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Theorem angpieqvdlem 20125
Description: Equivalence used in the proof of angpieqvd 20128. (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 9157 . . . . 5  |-  ( ph  ->  ( C  -  B
)  e.  CC )
4 angpieqvdlem.A . . . . . 6  |-  ( ph  ->  A  e.  CC )
54, 2subcld 9157 . . . . 5  |-  ( ph  ->  ( A  -  B
)  e.  CC )
6 angpieqvdlem.AneB . . . . . 6  |-  ( ph  ->  A  =/=  B )
74, 2, 6subne0d 9166 . . . . 5  |-  ( ph  ->  ( A  -  B
)  =/=  0 )
83, 5, 7divcld 9536 . . . 4  |-  ( ph  ->  ( ( C  -  B )  /  ( A  -  B )
)  e.  CC )
98negcld 9144 . . 3  |-  ( ph  -> 
-u ( ( C  -  B )  / 
( A  -  B
) )  e.  CC )
10 ax-1cn 8795 . . . . 5  |-  1  e.  CC
1110a1i 10 . . . 4  |-  ( ph  ->  1  e.  CC )
12 angpieqvdlem.AneC . . . . . . 7  |-  ( ph  ->  A  =/=  C )
1312necomd 2529 . . . . . 6  |-  ( ph  ->  C  =/=  A )
141, 4, 2, 13subneintr2d 9203 . . . . 5  |-  ( ph  ->  ( C  -  B
)  =/=  ( A  -  B ) )
153, 5, 7, 14divne1d 9547 . . . 4  |-  ( ph  ->  ( ( C  -  B )  /  ( A  -  B )
)  =/=  1 )
168, 11, 15negned 9154 . . 3  |-  ( ph  -> 
-u ( ( C  -  B )  / 
( A  -  B
) )  =/=  -u 1
)
179, 16xov1plusxeqvd 10780 . 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 9549 . . . . . 6  |-  ( ph  -> 
-u ( ( C  -  B )  / 
( A  -  B
) )  =  (
-u ( C  -  B )  /  ( A  -  B )
) )
191, 2negsubdi2d 9173 . . . . . . 7  |-  ( ph  -> 
-u ( C  -  B )  =  ( B  -  C ) )
2019oveq1d 5873 . . . . . 6  |-  ( ph  ->  ( -u ( C  -  B )  / 
( A  -  B
) )  =  ( ( B  -  C
)  /  ( A  -  B ) ) )
2118, 20eqtrd 2315 . . . . 5  |-  ( ph  -> 
-u ( ( C  -  B )  / 
( A  -  B
) )  =  ( ( B  -  C
)  /  ( A  -  B ) ) )
225, 7dividd 9534 . . . . . . . 8  |-  ( ph  ->  ( ( A  -  B )  /  ( A  -  B )
)  =  1 )
2322oveq1d 5873 . . . . . . 7  |-  ( ph  ->  ( ( ( A  -  B )  / 
( A  -  B
) )  -  (
( C  -  B
)  /  ( A  -  B ) ) )  =  ( 1  -  ( ( C  -  B )  / 
( A  -  B
) ) ) )
245, 3, 5, 7divsubdird 9575 . . . . . . 7  |-  ( ph  ->  ( ( ( A  -  B )  -  ( C  -  B
) )  /  ( A  -  B )
)  =  ( ( ( A  -  B
)  /  ( A  -  B ) )  -  ( ( C  -  B )  / 
( A  -  B
) ) ) )
2511, 8negsubd 9163 . . . . . . 7  |-  ( ph  ->  ( 1  +  -u ( ( C  -  B )  /  ( A  -  B )
) )  =  ( 1  -  ( ( C  -  B )  /  ( A  -  B ) ) ) )
2623, 24, 253eqtr4rd 2326 . . . . . 6  |-  ( ph  ->  ( 1  +  -u ( ( C  -  B )  /  ( A  -  B )
) )  =  ( ( ( A  -  B )  -  ( C  -  B )
)  /  ( A  -  B ) ) )
274, 1, 2nnncan2d 9192 . . . . . . 7  |-  ( ph  ->  ( ( A  -  B )  -  ( C  -  B )
)  =  ( A  -  C ) )
2827oveq1d 5873 . . . . . 6  |-  ( ph  ->  ( ( ( A  -  B )  -  ( C  -  B
) )  /  ( A  -  B )
)  =  ( ( A  -  C )  /  ( A  -  B ) ) )
2926, 28eqtrd 2315 . . . . 5  |-  ( ph  ->  ( 1  +  -u ( ( C  -  B )  /  ( A  -  B )
) )  =  ( ( A  -  C
)  /  ( A  -  B ) ) )
3021, 29oveq12d 5876 . . . 4  |-  ( ph  ->  ( -u ( ( C  -  B )  /  ( A  -  B ) )  / 
( 1  +  -u ( ( C  -  B )  /  ( A  -  B )
) ) )  =  ( ( ( B  -  C )  / 
( A  -  B
) )  /  (
( A  -  C
)  /  ( A  -  B ) ) ) )
312, 1subcld 9157 . . . . 5  |-  ( ph  ->  ( B  -  C
)  e.  CC )
324, 1subcld 9157 . . . . 5  |-  ( ph  ->  ( A  -  C
)  e.  CC )
334, 1, 12subne0d 9166 . . . . 5  |-  ( ph  ->  ( A  -  C
)  =/=  0 )
3431, 32, 5, 33, 7divcan7d 9564 . . . 4  |-  ( ph  ->  ( ( ( B  -  C )  / 
( A  -  B
) )  /  (
( A  -  C
)  /  ( A  -  B ) ) )  =  ( ( B  -  C )  /  ( A  -  C ) ) )
352, 1, 4, 1, 12div2subd 9586 . . . 4  |-  ( ph  ->  ( ( B  -  C )  /  ( A  -  C )
)  =  ( ( C  -  B )  /  ( C  -  A ) ) )
3630, 34, 353eqtrrd 2320 . . 3  |-  ( ph  ->  ( ( C  -  B )  /  ( C  -  A )
)  =  ( -u ( ( C  -  B )  /  ( A  -  B )
)  /  ( 1  +  -u ( ( C  -  B )  / 
( A  -  B
) ) ) ) )
3736eleq1d 2349 . 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 1684    =/= wne 2446  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    - cmin 9037   -ucneg 9038    / cdiv 9423   RR+crp 10354   (,)cioo 10656
This theorem is referenced by:  angpieqvd  20128
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-cnex 8793  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  ax-pre-mulgt0 8814
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-rmo 2551  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-iun 3907  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-1st 6122  df-2nd 6123  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-rp 10355  df-ioo 10660
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