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Theorem bj-bary1lem 33760
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 9615 . . . . . . . . 9  |-  ( ph  ->  ( B  x.  A
)  e.  CC )
4 bj-bary1.x . . . . . . . . . 10  |-  ( ph  ->  X  e.  CC )
54, 2mulcld 9615 . . . . . . . . 9  |-  ( ph  ->  ( X  x.  A
)  e.  CC )
63, 5subcld 9929 . . . . . . . 8  |-  ( ph  ->  ( ( B  x.  A )  -  ( X  x.  A )
)  e.  CC )
74, 1mulcld 9615 . . . . . . . 8  |-  ( ph  ->  ( X  x.  B
)  e.  CC )
82, 1mulcld 9615 . . . . . . . 8  |-  ( ph  ->  ( A  x.  B
)  e.  CC )
96, 7, 8addsub12d 9952 . . . . . . 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 9961 . . . . . . . . 9  |-  ( ph  ->  ( ( ( B  x.  A )  -  ( X  x.  A
) )  -  ( A  x.  B )
)  =  ( ( ( B  x.  A
)  -  ( A  x.  B ) )  -  ( X  x.  A ) ) )
111, 2mulcomd 9616 . . . . . . . . . . 11  |-  ( ph  ->  ( B  x.  A
)  =  ( A  x.  B ) )
123, 11subeq0bd 9984 . . . . . . . . . 10  |-  ( ph  ->  ( ( B  x.  A )  -  ( A  x.  B )
)  =  0 )
1312oveq1d 6298 . . . . . . . . 9  |-  ( ph  ->  ( ( ( B  x.  A )  -  ( A  x.  B
) )  -  ( X  x.  A )
)  =  ( 0  -  ( X  x.  A ) ) )
1410, 13eqtrd 2508 . . . . . . . 8  |-  ( ph  ->  ( ( ( B  x.  A )  -  ( X  x.  A
) )  -  ( A  x.  B )
)  =  ( 0  -  ( X  x.  A ) ) )
1514oveq2d 6299 . . . . . . 7  |-  ( ph  ->  ( ( X  x.  B )  +  ( ( ( B  x.  A )  -  ( X  x.  A )
)  -  ( A  x.  B ) ) )  =  ( ( X  x.  B )  +  ( 0  -  ( X  x.  A
) ) ) )
169, 15eqtrd 2508 . . . . . 6  |-  ( ph  ->  ( ( ( B  x.  A )  -  ( X  x.  A
) )  +  ( ( X  x.  B
)  -  ( A  x.  B ) ) )  =  ( ( X  x.  B )  +  ( 0  -  ( X  x.  A
) ) ) )
17 0cnd 9588 . . . . . . 7  |-  ( ph  ->  0  e.  CC )
187, 17, 5addsubassd 9949 . . . . . 6  |-  ( ph  ->  ( ( ( X  x.  B )  +  0 )  -  ( X  x.  A )
)  =  ( ( X  x.  B )  +  ( 0  -  ( X  x.  A
) ) ) )
197addid1d 9778 . . . . . . 7  |-  ( ph  ->  ( ( X  x.  B )  +  0 )  =  ( X  x.  B ) )
2019oveq1d 6298 . . . . . 6  |-  ( ph  ->  ( ( ( X  x.  B )  +  0 )  -  ( X  x.  A )
)  =  ( ( X  x.  B )  -  ( X  x.  A ) ) )
2116, 18, 203eqtr2d 2514 . . . . 5  |-  ( ph  ->  ( ( ( B  x.  A )  -  ( X  x.  A
) )  +  ( ( X  x.  B
)  -  ( A  x.  B ) ) )  =  ( ( X  x.  B )  -  ( X  x.  A ) ) )
221, 4, 2subdird 10012 . . . . . 6  |-  ( ph  ->  ( ( B  -  X )  x.  A
)  =  ( ( B  x.  A )  -  ( X  x.  A ) ) )
234, 2, 1subdird 10012 . . . . . 6  |-  ( ph  ->  ( ( X  -  A )  x.  B
)  =  ( ( X  x.  B )  -  ( A  x.  B ) ) )
2422, 23oveq12d 6301 . . . . 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 10011 . . . . 5  |-  ( ph  ->  ( X  x.  ( B  -  A )
)  =  ( ( X  x.  B )  -  ( X  x.  A ) ) )
2621, 24, 253eqtr4rd 2519 . . . 4  |-  ( ph  ->  ( X  x.  ( B  -  A )
)  =  ( ( ( B  -  X
)  x.  A )  +  ( ( X  -  A )  x.  B ) ) )
2726oveq1d 6298 . . 3  |-  ( ph  ->  ( ( X  x.  ( B  -  A
) )  /  ( B  -  A )
)  =  ( ( ( ( B  -  X )  x.  A
)  +  ( ( X  -  A )  x.  B ) )  /  ( B  -  A ) ) )
281, 4subcld 9929 . . . . 5  |-  ( ph  ->  ( B  -  X
)  e.  CC )
2928, 2mulcld 9615 . . . 4  |-  ( ph  ->  ( ( B  -  X )  x.  A
)  e.  CC )
304, 2subcld 9929 . . . . 5  |-  ( ph  ->  ( X  -  A
)  e.  CC )
3130, 1mulcld 9615 . . . 4  |-  ( ph  ->  ( ( X  -  A )  x.  B
)  e.  CC )
321, 2subcld 9929 . . . 4  |-  ( ph  ->  ( B  -  A
)  e.  CC )
33 bj-bary1.neq . . . . . 6  |-  ( ph  ->  A  =/=  B )
3433necomd 2738 . . . . 5  |-  ( ph  ->  B  =/=  A )
351, 2, 34subne0d 9938 . . . 4  |-  ( ph  ->  ( B  -  A
)  =/=  0 )
3629, 31, 32, 35divdird 10357 . . 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 2508 . 2  |-  ( ph  ->  ( ( X  x.  ( B  -  A
) )  /  ( B  -  A )
)  =  ( ( ( ( B  -  X )  x.  A
)  /  ( B  -  A ) )  +  ( ( ( X  -  A )  x.  B )  / 
( B  -  A
) ) ) )
384, 32, 35divcan4d 10325 . 2  |-  ( ph  ->  ( ( X  x.  ( B  -  A
) )  /  ( B  -  A )
)  =  X )
3928, 2, 32, 35div23d 10356 . . 3  |-  ( ph  ->  ( ( ( B  -  X )  x.  A )  /  ( B  -  A )
)  =  ( ( ( B  -  X
)  /  ( B  -  A ) )  x.  A ) )
4030, 1, 32, 35div23d 10356 . . 3  |-  ( ph  ->  ( ( ( X  -  A )  x.  B )  /  ( B  -  A )
)  =  ( ( ( X  -  A
)  /  ( B  -  A ) )  x.  B ) )
4139, 40oveq12d 6301 . 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 2516 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 1379    e. wcel 1767    =/= wne 2662  (class class class)co 6283   CCcc 9489   0cc0 9491    + caddc 9494    x. cmul 9496    - cmin 9804    / cdiv 10205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206
This theorem is referenced by:  bj-bary1  33762
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