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Theorem bj-bary1lem 35076
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 9605 . . . . . . . . 9  |-  ( ph  ->  ( B  x.  A
)  e.  CC )
4 bj-bary1.x . . . . . . . . . 10  |-  ( ph  ->  X  e.  CC )
54, 2mulcld 9605 . . . . . . . . 9  |-  ( ph  ->  ( X  x.  A
)  e.  CC )
63, 5subcld 9922 . . . . . . . 8  |-  ( ph  ->  ( ( B  x.  A )  -  ( X  x.  A )
)  e.  CC )
74, 1mulcld 9605 . . . . . . . 8  |-  ( ph  ->  ( X  x.  B
)  e.  CC )
82, 1mulcld 9605 . . . . . . . 8  |-  ( ph  ->  ( A  x.  B
)  e.  CC )
96, 7, 8addsub12d 9945 . . . . . . 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 9954 . . . . . . . . 9  |-  ( ph  ->  ( ( ( B  x.  A )  -  ( X  x.  A
) )  -  ( A  x.  B )
)  =  ( ( ( B  x.  A
)  -  ( A  x.  B ) )  -  ( X  x.  A ) ) )
111, 2bj-subcom 35070 . . . . . . . . . 10  |-  ( ph  ->  ( ( B  x.  A )  -  ( A  x.  B )
)  =  0 )
1211oveq1d 6285 . . . . . . . . 9  |-  ( ph  ->  ( ( ( B  x.  A )  -  ( A  x.  B
) )  -  ( X  x.  A )
)  =  ( 0  -  ( X  x.  A ) ) )
1310, 12eqtrd 2495 . . . . . . . 8  |-  ( ph  ->  ( ( ( B  x.  A )  -  ( X  x.  A
) )  -  ( A  x.  B )
)  =  ( 0  -  ( X  x.  A ) ) )
1413oveq2d 6286 . . . . . . 7  |-  ( ph  ->  ( ( X  x.  B )  +  ( ( ( B  x.  A )  -  ( X  x.  A )
)  -  ( A  x.  B ) ) )  =  ( ( X  x.  B )  +  ( 0  -  ( X  x.  A
) ) ) )
159, 14eqtrd 2495 . . . . . 6  |-  ( ph  ->  ( ( ( B  x.  A )  -  ( X  x.  A
) )  +  ( ( X  x.  B
)  -  ( A  x.  B ) ) )  =  ( ( X  x.  B )  +  ( 0  -  ( X  x.  A
) ) ) )
16 0cnd 9578 . . . . . . 7  |-  ( ph  ->  0  e.  CC )
177, 16, 5addsubassd 9942 . . . . . 6  |-  ( ph  ->  ( ( ( X  x.  B )  +  0 )  -  ( X  x.  A )
)  =  ( ( X  x.  B )  +  ( 0  -  ( X  x.  A
) ) ) )
187addid1d 9769 . . . . . . 7  |-  ( ph  ->  ( ( X  x.  B )  +  0 )  =  ( X  x.  B ) )
1918oveq1d 6285 . . . . . 6  |-  ( ph  ->  ( ( ( X  x.  B )  +  0 )  -  ( X  x.  A )
)  =  ( ( X  x.  B )  -  ( X  x.  A ) ) )
2015, 17, 193eqtr2d 2501 . . . . 5  |-  ( ph  ->  ( ( ( B  x.  A )  -  ( X  x.  A
) )  +  ( ( X  x.  B
)  -  ( A  x.  B ) ) )  =  ( ( X  x.  B )  -  ( X  x.  A ) ) )
211, 4, 2subdird 10009 . . . . . 6  |-  ( ph  ->  ( ( B  -  X )  x.  A
)  =  ( ( B  x.  A )  -  ( X  x.  A ) ) )
224, 2, 1subdird 10009 . . . . . 6  |-  ( ph  ->  ( ( X  -  A )  x.  B
)  =  ( ( X  x.  B )  -  ( A  x.  B ) ) )
2321, 22oveq12d 6288 . . . . 5  |-  ( ph  ->  ( ( ( B  -  X )  x.  A )  +  ( ( X  -  A
)  x.  B ) )  =  ( ( ( B  x.  A
)  -  ( X  x.  A ) )  +  ( ( X  x.  B )  -  ( A  x.  B
) ) ) )
244, 1, 2subdid 10008 . . . . 5  |-  ( ph  ->  ( X  x.  ( B  -  A )
)  =  ( ( X  x.  B )  -  ( X  x.  A ) ) )
2520, 23, 243eqtr4rd 2506 . . . 4  |-  ( ph  ->  ( X  x.  ( B  -  A )
)  =  ( ( ( B  -  X
)  x.  A )  +  ( ( X  -  A )  x.  B ) ) )
2625oveq1d 6285 . . 3  |-  ( ph  ->  ( ( X  x.  ( B  -  A
) )  /  ( B  -  A )
)  =  ( ( ( ( B  -  X )  x.  A
)  +  ( ( X  -  A )  x.  B ) )  /  ( B  -  A ) ) )
271, 4subcld 9922 . . . . 5  |-  ( ph  ->  ( B  -  X
)  e.  CC )
2827, 2mulcld 9605 . . . 4  |-  ( ph  ->  ( ( B  -  X )  x.  A
)  e.  CC )
294, 2subcld 9922 . . . . 5  |-  ( ph  ->  ( X  -  A
)  e.  CC )
3029, 1mulcld 9605 . . . 4  |-  ( ph  ->  ( ( X  -  A )  x.  B
)  e.  CC )
311, 2subcld 9922 . . . 4  |-  ( ph  ->  ( B  -  A
)  e.  CC )
32 bj-bary1.neq . . . . . 6  |-  ( ph  ->  A  =/=  B )
3332necomd 2725 . . . . 5  |-  ( ph  ->  B  =/=  A )
341, 2, 33subne0d 9931 . . . 4  |-  ( ph  ->  ( B  -  A
)  =/=  0 )
3528, 30, 31, 34divdird 10354 . . 3  |-  ( ph  ->  ( ( ( ( B  -  X )  x.  A )  +  ( ( X  -  A )  x.  B
) )  /  ( B  -  A )
)  =  ( ( ( ( B  -  X )  x.  A
)  /  ( B  -  A ) )  +  ( ( ( X  -  A )  x.  B )  / 
( B  -  A
) ) ) )
3626, 35eqtrd 2495 . 2  |-  ( ph  ->  ( ( X  x.  ( B  -  A
) )  /  ( B  -  A )
)  =  ( ( ( ( B  -  X )  x.  A
)  /  ( B  -  A ) )  +  ( ( ( X  -  A )  x.  B )  / 
( B  -  A
) ) ) )
374, 31, 34divcan4d 10322 . 2  |-  ( ph  ->  ( ( X  x.  ( B  -  A
) )  /  ( B  -  A )
)  =  X )
3827, 2, 31, 34div23d 10353 . . 3  |-  ( ph  ->  ( ( ( B  -  X )  x.  A )  /  ( B  -  A )
)  =  ( ( ( B  -  X
)  /  ( B  -  A ) )  x.  A ) )
3929, 1, 31, 34div23d 10353 . . 3  |-  ( ph  ->  ( ( ( X  -  A )  x.  B )  /  ( B  -  A )
)  =  ( ( ( X  -  A
)  /  ( B  -  A ) )  x.  B ) )
4038, 39oveq12d 6288 . 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 ) ) )
4136, 37, 403eqtr3d 2503 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 1398    e. wcel 1823    =/= wne 2649  (class class class)co 6270   CCcc 9479   0cc0 9481    + caddc 9484    x. cmul 9486    - cmin 9796    / cdiv 10202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203
This theorem is referenced by:  bj-bary1  35078
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