Theorem angpieqvdlem 24555

Description: Equivalence used in the proof of angpieqvd 24558. (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

Step	Hyp	Ref	Expression
1		angpieqvdlem.C	. . . . . 6 |- ( ph -> C e. CC )
2		angpieqvdlem.B	. . . . . 6 |- ( ph -> B e. CC )
3	1, 2	subcld 10392	. . . . 5 |- ( ph -> ( C - B ) e. CC )
4		angpieqvdlem.A	. . . . . 6 |- ( ph -> A e. CC )
5	4, 2	subcld 10392	. . . . 5 |- ( ph -> ( A - B ) e. CC )
6		angpieqvdlem.AneB	. . . . . 6 |- ( ph -> A =/= B )
7	4, 2, 6	subne0d 10401	. . . . 5 |- ( ph -> ( A - B ) =/= 0 )
8	3, 5, 7	divcld 10801	. . . 4 |- ( ph -> ( ( C - B ) / ( A - B ) ) e. CC )
9	8	negcld 10379	. . 3 |- ( ph -> -u ( ( C - B ) / ( A - B ) ) e. CC )
10		1cnd 10056	. . . 4 |- ( ph -> 1 e. CC )
11		angpieqvdlem.AneC	. . . . . . 7 |- ( ph -> A =/= C )
12	11	necomd 2849	. . . . . 6 |- ( ph -> C =/= A )
13	1, 4, 2, 12	subneintr2d 10438	. . . . 5 |- ( ph -> ( C - B ) =/= ( A - B ) )
14	3, 5, 7, 13	divne1d 10812	. . . 4 |- ( ph -> ( ( C - B ) / ( A - B ) ) =/= 1 )
15	8, 10, 14	negned 10389	. . 3 |- ( ph -> -u ( ( C - B ) / ( A - B ) ) =/= -u 1 )
16	9, 15	xov1plusxeqvd 12318	. 2 |- ( ph -> ( -u ( ( C - B ) / ( A - B ) ) e. RR+ <-> ( -u ( ( C - B ) / ( A - B ) ) / ( 1 + -u ( ( C - B ) / ( A - B ) ) ) ) e. ( 0 (,) 1 ) ) )
17	3, 5, 7	divnegd 10814	. . . . . 6 |- ( ph -> -u ( ( C - B ) / ( A - B ) ) = ( -u ( C - B ) / ( A - B ) ) )
18	1, 2	negsubdi2d 10408	. . . . . . 7 |- ( ph -> -u ( C - B ) = ( B - C ) )
19	18	oveq1d 6665	. . . . . 6 |- ( ph -> ( -u ( C - B ) / ( A - B ) ) = ( ( B - C ) / ( A - B ) ) )
20	17, 19	eqtrd 2656	. . . . 5 |- ( ph -> -u ( ( C - B ) / ( A - B ) ) = ( ( B - C ) / ( A - B ) ) )
21	5, 7	dividd 10799	. . . . . . . 8 |- ( ph -> ( ( A - B ) / ( A - B ) ) = 1 )
22	21	oveq1d 6665	. . . . . . 7 |- ( ph -> ( ( ( A - B ) / ( A - B ) ) - ( ( C - B ) / ( A - B ) ) ) = ( 1 - ( ( C - B ) / ( A - B ) ) ) )
23	5, 3, 5, 7	divsubdird 10840	. . . . . . 7 |- ( ph -> ( ( ( A - B ) - ( C - B ) ) / ( A - B ) ) = ( ( ( A - B ) / ( A - B ) ) - ( ( C - B ) / ( A - B ) ) ) )
24	10, 8	negsubd 10398	. . . . . . 7 |- ( ph -> ( 1 + -u ( ( C - B ) / ( A - B ) ) ) = ( 1 - ( ( C - B ) / ( A - B ) ) ) )
25	22, 23, 24	3eqtr4rd 2667	. . . . . 6 |- ( ph -> ( 1 + -u ( ( C - B ) / ( A - B ) ) ) = ( ( ( A - B ) - ( C - B ) ) / ( A - B ) ) )
26	4, 1, 2	nnncan2d 10427	. . . . . . 7 |- ( ph -> ( ( A - B ) - ( C - B ) ) = ( A - C ) )
27	26	oveq1d 6665	. . . . . 6 |- ( ph -> ( ( ( A - B ) - ( C - B ) ) / ( A - B ) ) = ( ( A - C ) / ( A - B ) ) )
28	25, 27	eqtrd 2656	. . . . 5 |- ( ph -> ( 1 + -u ( ( C - B ) / ( A - B ) ) ) = ( ( A - C ) / ( A - B ) ) )
29	20, 28	oveq12d 6668	. . . 4 |- ( ph -> ( -u ( ( C - B ) / ( A - B ) ) / ( 1 + -u ( ( C - B ) / ( A - B ) ) ) ) = ( ( ( B - C ) / ( A - B ) ) / ( ( A - C ) / ( A - B ) ) ) )
30	2, 1	subcld 10392	. . . . 5 |- ( ph -> ( B - C ) e. CC )
31	4, 1	subcld 10392	. . . . 5 |- ( ph -> ( A - C ) e. CC )
32	4, 1, 11	subne0d 10401	. . . . 5 |- ( ph -> ( A - C ) =/= 0 )
33	30, 31, 5, 32, 7	divcan7d 10829	. . . 4 |- ( ph -> ( ( ( B - C ) / ( A - B ) ) / ( ( A - C ) / ( A - B ) ) ) = ( ( B - C ) / ( A - C ) ) )
34	2, 1, 4, 1, 11	div2subd 10851	. . . 4 |- ( ph -> ( ( B - C ) / ( A - C ) ) = ( ( C - B ) / ( C - A ) ) )
35	29, 33, 34	3eqtrrd 2661	. . 3 |- ( ph -> ( ( C - B ) / ( C - A ) ) = ( -u ( ( C - B ) / ( A - B ) ) / ( 1 + -u ( ( C - B ) / ( A - B ) ) ) ) )
36	35	eleq1d 2686	. 2 |- ( ph -> ( ( ( C - B ) / ( C - A ) ) e. ( 0 (,) 1 ) <-> ( -u ( ( C - B ) / ( A - B ) ) / ( 1 + -u ( ( C - B ) / ( A - B ) ) ) ) e. ( 0 (,) 1 ) ) )
37	16, 36	bitr4d 271	1 |- ( ph -> ( -u ( ( C - B ) / ( A - B ) ) e. RR+ <-> ( ( C - B ) / ( C - A ) ) e. ( 0 (,) 1 ) ) )

Syntax hints:   -> wi 4   <-> wb 196   e. wcel 1990   =/= wne 2794  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266   -ucneg 10267   / cdiv 10684   RR+crp 11832   (,)cioo 12175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-rp 11833  df-ioo 12179

This theorem is referenced by:  angpieqvd  24558