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Theorem affineequiv 20228
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 8942 . . . . . . . 8  |-  ( ph  ->  ( D  x.  C
)  e.  CC )
4 affineequiv.A . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
52, 4mulcld 8942 . . . . . . . 8  |-  ( ph  ->  ( D  x.  A
)  e.  CC )
61, 3, 5subsubd 9272 . . . . . . 7  |-  ( ph  ->  ( C  -  (
( D  x.  C
)  -  ( D  x.  A ) ) )  =  ( ( C  -  ( D  x.  C ) )  +  ( D  x.  A ) ) )
71, 3subcld 9244 . . . . . . . 8  |-  ( ph  ->  ( C  -  ( D  x.  C )
)  e.  CC )
87, 5addcomd 9101 . . . . . . 7  |-  ( ph  ->  ( ( C  -  ( D  x.  C
) )  +  ( D  x.  A ) )  =  ( ( D  x.  A )  +  ( C  -  ( D  x.  C
) ) ) )
96, 8eqtr2d 2391 . . . . . 6  |-  ( ph  ->  ( ( D  x.  A )  +  ( C  -  ( D  x.  C ) ) )  =  ( C  -  ( ( D  x.  C )  -  ( D  x.  A
) ) ) )
10 ax-1cn 8882 . . . . . . . . . 10  |-  1  e.  CC
1110a1i 10 . . . . . . . . 9  |-  ( ph  ->  1  e.  CC )
1211, 2, 1subdird 9323 . . . . . . . 8  |-  ( ph  ->  ( ( 1  -  D )  x.  C
)  =  ( ( 1  x.  C )  -  ( D  x.  C ) ) )
131mulid2d 8940 . . . . . . . . 9  |-  ( ph  ->  ( 1  x.  C
)  =  C )
1413oveq1d 5957 . . . . . . . 8  |-  ( ph  ->  ( ( 1  x.  C )  -  ( D  x.  C )
)  =  ( C  -  ( D  x.  C ) ) )
1512, 14eqtrd 2390 . . . . . . 7  |-  ( ph  ->  ( ( 1  -  D )  x.  C
)  =  ( C  -  ( D  x.  C ) ) )
1615oveq2d 5958 . . . . . 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 9244 . . . . . . . 8  |-  ( ph  ->  ( C  -  B
)  e.  CC )
191, 4subcld 9244 . . . . . . . . 9  |-  ( ph  ->  ( C  -  A
)  e.  CC )
202, 19mulcld 8942 . . . . . . . 8  |-  ( ph  ->  ( D  x.  ( C  -  A )
)  e.  CC )
2117, 18, 20addsubassd 9264 . . . . . . 7  |-  ( ph  ->  ( ( B  +  ( C  -  B
) )  -  ( D  x.  ( C  -  A ) ) )  =  ( B  +  ( ( C  -  B )  -  ( D  x.  ( C  -  A ) ) ) ) )
2217, 1pncan3d 9247 . . . . . . . 8  |-  ( ph  ->  ( B  +  ( C  -  B ) )  =  C )
232, 1, 4subdid 9322 . . . . . . . 8  |-  ( ph  ->  ( D  x.  ( C  -  A )
)  =  ( ( D  x.  C )  -  ( D  x.  A ) ) )
2422, 23oveq12d 5960 . . . . . . 7  |-  ( ph  ->  ( ( B  +  ( C  -  B
) )  -  ( D  x.  ( C  -  A ) ) )  =  ( C  -  ( ( D  x.  C )  -  ( D  x.  A )
) ) )
2521, 24eqtr3d 2392 . . . . . 6  |-  ( ph  ->  ( B  +  ( ( C  -  B
)  -  ( D  x.  ( C  -  A ) ) ) )  =  ( C  -  ( ( D  x.  C )  -  ( D  x.  A
) ) ) )
269, 16, 253eqtr4d 2400 . . . . 5  |-  ( ph  ->  ( ( D  x.  A )  +  ( ( 1  -  D
)  x.  C ) )  =  ( B  +  ( ( C  -  B )  -  ( D  x.  ( C  -  A )
) ) ) )
2726eqeq2d 2369 . . . 4  |-  ( ph  ->  ( B  =  ( ( D  x.  A
)  +  ( ( 1  -  D )  x.  C ) )  <-> 
B  =  ( B  +  ( ( C  -  B )  -  ( D  x.  ( C  -  A )
) ) ) ) )
2817addid1d 9099 . . . . 5  |-  ( ph  ->  ( B  +  0 )  =  B )
2928eqeq1d 2366 . . . 4  |-  ( ph  ->  ( ( B  + 
0 )  =  ( B  +  ( ( C  -  B )  -  ( D  x.  ( C  -  A
) ) ) )  <-> 
B  =  ( B  +  ( ( C  -  B )  -  ( D  x.  ( C  -  A )
) ) ) ) )
30 0cn 8918 . . . . . 6  |-  0  e.  CC
3130a1i 10 . . . . 5  |-  ( ph  ->  0  e.  CC )
3218, 20subcld 9244 . . . . 5  |-  ( ph  ->  ( ( C  -  B )  -  ( D  x.  ( C  -  A ) ) )  e.  CC )
3317, 31, 32addcand 9102 . . . 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 2360 . . 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 9254 . 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 1642    e. wcel 1710  (class class class)co 5942   CCcc 8822   0cc0 8824   1c1 8825    + caddc 8827    x. cmul 8829    - cmin 9124
This theorem is referenced by:  affineequiv2  20229  angpieqvd  20233  chordthmlem2  20235  chordthmlem4  20237
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-po 4393  df-so 4394  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-riota 6388  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-pnf 8956  df-mnf 8957  df-ltxr 8959  df-sub 9126
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