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Theorem angpieqvdlem 20073
Description: Equivalence used in the proof of angpieqvd 20076. (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 9111 . . . . 5  |-  ( ph  ->  ( C  -  B
)  e.  CC )
4 angpieqvdlem.A . . . . . 6  |-  ( ph  ->  A  e.  CC )
54, 2subcld 9111 . . . . 5  |-  ( ph  ->  ( A  -  B
)  e.  CC )
6 angpieqvdlem.AneB . . . . . 6  |-  ( ph  ->  A  =/=  B )
74, 2, 6subne0d 9120 . . . . 5  |-  ( ph  ->  ( A  -  B
)  =/=  0 )
83, 5, 7divcld 9490 . . . 4  |-  ( ph  ->  ( ( C  -  B )  /  ( A  -  B )
)  e.  CC )
98negcld 9098 . . 3  |-  ( ph  -> 
-u ( ( C  -  B )  / 
( A  -  B
) )  e.  CC )
10 ax-1cn 8749 . . . . 5  |-  1  e.  CC
1110a1i 12 . . . 4  |-  ( ph  ->  1  e.  CC )
12 angpieqvdlem.AneC . . . . . . 7  |-  ( ph  ->  A  =/=  C )
1312necomd 2502 . . . . . 6  |-  ( ph  ->  C  =/=  A )
141, 4, 2, 13subneintr2d 9157 . . . . 5  |-  ( ph  ->  ( C  -  B
)  =/=  ( A  -  B ) )
153, 5, 7, 14divne1d 9501 . . . 4  |-  ( ph  ->  ( ( C  -  B )  /  ( A  -  B )
)  =/=  1 )
168, 11, 15negned 9108 . . 3  |-  ( ph  -> 
-u ( ( C  -  B )  / 
( A  -  B
) )  =/=  -u 1
)
179, 16xov1plusxeqvd 10732 . 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 9503 . . . . . 6  |-  ( ph  -> 
-u ( ( C  -  B )  / 
( A  -  B
) )  =  (
-u ( C  -  B )  /  ( A  -  B )
) )
191, 2negsubdi2d 9127 . . . . . . 7  |-  ( ph  -> 
-u ( C  -  B )  =  ( B  -  C ) )
2019oveq1d 5793 . . . . . 6  |-  ( ph  ->  ( -u ( C  -  B )  / 
( A  -  B
) )  =  ( ( B  -  C
)  /  ( A  -  B ) ) )
2118, 20eqtrd 2288 . . . . 5  |-  ( ph  -> 
-u ( ( C  -  B )  / 
( A  -  B
) )  =  ( ( B  -  C
)  /  ( A  -  B ) ) )
225, 7dividd 9488 . . . . . . . 8  |-  ( ph  ->  ( ( A  -  B )  /  ( A  -  B )
)  =  1 )
2322oveq1d 5793 . . . . . . 7  |-  ( ph  ->  ( ( ( A  -  B )  / 
( A  -  B
) )  -  (
( C  -  B
)  /  ( A  -  B ) ) )  =  ( 1  -  ( ( C  -  B )  / 
( A  -  B
) ) ) )
245, 3, 5, 7divsubdird 9529 . . . . . . 7  |-  ( ph  ->  ( ( ( A  -  B )  -  ( C  -  B
) )  /  ( A  -  B )
)  =  ( ( ( A  -  B
)  /  ( A  -  B ) )  -  ( ( C  -  B )  / 
( A  -  B
) ) ) )
2511, 8negsubd 9117 . . . . . . 7  |-  ( ph  ->  ( 1  +  -u ( ( C  -  B )  /  ( A  -  B )
) )  =  ( 1  -  ( ( C  -  B )  /  ( A  -  B ) ) ) )
2623, 24, 253eqtr4rd 2299 . . . . . 6  |-  ( ph  ->  ( 1  +  -u ( ( C  -  B )  /  ( A  -  B )
) )  =  ( ( ( A  -  B )  -  ( C  -  B )
)  /  ( A  -  B ) ) )
274, 1, 2nnncan2d 9146 . . . . . . 7  |-  ( ph  ->  ( ( A  -  B )  -  ( C  -  B )
)  =  ( A  -  C ) )
2827oveq1d 5793 . . . . . 6  |-  ( ph  ->  ( ( ( A  -  B )  -  ( C  -  B
) )  /  ( A  -  B )
)  =  ( ( A  -  C )  /  ( A  -  B ) ) )
2926, 28eqtrd 2288 . . . . 5  |-  ( ph  ->  ( 1  +  -u ( ( C  -  B )  /  ( A  -  B )
) )  =  ( ( A  -  C
)  /  ( A  -  B ) ) )
3021, 29oveq12d 5796 . . . 4  |-  ( ph  ->  ( -u ( ( C  -  B )  /  ( A  -  B ) )  / 
( 1  +  -u ( ( C  -  B )  /  ( A  -  B )
) ) )  =  ( ( ( B  -  C )  / 
( A  -  B
) )  /  (
( A  -  C
)  /  ( A  -  B ) ) ) )
312, 1subcld 9111 . . . . 5  |-  ( ph  ->  ( B  -  C
)  e.  CC )
324, 1subcld 9111 . . . . 5  |-  ( ph  ->  ( A  -  C
)  e.  CC )
334, 1, 12subne0d 9120 . . . . 5  |-  ( ph  ->  ( A  -  C
)  =/=  0 )
3431, 32, 5, 33, 7divcan7d 9518 . . . 4  |-  ( ph  ->  ( ( ( B  -  C )  / 
( A  -  B
) )  /  (
( A  -  C
)  /  ( A  -  B ) ) )  =  ( ( B  -  C )  /  ( A  -  C ) ) )
352, 1, 4, 1, 12div2subd 9540 . . . 4  |-  ( ph  ->  ( ( B  -  C )  /  ( A  -  C )
)  =  ( ( C  -  B )  /  ( C  -  A ) ) )
3630, 34, 353eqtrrd 2293 . . 3  |-  ( ph  ->  ( ( C  -  B )  /  ( C  -  A )
)  =  ( -u ( ( C  -  B )  /  ( A  -  B )
)  /  ( 1  +  -u ( ( C  -  B )  / 
( A  -  B
) ) ) ) )
3736eleq1d 2322 . 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 249 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 6    <-> wb 178    e. wcel 1621    =/= wne 2419  (class class class)co 5778   CCcc 8689   0cc0 8691   1c1 8692    + caddc 8694    - cmin 8991   -ucneg 8992    / cdiv 9377   RR+crp 10307   (,)cioo 10608
This theorem is referenced by:  angpieqvd  20076
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-cnex 8747  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-po 4272  df-so 4273  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-riota 6258  df-er 6614  df-en 6818  df-dom 6819  df-sdom 6820  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-div 9378  df-rp 10308  df-ioo 10612
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