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Theorem angpieqvdlem 20141
Description: Equivalence used in the proof of angpieqvd 20144. (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 9173 . . . . 5  |-  ( ph  ->  ( C  -  B
)  e.  CC )
4 angpieqvdlem.A . . . . . 6  |-  ( ph  ->  A  e.  CC )
54, 2subcld 9173 . . . . 5  |-  ( ph  ->  ( A  -  B
)  e.  CC )
6 angpieqvdlem.AneB . . . . . 6  |-  ( ph  ->  A  =/=  B )
74, 2, 6subne0d 9182 . . . . 5  |-  ( ph  ->  ( A  -  B
)  =/=  0 )
83, 5, 7divcld 9552 . . . 4  |-  ( ph  ->  ( ( C  -  B )  /  ( A  -  B )
)  e.  CC )
98negcld 9160 . . 3  |-  ( ph  -> 
-u ( ( C  -  B )  / 
( A  -  B
) )  e.  CC )
10 ax-1cn 8811 . . . . 5  |-  1  e.  CC
1110a1i 10 . . . 4  |-  ( ph  ->  1  e.  CC )
12 angpieqvdlem.AneC . . . . . . 7  |-  ( ph  ->  A  =/=  C )
1312necomd 2542 . . . . . 6  |-  ( ph  ->  C  =/=  A )
141, 4, 2, 13subneintr2d 9219 . . . . 5  |-  ( ph  ->  ( C  -  B
)  =/=  ( A  -  B ) )
153, 5, 7, 14divne1d 9563 . . . 4  |-  ( ph  ->  ( ( C  -  B )  /  ( A  -  B )
)  =/=  1 )
168, 11, 15negned 9170 . . 3  |-  ( ph  -> 
-u ( ( C  -  B )  / 
( A  -  B
) )  =/=  -u 1
)
179, 16xov1plusxeqvd 10796 . 2  |-  ( ph  ->  ( -u ( ( C  -  B )  /  ( A  -  B ) )  e.  RR+ 
<->  ( -u ( ( C  -  B )  /  ( A  -  B ) )  / 
( 1  +  -u ( ( C  -  B )  /  ( A  -  B )
) ) )  e.  ( 0 (,) 1
) ) )
183, 5, 7divnegd 9565 . . . . . 6  |-  ( ph  -> 
-u ( ( C  -  B )  / 
( A  -  B
) )  =  (
-u ( C  -  B )  /  ( A  -  B )
) )
191, 2negsubdi2d 9189 . . . . . . 7  |-  ( ph  -> 
-u ( C  -  B )  =  ( B  -  C ) )
2019oveq1d 5889 . . . . . 6  |-  ( ph  ->  ( -u ( C  -  B )  / 
( A  -  B
) )  =  ( ( B  -  C
)  /  ( A  -  B ) ) )
2118, 20eqtrd 2328 . . . . 5  |-  ( ph  -> 
-u ( ( C  -  B )  / 
( A  -  B
) )  =  ( ( B  -  C
)  /  ( A  -  B ) ) )
225, 7dividd 9550 . . . . . . . 8  |-  ( ph  ->  ( ( A  -  B )  /  ( A  -  B )
)  =  1 )
2322oveq1d 5889 . . . . . . 7  |-  ( ph  ->  ( ( ( A  -  B )  / 
( A  -  B
) )  -  (
( C  -  B
)  /  ( A  -  B ) ) )  =  ( 1  -  ( ( C  -  B )  / 
( A  -  B
) ) ) )
245, 3, 5, 7divsubdird 9591 . . . . . . 7  |-  ( ph  ->  ( ( ( A  -  B )  -  ( C  -  B
) )  /  ( A  -  B )
)  =  ( ( ( A  -  B
)  /  ( A  -  B ) )  -  ( ( C  -  B )  / 
( A  -  B
) ) ) )
2511, 8negsubd 9179 . . . . . . 7  |-  ( ph  ->  ( 1  +  -u ( ( C  -  B )  /  ( A  -  B )
) )  =  ( 1  -  ( ( C  -  B )  /  ( A  -  B ) ) ) )
2623, 24, 253eqtr4rd 2339 . . . . . 6  |-  ( ph  ->  ( 1  +  -u ( ( C  -  B )  /  ( A  -  B )
) )  =  ( ( ( A  -  B )  -  ( C  -  B )
)  /  ( A  -  B ) ) )
274, 1, 2nnncan2d 9208 . . . . . . 7  |-  ( ph  ->  ( ( A  -  B )  -  ( C  -  B )
)  =  ( A  -  C ) )
2827oveq1d 5889 . . . . . 6  |-  ( ph  ->  ( ( ( A  -  B )  -  ( C  -  B
) )  /  ( A  -  B )
)  =  ( ( A  -  C )  /  ( A  -  B ) ) )
2926, 28eqtrd 2328 . . . . 5  |-  ( ph  ->  ( 1  +  -u ( ( C  -  B )  /  ( A  -  B )
) )  =  ( ( A  -  C
)  /  ( A  -  B ) ) )
3021, 29oveq12d 5892 . . . 4  |-  ( ph  ->  ( -u ( ( C  -  B )  /  ( A  -  B ) )  / 
( 1  +  -u ( ( C  -  B )  /  ( A  -  B )
) ) )  =  ( ( ( B  -  C )  / 
( A  -  B
) )  /  (
( A  -  C
)  /  ( A  -  B ) ) ) )
312, 1subcld 9173 . . . . 5  |-  ( ph  ->  ( B  -  C
)  e.  CC )
324, 1subcld 9173 . . . . 5  |-  ( ph  ->  ( A  -  C
)  e.  CC )
334, 1, 12subne0d 9182 . . . . 5  |-  ( ph  ->  ( A  -  C
)  =/=  0 )
3431, 32, 5, 33, 7divcan7d 9580 . . . 4  |-  ( ph  ->  ( ( ( B  -  C )  / 
( A  -  B
) )  /  (
( A  -  C
)  /  ( A  -  B ) ) )  =  ( ( B  -  C )  /  ( A  -  C ) ) )
352, 1, 4, 1, 12div2subd 9602 . . . 4  |-  ( ph  ->  ( ( B  -  C )  /  ( A  -  C )
)  =  ( ( C  -  B )  /  ( C  -  A ) ) )
3630, 34, 353eqtrrd 2333 . . 3  |-  ( ph  ->  ( ( C  -  B )  /  ( C  -  A )
)  =  ( -u ( ( C  -  B )  /  ( A  -  B )
)  /  ( 1  +  -u ( ( C  -  B )  / 
( A  -  B
) ) ) ) )
3736eleq1d 2362 . 2  |-  ( ph  ->  ( ( ( C  -  B )  / 
( C  -  A
) )  e.  ( 0 (,) 1 )  <-> 
( -u ( ( C  -  B )  / 
( A  -  B
) )  /  (
1  +  -u (
( C  -  B
)  /  ( A  -  B ) ) ) )  e.  ( 0 (,) 1 ) ) )
3817, 37bitr4d 247 1  |-  ( ph  ->  ( -u ( ( C  -  B )  /  ( A  -  B ) )  e.  RR+ 
<->  ( ( C  -  B )  /  ( C  -  A )
)  e.  ( 0 (,) 1 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1696    =/= wne 2459  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    + caddc 8756    - cmin 9053   -ucneg 9054    / cdiv 9439   RR+crp 10370   (,)cioo 10672
This theorem is referenced by:  angpieqvd  20144
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-rp 10371  df-ioo 10676
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