MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  affineequiv Structured version   Unicode version

Theorem affineequiv 20667
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 9108 . . . . . . . 8  |-  ( ph  ->  ( D  x.  C
)  e.  CC )
4 affineequiv.A . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
52, 4mulcld 9108 . . . . . . . 8  |-  ( ph  ->  ( D  x.  A
)  e.  CC )
61, 3, 5subsubd 9439 . . . . . . 7  |-  ( ph  ->  ( C  -  (
( D  x.  C
)  -  ( D  x.  A ) ) )  =  ( ( C  -  ( D  x.  C ) )  +  ( D  x.  A ) ) )
71, 3subcld 9411 . . . . . . . 8  |-  ( ph  ->  ( C  -  ( D  x.  C )
)  e.  CC )
87, 5addcomd 9268 . . . . . . 7  |-  ( ph  ->  ( ( C  -  ( D  x.  C
) )  +  ( D  x.  A ) )  =  ( ( D  x.  A )  +  ( C  -  ( D  x.  C
) ) ) )
96, 8eqtr2d 2469 . . . . . 6  |-  ( ph  ->  ( ( D  x.  A )  +  ( C  -  ( D  x.  C ) ) )  =  ( C  -  ( ( D  x.  C )  -  ( D  x.  A
) ) ) )
10 ax-1cn 9048 . . . . . . . . . 10  |-  1  e.  CC
1110a1i 11 . . . . . . . . 9  |-  ( ph  ->  1  e.  CC )
1211, 2, 1subdird 9490 . . . . . . . 8  |-  ( ph  ->  ( ( 1  -  D )  x.  C
)  =  ( ( 1  x.  C )  -  ( D  x.  C ) ) )
131mulid2d 9106 . . . . . . . . 9  |-  ( ph  ->  ( 1  x.  C
)  =  C )
1413oveq1d 6096 . . . . . . . 8  |-  ( ph  ->  ( ( 1  x.  C )  -  ( D  x.  C )
)  =  ( C  -  ( D  x.  C ) ) )
1512, 14eqtrd 2468 . . . . . . 7  |-  ( ph  ->  ( ( 1  -  D )  x.  C
)  =  ( C  -  ( D  x.  C ) ) )
1615oveq2d 6097 . . . . . 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 9411 . . . . . . . 8  |-  ( ph  ->  ( C  -  B
)  e.  CC )
191, 4subcld 9411 . . . . . . . . 9  |-  ( ph  ->  ( C  -  A
)  e.  CC )
202, 19mulcld 9108 . . . . . . . 8  |-  ( ph  ->  ( D  x.  ( C  -  A )
)  e.  CC )
2117, 18, 20addsubassd 9431 . . . . . . 7  |-  ( ph  ->  ( ( B  +  ( C  -  B
) )  -  ( D  x.  ( C  -  A ) ) )  =  ( B  +  ( ( C  -  B )  -  ( D  x.  ( C  -  A ) ) ) ) )
2217, 1pncan3d 9414 . . . . . . . 8  |-  ( ph  ->  ( B  +  ( C  -  B ) )  =  C )
232, 1, 4subdid 9489 . . . . . . . 8  |-  ( ph  ->  ( D  x.  ( C  -  A )
)  =  ( ( D  x.  C )  -  ( D  x.  A ) ) )
2422, 23oveq12d 6099 . . . . . . 7  |-  ( ph  ->  ( ( B  +  ( C  -  B
) )  -  ( D  x.  ( C  -  A ) ) )  =  ( C  -  ( ( D  x.  C )  -  ( D  x.  A )
) ) )
2521, 24eqtr3d 2470 . . . . . 6  |-  ( ph  ->  ( B  +  ( ( C  -  B
)  -  ( D  x.  ( C  -  A ) ) ) )  =  ( C  -  ( ( D  x.  C )  -  ( D  x.  A
) ) ) )
269, 16, 253eqtr4d 2478 . . . . 5  |-  ( ph  ->  ( ( D  x.  A )  +  ( ( 1  -  D
)  x.  C ) )  =  ( B  +  ( ( C  -  B )  -  ( D  x.  ( C  -  A )
) ) ) )
2726eqeq2d 2447 . . . 4  |-  ( ph  ->  ( B  =  ( ( D  x.  A
)  +  ( ( 1  -  D )  x.  C ) )  <-> 
B  =  ( B  +  ( ( C  -  B )  -  ( D  x.  ( C  -  A )
) ) ) ) )
2817addid1d 9266 . . . . 5  |-  ( ph  ->  ( B  +  0 )  =  B )
2928eqeq1d 2444 . . . 4  |-  ( ph  ->  ( ( B  + 
0 )  =  ( B  +  ( ( C  -  B )  -  ( D  x.  ( C  -  A
) ) ) )  <-> 
B  =  ( B  +  ( ( C  -  B )  -  ( D  x.  ( C  -  A )
) ) ) ) )
30 0cn 9084 . . . . . 6  |-  0  e.  CC
3130a1i 11 . . . . 5  |-  ( ph  ->  0  e.  CC )
3218, 20subcld 9411 . . . . 5  |-  ( ph  ->  ( ( C  -  B )  -  ( D  x.  ( C  -  A ) ) )  e.  CC )
3317, 31, 32addcand 9269 . . . 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 2438 . . 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 9421 . 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 1652    e. wcel 1725  (class class class)co 6081   CCcc 8988   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995    - cmin 9291
This theorem is referenced by:  affineequiv2  20668  angpieqvd  20672  chordthmlem2  20674  chordthmlem4  20676
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-ltxr 9125  df-sub 9293
  Copyright terms: Public domain W3C validator