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Theorem affineequiv 20620
Description: Equivalence between two ways of expressing  B as an affine combination of  A and  C. (Contributed by David Moews, 28-Feb-2017.)
Hypotheses
Ref Expression
affineequiv.A  |-  ( ph  ->  A  e.  CC )
affineequiv.B  |-  ( ph  ->  B  e.  CC )
affineequiv.C  |-  ( ph  ->  C  e.  CC )
affineequiv.D  |-  ( ph  ->  D  e.  CC )
Assertion
Ref Expression
affineequiv  |-  ( ph  ->  ( B  =  ( ( D  x.  A
)  +  ( ( 1  -  D )  x.  C ) )  <-> 
( C  -  B
)  =  ( D  x.  ( C  -  A ) ) ) )

Proof of Theorem affineequiv
StepHypRef Expression
1 affineequiv.C . . . . . . . 8  |-  ( ph  ->  C  e.  CC )
2 affineequiv.D . . . . . . . . 9  |-  ( ph  ->  D  e.  CC )
32, 1mulcld 9064 . . . . . . . 8  |-  ( ph  ->  ( D  x.  C
)  e.  CC )
4 affineequiv.A . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
52, 4mulcld 9064 . . . . . . . 8  |-  ( ph  ->  ( D  x.  A
)  e.  CC )
61, 3, 5subsubd 9395 . . . . . . 7  |-  ( ph  ->  ( C  -  (
( D  x.  C
)  -  ( D  x.  A ) ) )  =  ( ( C  -  ( D  x.  C ) )  +  ( D  x.  A ) ) )
71, 3subcld 9367 . . . . . . . 8  |-  ( ph  ->  ( C  -  ( D  x.  C )
)  e.  CC )
87, 5addcomd 9224 . . . . . . 7  |-  ( ph  ->  ( ( C  -  ( D  x.  C
) )  +  ( D  x.  A ) )  =  ( ( D  x.  A )  +  ( C  -  ( D  x.  C
) ) ) )
96, 8eqtr2d 2437 . . . . . 6  |-  ( ph  ->  ( ( D  x.  A )  +  ( C  -  ( D  x.  C ) ) )  =  ( C  -  ( ( D  x.  C )  -  ( D  x.  A
) ) ) )
10 ax-1cn 9004 . . . . . . . . . 10  |-  1  e.  CC
1110a1i 11 . . . . . . . . 9  |-  ( ph  ->  1  e.  CC )
1211, 2, 1subdird 9446 . . . . . . . 8  |-  ( ph  ->  ( ( 1  -  D )  x.  C
)  =  ( ( 1  x.  C )  -  ( D  x.  C ) ) )
131mulid2d 9062 . . . . . . . . 9  |-  ( ph  ->  ( 1  x.  C
)  =  C )
1413oveq1d 6055 . . . . . . . 8  |-  ( ph  ->  ( ( 1  x.  C )  -  ( D  x.  C )
)  =  ( C  -  ( D  x.  C ) ) )
1512, 14eqtrd 2436 . . . . . . 7  |-  ( ph  ->  ( ( 1  -  D )  x.  C
)  =  ( C  -  ( D  x.  C ) ) )
1615oveq2d 6056 . . . . . 6  |-  ( ph  ->  ( ( D  x.  A )  +  ( ( 1  -  D
)  x.  C ) )  =  ( ( D  x.  A )  +  ( C  -  ( D  x.  C
) ) ) )
17 affineequiv.B . . . . . . . 8  |-  ( ph  ->  B  e.  CC )
181, 17subcld 9367 . . . . . . . 8  |-  ( ph  ->  ( C  -  B
)  e.  CC )
191, 4subcld 9367 . . . . . . . . 9  |-  ( ph  ->  ( C  -  A
)  e.  CC )
202, 19mulcld 9064 . . . . . . . 8  |-  ( ph  ->  ( D  x.  ( C  -  A )
)  e.  CC )
2117, 18, 20addsubassd 9387 . . . . . . 7  |-  ( ph  ->  ( ( B  +  ( C  -  B
) )  -  ( D  x.  ( C  -  A ) ) )  =  ( B  +  ( ( C  -  B )  -  ( D  x.  ( C  -  A ) ) ) ) )
2217, 1pncan3d 9370 . . . . . . . 8  |-  ( ph  ->  ( B  +  ( C  -  B ) )  =  C )
232, 1, 4subdid 9445 . . . . . . . 8  |-  ( ph  ->  ( D  x.  ( C  -  A )
)  =  ( ( D  x.  C )  -  ( D  x.  A ) ) )
2422, 23oveq12d 6058 . . . . . . 7  |-  ( ph  ->  ( ( B  +  ( C  -  B
) )  -  ( D  x.  ( C  -  A ) ) )  =  ( C  -  ( ( D  x.  C )  -  ( D  x.  A )
) ) )
2521, 24eqtr3d 2438 . . . . . 6  |-  ( ph  ->  ( B  +  ( ( C  -  B
)  -  ( D  x.  ( C  -  A ) ) ) )  =  ( C  -  ( ( D  x.  C )  -  ( D  x.  A
) ) ) )
269, 16, 253eqtr4d 2446 . . . . 5  |-  ( ph  ->  ( ( D  x.  A )  +  ( ( 1  -  D
)  x.  C ) )  =  ( B  +  ( ( C  -  B )  -  ( D  x.  ( C  -  A )
) ) ) )
2726eqeq2d 2415 . . . 4  |-  ( ph  ->  ( B  =  ( ( D  x.  A
)  +  ( ( 1  -  D )  x.  C ) )  <-> 
B  =  ( B  +  ( ( C  -  B )  -  ( D  x.  ( C  -  A )
) ) ) ) )
2817addid1d 9222 . . . . 5  |-  ( ph  ->  ( B  +  0 )  =  B )
2928eqeq1d 2412 . . . 4  |-  ( ph  ->  ( ( B  + 
0 )  =  ( B  +  ( ( C  -  B )  -  ( D  x.  ( C  -  A
) ) ) )  <-> 
B  =  ( B  +  ( ( C  -  B )  -  ( D  x.  ( C  -  A )
) ) ) ) )
30 0cn 9040 . . . . . 6  |-  0  e.  CC
3130a1i 11 . . . . 5  |-  ( ph  ->  0  e.  CC )
3218, 20subcld 9367 . . . . 5  |-  ( ph  ->  ( ( C  -  B )  -  ( D  x.  ( C  -  A ) ) )  e.  CC )
3317, 31, 32addcand 9225 . . . 4  |-  ( ph  ->  ( ( B  + 
0 )  =  ( B  +  ( ( C  -  B )  -  ( D  x.  ( C  -  A
) ) ) )  <->  0  =  ( ( C  -  B )  -  ( D  x.  ( C  -  A
) ) ) ) )
3427, 29, 333bitr2d 273 . . 3  |-  ( ph  ->  ( B  =  ( ( D  x.  A
)  +  ( ( 1  -  D )  x.  C ) )  <->  0  =  ( ( C  -  B )  -  ( D  x.  ( C  -  A
) ) ) ) )
35 eqcom 2406 . . 3  |-  ( 0  =  ( ( C  -  B )  -  ( D  x.  ( C  -  A )
) )  <->  ( ( C  -  B )  -  ( D  x.  ( C  -  A
) ) )  =  0 )
3634, 35syl6bb 253 . 2  |-  ( ph  ->  ( B  =  ( ( D  x.  A
)  +  ( ( 1  -  D )  x.  C ) )  <-> 
( ( C  -  B )  -  ( D  x.  ( C  -  A ) ) )  =  0 ) )
3718, 20subeq0ad 9377 . 2  |-  ( ph  ->  ( ( ( C  -  B )  -  ( D  x.  ( C  -  A )
) )  =  0  <-> 
( C  -  B
)  =  ( D  x.  ( C  -  A ) ) ) )
3836, 37bitrd 245 1  |-  ( ph  ->  ( B  =  ( ( D  x.  A
)  +  ( ( 1  -  D )  x.  C ) )  <-> 
( C  -  B
)  =  ( D  x.  ( C  -  A ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1721  (class class class)co 6040   CCcc 8944   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    - cmin 9247
This theorem is referenced by:  affineequiv2  20621  angpieqvd  20625  chordthmlem2  20627  chordthmlem4  20629
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-ltxr 9081  df-sub 9249
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