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Theorem affineequiv 20705
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 9146 . . . . . . . 8  |-  ( ph  ->  ( D  x.  C
)  e.  CC )
4 affineequiv.A . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
52, 4mulcld 9146 . . . . . . . 8  |-  ( ph  ->  ( D  x.  A
)  e.  CC )
61, 3, 5subsubd 9477 . . . . . . 7  |-  ( ph  ->  ( C  -  (
( D  x.  C
)  -  ( D  x.  A ) ) )  =  ( ( C  -  ( D  x.  C ) )  +  ( D  x.  A ) ) )
71, 3subcld 9449 . . . . . . . 8  |-  ( ph  ->  ( C  -  ( D  x.  C )
)  e.  CC )
87, 5addcomd 9306 . . . . . . 7  |-  ( ph  ->  ( ( C  -  ( D  x.  C
) )  +  ( D  x.  A ) )  =  ( ( D  x.  A )  +  ( C  -  ( D  x.  C
) ) ) )
96, 8eqtr2d 2476 . . . . . 6  |-  ( ph  ->  ( ( D  x.  A )  +  ( C  -  ( D  x.  C ) ) )  =  ( C  -  ( ( D  x.  C )  -  ( D  x.  A
) ) ) )
10 ax-1cn 9086 . . . . . . . . . 10  |-  1  e.  CC
1110a1i 11 . . . . . . . . 9  |-  ( ph  ->  1  e.  CC )
1211, 2, 1subdird 9528 . . . . . . . 8  |-  ( ph  ->  ( ( 1  -  D )  x.  C
)  =  ( ( 1  x.  C )  -  ( D  x.  C ) ) )
131mulid2d 9144 . . . . . . . . 9  |-  ( ph  ->  ( 1  x.  C
)  =  C )
1413oveq1d 6132 . . . . . . . 8  |-  ( ph  ->  ( ( 1  x.  C )  -  ( D  x.  C )
)  =  ( C  -  ( D  x.  C ) ) )
1512, 14eqtrd 2475 . . . . . . 7  |-  ( ph  ->  ( ( 1  -  D )  x.  C
)  =  ( C  -  ( D  x.  C ) ) )
1615oveq2d 6133 . . . . . 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 9449 . . . . . . . 8  |-  ( ph  ->  ( C  -  B
)  e.  CC )
191, 4subcld 9449 . . . . . . . . 9  |-  ( ph  ->  ( C  -  A
)  e.  CC )
202, 19mulcld 9146 . . . . . . . 8  |-  ( ph  ->  ( D  x.  ( C  -  A )
)  e.  CC )
2117, 18, 20addsubassd 9469 . . . . . . 7  |-  ( ph  ->  ( ( B  +  ( C  -  B
) )  -  ( D  x.  ( C  -  A ) ) )  =  ( B  +  ( ( C  -  B )  -  ( D  x.  ( C  -  A ) ) ) ) )
2217, 1pncan3d 9452 . . . . . . . 8  |-  ( ph  ->  ( B  +  ( C  -  B ) )  =  C )
232, 1, 4subdid 9527 . . . . . . . 8  |-  ( ph  ->  ( D  x.  ( C  -  A )
)  =  ( ( D  x.  C )  -  ( D  x.  A ) ) )
2422, 23oveq12d 6135 . . . . . . 7  |-  ( ph  ->  ( ( B  +  ( C  -  B
) )  -  ( D  x.  ( C  -  A ) ) )  =  ( C  -  ( ( D  x.  C )  -  ( D  x.  A )
) ) )
2521, 24eqtr3d 2477 . . . . . 6  |-  ( ph  ->  ( B  +  ( ( C  -  B
)  -  ( D  x.  ( C  -  A ) ) ) )  =  ( C  -  ( ( D  x.  C )  -  ( D  x.  A
) ) ) )
269, 16, 253eqtr4d 2485 . . . . 5  |-  ( ph  ->  ( ( D  x.  A )  +  ( ( 1  -  D
)  x.  C ) )  =  ( B  +  ( ( C  -  B )  -  ( D  x.  ( C  -  A )
) ) ) )
2726eqeq2d 2454 . . . 4  |-  ( ph  ->  ( B  =  ( ( D  x.  A
)  +  ( ( 1  -  D )  x.  C ) )  <-> 
B  =  ( B  +  ( ( C  -  B )  -  ( D  x.  ( C  -  A )
) ) ) ) )
2817addid1d 9304 . . . . 5  |-  ( ph  ->  ( B  +  0 )  =  B )
2928eqeq1d 2451 . . . 4  |-  ( ph  ->  ( ( B  + 
0 )  =  ( B  +  ( ( C  -  B )  -  ( D  x.  ( C  -  A
) ) ) )  <-> 
B  =  ( B  +  ( ( C  -  B )  -  ( D  x.  ( C  -  A )
) ) ) ) )
30 0cn 9122 . . . . . 6  |-  0  e.  CC
3130a1i 11 . . . . 5  |-  ( ph  ->  0  e.  CC )
3218, 20subcld 9449 . . . . 5  |-  ( ph  ->  ( ( C  -  B )  -  ( D  x.  ( C  -  A ) ) )  e.  CC )
3317, 31, 32addcand 9307 . . . 4  |-  ( ph  ->  ( ( B  + 
0 )  =  ( B  +  ( ( C  -  B )  -  ( D  x.  ( C  -  A
) ) ) )  <->  0  =  ( ( C  -  B )  -  ( D  x.  ( C  -  A
) ) ) ) )
3427, 29, 333bitr2d 274 . . 3  |-  ( ph  ->  ( B  =  ( ( D  x.  A
)  +  ( ( 1  -  D )  x.  C ) )  <->  0  =  ( ( C  -  B )  -  ( D  x.  ( C  -  A
) ) ) ) )
35 eqcom 2445 . . 3  |-  ( 0  =  ( ( C  -  B )  -  ( D  x.  ( C  -  A )
) )  <->  ( ( C  -  B )  -  ( D  x.  ( C  -  A
) ) )  =  0 )
3634, 35syl6bb 254 . 2  |-  ( ph  ->  ( B  =  ( ( D  x.  A
)  +  ( ( 1  -  D )  x.  C ) )  <-> 
( ( C  -  B )  -  ( D  x.  ( C  -  A ) ) )  =  0 ) )
3718, 20subeq0ad 9459 . 2  |-  ( ph  ->  ( ( ( C  -  B )  -  ( D  x.  ( C  -  A )
) )  =  0  <-> 
( C  -  B
)  =  ( D  x.  ( C  -  A ) ) ) )
3836, 37bitrd 246 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 178    = wceq 1654    e. wcel 1728  (class class class)co 6117   CCcc 9026   0cc0 9028   1c1 9029    + caddc 9031    x. cmul 9033    - cmin 9329
This theorem is referenced by:  affineequiv2  20706  angpieqvd  20710  chordthmlem2  20712  chordthmlem4  20714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736  ax-resscn 9085  ax-1cn 9086  ax-icn 9087  ax-addcl 9088  ax-addrcl 9089  ax-mulcl 9090  ax-mulrcl 9091  ax-mulcom 9092  ax-addass 9093  ax-mulass 9094  ax-distr 9095  ax-i2m1 9096  ax-1ne0 9097  ax-1rid 9098  ax-rnegex 9099  ax-rrecex 9100  ax-cnre 9101  ax-pre-lttri 9102  ax-pre-lttrn 9103  ax-pre-ltadd 9104
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2717  df-rex 2718  df-reu 2719  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-po 4538  df-so 4539  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-riota 6585  df-er 6941  df-en 7146  df-dom 7147  df-sdom 7148  df-pnf 9160  df-mnf 9161  df-ltxr 9163  df-sub 9331
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