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Theorem angpieqvdlem 23619
Description: Equivalence used in the proof of angpieqvd 23622. (Contributed by David Moews, 28-Feb-2017.)
Hypotheses
Ref Expression
angpieqvdlem.A  |-  ( ph  ->  A  e.  CC )
angpieqvdlem.B  |-  ( ph  ->  B  e.  CC )
angpieqvdlem.C  |-  ( ph  ->  C  e.  CC )
angpieqvdlem.AneB  |-  ( ph  ->  A  =/=  B )
angpieqvdlem.AneC  |-  ( ph  ->  A  =/=  C )
Assertion
Ref Expression
angpieqvdlem  |-  ( ph  ->  ( -u ( ( C  -  B )  /  ( A  -  B ) )  e.  RR+ 
<->  ( ( C  -  B )  /  ( C  -  A )
)  e.  ( 0 (,) 1 ) ) )

Proof of Theorem angpieqvdlem
StepHypRef Expression
1 angpieqvdlem.C . . . . . 6  |-  ( ph  ->  C  e.  CC )
2 angpieqvdlem.B . . . . . 6  |-  ( ph  ->  B  e.  CC )
31, 2subcld 9985 . . . . 5  |-  ( ph  ->  ( C  -  B
)  e.  CC )
4 angpieqvdlem.A . . . . . 6  |-  ( ph  ->  A  e.  CC )
54, 2subcld 9985 . . . . 5  |-  ( ph  ->  ( A  -  B
)  e.  CC )
6 angpieqvdlem.AneB . . . . . 6  |-  ( ph  ->  A  =/=  B )
74, 2, 6subne0d 9994 . . . . 5  |-  ( ph  ->  ( A  -  B
)  =/=  0 )
83, 5, 7divcld 10382 . . . 4  |-  ( ph  ->  ( ( C  -  B )  /  ( A  -  B )
)  e.  CC )
98negcld 9972 . . 3  |-  ( ph  -> 
-u ( ( C  -  B )  / 
( A  -  B
) )  e.  CC )
10 1cnd 9658 . . . 4  |-  ( ph  ->  1  e.  CC )
11 angpieqvdlem.AneC . . . . . . 7  |-  ( ph  ->  A  =/=  C )
1211necomd 2702 . . . . . 6  |-  ( ph  ->  C  =/=  A )
131, 4, 2, 12subneintr2d 10031 . . . . 5  |-  ( ph  ->  ( C  -  B
)  =/=  ( A  -  B ) )
143, 5, 7, 13divne1d 10393 . . . 4  |-  ( ph  ->  ( ( C  -  B )  /  ( A  -  B )
)  =/=  1 )
158, 10, 14negned 9982 . . 3  |-  ( ph  -> 
-u ( ( C  -  B )  / 
( A  -  B
) )  =/=  -u 1
)
169, 15xov1plusxeqvd 11776 . 2  |-  ( ph  ->  ( -u ( ( C  -  B )  /  ( A  -  B ) )  e.  RR+ 
<->  ( -u ( ( C  -  B )  /  ( A  -  B ) )  / 
( 1  +  -u ( ( C  -  B )  /  ( A  -  B )
) ) )  e.  ( 0 (,) 1
) ) )
173, 5, 7divnegd 10395 . . . . . 6  |-  ( ph  -> 
-u ( ( C  -  B )  / 
( A  -  B
) )  =  (
-u ( C  -  B )  /  ( A  -  B )
) )
181, 2negsubdi2d 10001 . . . . . . 7  |-  ( ph  -> 
-u ( C  -  B )  =  ( B  -  C ) )
1918oveq1d 6320 . . . . . 6  |-  ( ph  ->  ( -u ( C  -  B )  / 
( A  -  B
) )  =  ( ( B  -  C
)  /  ( A  -  B ) ) )
2017, 19eqtrd 2470 . . . . 5  |-  ( ph  -> 
-u ( ( C  -  B )  / 
( A  -  B
) )  =  ( ( B  -  C
)  /  ( A  -  B ) ) )
215, 7dividd 10380 . . . . . . . 8  |-  ( ph  ->  ( ( A  -  B )  /  ( A  -  B )
)  =  1 )
2221oveq1d 6320 . . . . . . 7  |-  ( ph  ->  ( ( ( A  -  B )  / 
( A  -  B
) )  -  (
( C  -  B
)  /  ( A  -  B ) ) )  =  ( 1  -  ( ( C  -  B )  / 
( A  -  B
) ) ) )
235, 3, 5, 7divsubdird 10421 . . . . . . 7  |-  ( ph  ->  ( ( ( A  -  B )  -  ( C  -  B
) )  /  ( A  -  B )
)  =  ( ( ( A  -  B
)  /  ( A  -  B ) )  -  ( ( C  -  B )  / 
( A  -  B
) ) ) )
2410, 8negsubd 9991 . . . . . . 7  |-  ( ph  ->  ( 1  +  -u ( ( C  -  B )  /  ( A  -  B )
) )  =  ( 1  -  ( ( C  -  B )  /  ( A  -  B ) ) ) )
2522, 23, 243eqtr4rd 2481 . . . . . 6  |-  ( ph  ->  ( 1  +  -u ( ( C  -  B )  /  ( A  -  B )
) )  =  ( ( ( A  -  B )  -  ( C  -  B )
)  /  ( A  -  B ) ) )
264, 1, 2nnncan2d 10020 . . . . . . 7  |-  ( ph  ->  ( ( A  -  B )  -  ( C  -  B )
)  =  ( A  -  C ) )
2726oveq1d 6320 . . . . . 6  |-  ( ph  ->  ( ( ( A  -  B )  -  ( C  -  B
) )  /  ( A  -  B )
)  =  ( ( A  -  C )  /  ( A  -  B ) ) )
2825, 27eqtrd 2470 . . . . 5  |-  ( ph  ->  ( 1  +  -u ( ( C  -  B )  /  ( A  -  B )
) )  =  ( ( A  -  C
)  /  ( A  -  B ) ) )
2920, 28oveq12d 6323 . . . 4  |-  ( ph  ->  ( -u ( ( C  -  B )  /  ( A  -  B ) )  / 
( 1  +  -u ( ( C  -  B )  /  ( A  -  B )
) ) )  =  ( ( ( B  -  C )  / 
( A  -  B
) )  /  (
( A  -  C
)  /  ( A  -  B ) ) ) )
302, 1subcld 9985 . . . . 5  |-  ( ph  ->  ( B  -  C
)  e.  CC )
314, 1subcld 9985 . . . . 5  |-  ( ph  ->  ( A  -  C
)  e.  CC )
324, 1, 11subne0d 9994 . . . . 5  |-  ( ph  ->  ( A  -  C
)  =/=  0 )
3330, 31, 5, 32, 7divcan7d 10410 . . . 4  |-  ( ph  ->  ( ( ( B  -  C )  / 
( A  -  B
) )  /  (
( A  -  C
)  /  ( A  -  B ) ) )  =  ( ( B  -  C )  /  ( A  -  C ) ) )
342, 1, 4, 1, 11div2subd 10432 . . . 4  |-  ( ph  ->  ( ( B  -  C )  /  ( A  -  C )
)  =  ( ( C  -  B )  /  ( C  -  A ) ) )
3529, 33, 343eqtrrd 2475 . . 3  |-  ( ph  ->  ( ( C  -  B )  /  ( C  -  A )
)  =  ( -u ( ( C  -  B )  /  ( A  -  B )
)  /  ( 1  +  -u ( ( C  -  B )  / 
( A  -  B
) ) ) ) )
3635eleq1d 2498 . 2  |-  ( ph  ->  ( ( ( C  -  B )  / 
( C  -  A
) )  e.  ( 0 (,) 1 )  <-> 
( -u ( ( C  -  B )  / 
( A  -  B
) )  /  (
1  +  -u (
( C  -  B
)  /  ( A  -  B ) ) ) )  e.  ( 0 (,) 1 ) ) )
3716, 36bitr4d 259 1  |-  ( ph  ->  ( -u ( ( C  -  B )  /  ( A  -  B ) )  e.  RR+ 
<->  ( ( C  -  B )  /  ( C  -  A )
)  e.  ( 0 (,) 1 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    e. wcel 1870    =/= wne 2625  (class class class)co 6305   CCcc 9536   0cc0 9538   1c1 9539    + caddc 9541    - cmin 9859   -ucneg 9860    / cdiv 10268   RR+crp 11302   (,)cioo 11635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-po 4775  df-so 4776  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-rp 11303  df-ioo 11639
This theorem is referenced by:  angpieqvd  23622
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