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Theorem angpieqvdlem 22984
Description: Equivalence used in the proof of angpieqvd 22987. (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 9931 . . . . 5  |-  ( ph  ->  ( C  -  B
)  e.  CC )
4 angpieqvdlem.A . . . . . 6  |-  ( ph  ->  A  e.  CC )
54, 2subcld 9931 . . . . 5  |-  ( ph  ->  ( A  -  B
)  e.  CC )
6 angpieqvdlem.AneB . . . . . 6  |-  ( ph  ->  A  =/=  B )
74, 2, 6subne0d 9940 . . . . 5  |-  ( ph  ->  ( A  -  B
)  =/=  0 )
83, 5, 7divcld 10321 . . . 4  |-  ( ph  ->  ( ( C  -  B )  /  ( A  -  B )
)  e.  CC )
98negcld 9918 . . 3  |-  ( ph  -> 
-u ( ( C  -  B )  / 
( A  -  B
) )  e.  CC )
10 ax-1cn 9551 . . . . 5  |-  1  e.  CC
1110a1i 11 . . . 4  |-  ( ph  ->  1  e.  CC )
12 angpieqvdlem.AneC . . . . . . 7  |-  ( ph  ->  A  =/=  C )
1312necomd 2738 . . . . . 6  |-  ( ph  ->  C  =/=  A )
141, 4, 2, 13subneintr2d 9977 . . . . 5  |-  ( ph  ->  ( C  -  B
)  =/=  ( A  -  B ) )
153, 5, 7, 14divne1d 10332 . . . 4  |-  ( ph  ->  ( ( C  -  B )  /  ( A  -  B )
)  =/=  1 )
168, 11, 15negned 9928 . . 3  |-  ( ph  -> 
-u ( ( C  -  B )  / 
( A  -  B
) )  =/=  -u 1
)
179, 16xov1plusxeqvd 11667 . 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 10334 . . . . . 6  |-  ( ph  -> 
-u ( ( C  -  B )  / 
( A  -  B
) )  =  (
-u ( C  -  B )  /  ( A  -  B )
) )
191, 2negsubdi2d 9947 . . . . . . 7  |-  ( ph  -> 
-u ( C  -  B )  =  ( B  -  C ) )
2019oveq1d 6300 . . . . . 6  |-  ( ph  ->  ( -u ( C  -  B )  / 
( A  -  B
) )  =  ( ( B  -  C
)  /  ( A  -  B ) ) )
2118, 20eqtrd 2508 . . . . 5  |-  ( ph  -> 
-u ( ( C  -  B )  / 
( A  -  B
) )  =  ( ( B  -  C
)  /  ( A  -  B ) ) )
225, 7dividd 10319 . . . . . . . 8  |-  ( ph  ->  ( ( A  -  B )  /  ( A  -  B )
)  =  1 )
2322oveq1d 6300 . . . . . . 7  |-  ( ph  ->  ( ( ( A  -  B )  / 
( A  -  B
) )  -  (
( C  -  B
)  /  ( A  -  B ) ) )  =  ( 1  -  ( ( C  -  B )  / 
( A  -  B
) ) ) )
245, 3, 5, 7divsubdird 10360 . . . . . . 7  |-  ( ph  ->  ( ( ( A  -  B )  -  ( C  -  B
) )  /  ( A  -  B )
)  =  ( ( ( A  -  B
)  /  ( A  -  B ) )  -  ( ( C  -  B )  / 
( A  -  B
) ) ) )
2511, 8negsubd 9937 . . . . . . 7  |-  ( ph  ->  ( 1  +  -u ( ( C  -  B )  /  ( A  -  B )
) )  =  ( 1  -  ( ( C  -  B )  /  ( A  -  B ) ) ) )
2623, 24, 253eqtr4rd 2519 . . . . . 6  |-  ( ph  ->  ( 1  +  -u ( ( C  -  B )  /  ( A  -  B )
) )  =  ( ( ( A  -  B )  -  ( C  -  B )
)  /  ( A  -  B ) ) )
274, 1, 2nnncan2d 9966 . . . . . . 7  |-  ( ph  ->  ( ( A  -  B )  -  ( C  -  B )
)  =  ( A  -  C ) )
2827oveq1d 6300 . . . . . 6  |-  ( ph  ->  ( ( ( A  -  B )  -  ( C  -  B
) )  /  ( A  -  B )
)  =  ( ( A  -  C )  /  ( A  -  B ) ) )
2926, 28eqtrd 2508 . . . . 5  |-  ( ph  ->  ( 1  +  -u ( ( C  -  B )  /  ( A  -  B )
) )  =  ( ( A  -  C
)  /  ( A  -  B ) ) )
3021, 29oveq12d 6303 . . . 4  |-  ( ph  ->  ( -u ( ( C  -  B )  /  ( A  -  B ) )  / 
( 1  +  -u ( ( C  -  B )  /  ( A  -  B )
) ) )  =  ( ( ( B  -  C )  / 
( A  -  B
) )  /  (
( A  -  C
)  /  ( A  -  B ) ) ) )
312, 1subcld 9931 . . . . 5  |-  ( ph  ->  ( B  -  C
)  e.  CC )
324, 1subcld 9931 . . . . 5  |-  ( ph  ->  ( A  -  C
)  e.  CC )
334, 1, 12subne0d 9940 . . . . 5  |-  ( ph  ->  ( A  -  C
)  =/=  0 )
3431, 32, 5, 33, 7divcan7d 10349 . . . 4  |-  ( ph  ->  ( ( ( B  -  C )  / 
( A  -  B
) )  /  (
( A  -  C
)  /  ( A  -  B ) ) )  =  ( ( B  -  C )  /  ( A  -  C ) ) )
352, 1, 4, 1, 12div2subd 10371 . . . 4  |-  ( ph  ->  ( ( B  -  C )  /  ( A  -  C )
)  =  ( ( C  -  B )  /  ( C  -  A ) ) )
3630, 34, 353eqtrrd 2513 . . 3  |-  ( ph  ->  ( ( C  -  B )  /  ( C  -  A )
)  =  ( -u ( ( C  -  B )  /  ( A  -  B )
)  /  ( 1  +  -u ( ( C  -  B )  / 
( A  -  B
) ) ) ) )
3736eleq1d 2536 . 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 256 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 184    e. wcel 1767    =/= wne 2662  (class class class)co 6285   CCcc 9491   0cc0 9493   1c1 9494    + caddc 9496    - cmin 9806   -ucneg 9807    / cdiv 10207   RR+crp 11221   (,)cioo 11530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-1st 6785  df-2nd 6786  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-rp 11222  df-ioo 11534
This theorem is referenced by:  angpieqvd  22987
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