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

Theorem affineequiv 20123
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 8855 . . . . . . . 8  |-  ( ph  ->  ( D  x.  C
)  e.  CC )
4 affineequiv.A . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
52, 4mulcld 8855 . . . . . . . 8  |-  ( ph  ->  ( D  x.  A
)  e.  CC )
61, 3, 5subsubd 9185 . . . . . . 7  |-  ( ph  ->  ( C  -  (
( D  x.  C
)  -  ( D  x.  A ) ) )  =  ( ( C  -  ( D  x.  C ) )  +  ( D  x.  A ) ) )
71, 3subcld 9157 . . . . . . . 8  |-  ( ph  ->  ( C  -  ( D  x.  C )
)  e.  CC )
87, 5addcomd 9014 . . . . . . 7  |-  ( ph  ->  ( ( C  -  ( D  x.  C
) )  +  ( D  x.  A ) )  =  ( ( D  x.  A )  +  ( C  -  ( D  x.  C
) ) ) )
96, 8eqtr2d 2316 . . . . . 6  |-  ( ph  ->  ( ( D  x.  A )  +  ( C  -  ( D  x.  C ) ) )  =  ( C  -  ( ( D  x.  C )  -  ( D  x.  A
) ) ) )
10 ax-1cn 8795 . . . . . . . . . 10  |-  1  e.  CC
1110a1i 10 . . . . . . . . 9  |-  ( ph  ->  1  e.  CC )
1211, 2, 1subdird 9236 . . . . . . . 8  |-  ( ph  ->  ( ( 1  -  D )  x.  C
)  =  ( ( 1  x.  C )  -  ( D  x.  C ) ) )
131mulid2d 8853 . . . . . . . . 9  |-  ( ph  ->  ( 1  x.  C
)  =  C )
1413oveq1d 5873 . . . . . . . 8  |-  ( ph  ->  ( ( 1  x.  C )  -  ( D  x.  C )
)  =  ( C  -  ( D  x.  C ) ) )
1512, 14eqtrd 2315 . . . . . . 7  |-  ( ph  ->  ( ( 1  -  D )  x.  C
)  =  ( C  -  ( D  x.  C ) ) )
1615oveq2d 5874 . . . . . 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 9157 . . . . . . . 8  |-  ( ph  ->  ( C  -  B
)  e.  CC )
191, 4subcld 9157 . . . . . . . . 9  |-  ( ph  ->  ( C  -  A
)  e.  CC )
202, 19mulcld 8855 . . . . . . . 8  |-  ( ph  ->  ( D  x.  ( C  -  A )
)  e.  CC )
2117, 18, 20addsubassd 9177 . . . . . . 7  |-  ( ph  ->  ( ( B  +  ( C  -  B
) )  -  ( D  x.  ( C  -  A ) ) )  =  ( B  +  ( ( C  -  B )  -  ( D  x.  ( C  -  A ) ) ) ) )
2217, 1pncan3d 9160 . . . . . . . 8  |-  ( ph  ->  ( B  +  ( C  -  B ) )  =  C )
232, 1, 4subdid 9235 . . . . . . . 8  |-  ( ph  ->  ( D  x.  ( C  -  A )
)  =  ( ( D  x.  C )  -  ( D  x.  A ) ) )
2422, 23oveq12d 5876 . . . . . . 7  |-  ( ph  ->  ( ( B  +  ( C  -  B
) )  -  ( D  x.  ( C  -  A ) ) )  =  ( C  -  ( ( D  x.  C )  -  ( D  x.  A )
) ) )
2521, 24eqtr3d 2317 . . . . . 6  |-  ( ph  ->  ( B  +  ( ( C  -  B
)  -  ( D  x.  ( C  -  A ) ) ) )  =  ( C  -  ( ( D  x.  C )  -  ( D  x.  A
) ) ) )
269, 16, 253eqtr4d 2325 . . . . 5  |-  ( ph  ->  ( ( D  x.  A )  +  ( ( 1  -  D
)  x.  C ) )  =  ( B  +  ( ( C  -  B )  -  ( D  x.  ( C  -  A )
) ) ) )
2726eqeq2d 2294 . . . 4  |-  ( ph  ->  ( B  =  ( ( D  x.  A
)  +  ( ( 1  -  D )  x.  C ) )  <-> 
B  =  ( B  +  ( ( C  -  B )  -  ( D  x.  ( C  -  A )
) ) ) ) )
2817addid1d 9012 . . . . 5  |-  ( ph  ->  ( B  +  0 )  =  B )
2928eqeq1d 2291 . . . 4  |-  ( ph  ->  ( ( B  + 
0 )  =  ( B  +  ( ( C  -  B )  -  ( D  x.  ( C  -  A
) ) ) )  <-> 
B  =  ( B  +  ( ( C  -  B )  -  ( D  x.  ( C  -  A )
) ) ) ) )
30 0cn 8831 . . . . . 6  |-  0  e.  CC
3130a1i 10 . . . . 5  |-  ( ph  ->  0  e.  CC )
3218, 20subcld 9157 . . . . 5  |-  ( ph  ->  ( ( C  -  B )  -  ( D  x.  ( C  -  A ) ) )  e.  CC )
3317, 31, 32addcand 9015 . . . 4  |-  ( ph  ->  ( ( B  + 
0 )  =  ( B  +  ( ( C  -  B )  -  ( D  x.  ( C  -  A
) ) ) )  <->  0  =  ( ( C  -  B )  -  ( D  x.  ( C  -  A
) ) ) ) )
3427, 29, 333bitr2d 272 . . 3  |-  ( ph  ->  ( B  =  ( ( D  x.  A
)  +  ( ( 1  -  D )  x.  C ) )  <->  0  =  ( ( C  -  B )  -  ( D  x.  ( C  -  A
) ) ) ) )
35 eqcom 2285 . . 3  |-  ( 0  =  ( ( C  -  B )  -  ( D  x.  ( C  -  A )
) )  <->  ( ( C  -  B )  -  ( D  x.  ( C  -  A
) ) )  =  0 )
3634, 35syl6bb 252 . 2  |-  ( ph  ->  ( B  =  ( ( D  x.  A
)  +  ( ( 1  -  D )  x.  C ) )  <-> 
( ( C  -  B )  -  ( D  x.  ( C  -  A ) ) )  =  0 ) )
3718, 20subeq0ad 9167 . 2  |-  ( ph  ->  ( ( ( C  -  B )  -  ( D  x.  ( C  -  A )
) )  =  0  <-> 
( C  -  B
)  =  ( D  x.  ( C  -  A ) ) ) )
3836, 37bitrd 244 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 176    = wceq 1623    e. wcel 1684  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    - cmin 9037
This theorem is referenced by:  affineequiv2  20124  angpieqvd  20128  chordthmlem2  20130  chordthmlem4  20132
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-sub 9039
  Copyright terms: Public domain W3C validator