116, 18, 203eqtr2d 2481 . . . . 5  |-  ( ph  ->  ( ( ( B  x.  A )  -  ( X  x.  A
) )  +  ( ( X  x.  B
)  -  ( A  x.  B ) ) )  =  ( ( X  x.  B )  -  ( X  x.  A ) ) )
221, 4, 2subdird 9813 . . . . . 6  |-  ( ph  ->  ( ( B  -  X )  x.  A
)  =  ( ( B  x.  A )  -  ( X  x.  A ) ) )
234, 2, 1subdird 9813 . . . . . 6  |-  ( ph  ->  ( ( X  -  A )  x.  B
)  =  ( ( X  x.  B )  -  ( A  x.  B ) ) )
2422, 23oveq12d 6121 . . . . 5  |-  ( ph  ->  ( ( ( B  -  X )  x.  A )  +  ( ( X  -  A
)  x.  B ) )  =  ( ( ( B  x.  A
)  -  ( X  x.  A ) )  +  ( ( X  x.  B )  -  ( A  x.  B
) ) ) )
254, 1, 2subdid 9812 . . . . 5  |-  ( ph  ->  ( X  x.  ( B  -  A )
)  =  ( ( X  x.  B )  -  ( X  x.  A ) ) )
2621, 24, 253eqtr4rd 2486 . . . 4  |-  ( ph  ->  ( X  x.  ( B  -  A )
)  =  ( ( ( B  -  X
)  x.  A )  +  ( ( X  -  A )  x.  B ) ) )
2726oveq1d 6118 . . 3  |-  ( ph  ->  ( ( X  x.  ( B  -  A
) )  /  ( B  -  A )
)  =  ( ( ( ( B  -  X )  x.  A
)  +  ( ( X  -  A )  x.  B ) )  /  ( B  -  A ) ) )
281, 4subcld 9731 . . . . 5  |-  ( ph  ->  ( B  -  X
)  e.  CC )
2928, 2mulcld 9418 . . . 4  |-  ( ph  ->  ( ( B  -  X )  x.  A
)  e.  CC )
304, 2subcld 9731 . . . . 5  |-  ( ph  ->  ( X  -  A
)  e.  CC )
3130, 1mulcld 9418 . . . 4  |-  ( ph  ->  ( ( X  -  A )  x.  B
)  e.  CC )
321, 2subcld 9731 . . . 4  |-  ( ph  ->  ( B  -  A
)  e.  CC )
33 bj-bary1.neq . . . . . 6  |-  ( ph  ->  A  =/=  B )
3433necomd 2707 . . . . 5  |-  ( ph  ->  B  =/=  A )
351, 2, 34subne0d 9740 . . . 4  |-  ( ph  ->  ( B  -  A
)  =/=  0 )
3629, 31, 32, 35divdird 10157 . . 3  |-  ( ph  ->  ( ( ( ( B  -  X )  x.  A )  +  ( ( X  -  A )  x.  B
) )  /  ( B  -  A )
)  =  ( ( ( ( B  -  X )  x.  A
)  /  ( B  -  A ) )  +  ( ( ( X  -  A )  x.  B )  / 
( B  -  A
) ) ) )
3727, 36eqtrd 2475 . 2  |-  ( ph  ->  ( ( X  x.  ( B  -  A
) )  /  ( B  -  A )
)  =  ( ( ( ( B  -  X )  x.  A
)  /  ( B  -  A ) )  +  ( ( ( X  -  A )  x.  B )  / 
( B  -  A
) ) ) )
384, 32, 35divcan4d 10125 . 2  |-  ( ph  ->  ( ( X  x.  ( B  -  A
) )  /  ( B  -  A )
)  =  X )
3928, 2, 32, 35div23d 10156 . . 3  |-  ( ph  ->  ( ( ( B  -  X )  x.  A )  /  ( B  -  A )
)  =  ( ( ( B  -  X
)  /  ( B  -  A ) )  x.  A ) )
4030, 1, 32, 35div23d 10156 . . 3  |-  ( ph  ->  ( ( ( X  -  A )  x.  B )  /  ( B  -  A )
)  =  ( ( ( X  -  A
)  /  ( B  -  A ) )  x.  B ) )
4139, 40oveq12d 6121 . 2  |-  ( ph  ->  ( ( ( ( B  -  X )  x.  A )  / 
( B  -  A
) )  +  ( ( ( X  -  A )  x.  B
)  /  ( B  -  A ) ) )  =  ( ( ( ( B  -  X )  /  ( B  -  A )
)  x.  A )  +  ( ( ( X  -  A )  /  ( B  -  A ) )  x.  B ) ) )
4237, 38, 413eqtr3d 2483 1  |-  ( ph  ->  X  =  ( ( ( ( B  -  X )  /  ( B  -  A )
)  x.  A )  +  ( ( ( X  -  A )  /  ( B  -  A ) )  x.  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756    =/= wne 2618  (class class class)co 6103   CCcc 9292   0cc0 9294    + caddc 9297    x. cmul 9299    - cmin 9607    / cdiv 10005
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 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
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 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-po 4653  df-so 4654  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006
This theorem is referenced by:  bj-bary1  32613
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