Theorem chordthmlem2 20666

Description: If M is the midpoint of AB, AQ = BQ, and P is on the line AB, then QMP is a right angle. This is proven by reduction to the special case chordthmlem 20665, where P = B, and using angrtmuld 20642 to observe that QMP is right iff QMB is. (Contributed by David Moews, 28-Feb-2017.)

Hypotheses

Ref	Expression
chordthmlem2.angdef	|- F = ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) )
chordthmlem2.A	|- ( ph -> A e. CC )
chordthmlem2.B	|- ( ph -> B e. CC )
chordthmlem2.Q	|- ( ph -> Q e. CC )
chordthmlem2.X	|- ( ph -> X e. RR )
chordthmlem2.M	|- ( ph -> M = ( ( A + B ) / 2 ) )
chordthmlem2.P	|- ( ph -> P = ( ( X x. A ) + ( ( 1 - X ) x. B ) ) )
chordthmlem2.ABequidistQ	|- ( ph -> ( abs ` ( A - Q ) ) = ( abs ` ( B - Q ) ) )
chordthmlem2.PneM	|- ( ph -> P =/= M )
chordthmlem2.QneM	|- ( ph -> Q =/= M )

Assertion

Ref	Expression
chordthmlem2	|- ( ph -> ( ( Q - M ) F ( P - M ) ) e. { ( pi / 2 ) , -u ( pi / 2 ) } )

Distinct variable groups:   x, y, Q   x, P, y   x, M, y   x, B, y   x, A, y

Allowed substitution hints:   ph( x, y)   F( x, y)   X( x, y)

Proof of Theorem chordthmlem2

Step	Hyp	Ref	Expression
1		chordthmlem2.angdef	. . 3 |- F = ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) )
2		chordthmlem2.A	. . 3 |- ( ph -> A e. CC )
3		chordthmlem2.B	. . 3 |- ( ph -> B e. CC )
4		chordthmlem2.Q	. . 3 |- ( ph -> Q e. CC )
5		chordthmlem2.M	. . 3 |- ( ph -> M = ( ( A + B ) / 2 ) )
6		chordthmlem2.ABequidistQ	. . 3 |- ( ph -> ( abs ` ( A - Q ) ) = ( abs ` ( B - Q ) ) )
7		2re 10061	. . . . . . . . . 10 |- 2 e. RR
8	7	a1i 11	. . . . . . . . 9 |- ( ph -> 2 e. RR )
9		2ne0 10075	. . . . . . . . . 10 |- 2 =/= 0
10	9	a1i 11	. . . . . . . . 9 |- ( ph -> 2 =/= 0 )
11	8, 10	rereccld 9833	. . . . . . . 8 |- ( ph -> ( 1 / 2 ) e. RR )
12		chordthmlem2.X	. . . . . . . 8 |- ( ph -> X e. RR )
13	11, 12	resubcld 9457	. . . . . . 7 |- ( ph -> ( ( 1 / 2 ) - X ) e. RR )
14	13	recnd 9106	. . . . . 6 |- ( ph -> ( ( 1 / 2 ) - X ) e. CC )
15	3, 2	subcld 9403	. . . . . 6 |- ( ph -> ( B - A ) e. CC )
16	11	recnd 9106	. . . . . . . . 9 |- ( ph -> ( 1 / 2 ) e. CC )
17	12	recnd 9106	. . . . . . . . 9 |- ( ph -> X e. CC )
18	16, 17, 15	subdird 9482	. . . . . . . 8 |- ( ph -> ( ( ( 1 / 2 ) - X ) x. ( B - A ) ) = ( ( ( 1 / 2 ) x. ( B - A ) ) - ( X x. ( B - A ) ) ) )
19		2cn 10062	. . . . . . . . . . . . . . 15 |- 2 e. CC
20	19	a1i 11	. . . . . . . . . . . . . 14 |- ( ph -> 2 e. CC )
21	3, 20, 10	divcan4d 9788	. . . . . . . . . . . . 13 |- ( ph -> ( ( B x. 2 ) / 2 ) = B )
22	3	times2d 10203	. . . . . . . . . . . . . 14 |- ( ph -> ( B x. 2 ) = ( B + B ) )
23	22	oveq1d 6088	. . . . . . . . . . . . 13 |- ( ph -> ( ( B x. 2 ) / 2 ) = ( ( B + B ) / 2 ) )
24	21, 23	eqtr3d 2469	. . . . . . . . . . . 12 |- ( ph -> B = ( ( B + B ) / 2 ) )
25	24, 5	oveq12d 6091	. . . . . . . . . . 11 |- ( ph -> ( B - M ) = ( ( ( B + B ) / 2 ) - ( ( A + B ) / 2 ) ) )
26	3, 3	addcld 9099	. . . . . . . . . . . 12 |- ( ph -> ( B + B ) e. CC )
27	2, 3	addcld 9099	. . . . . . . . . . . 12 |- ( ph -> ( A + B ) e. CC )
28	26, 27, 20, 10	divsubdird 9821	. . . . . . . . . . 11 |- ( ph -> ( ( ( B + B ) - ( A + B ) ) / 2 ) = ( ( ( B + B ) / 2 ) - ( ( A + B ) / 2 ) ) )
29	3, 2, 3	pnpcan2d 9441	. . . . . . . . . . . 12 |- ( ph -> ( ( B + B ) - ( A + B ) ) = ( B - A ) )
30	29	oveq1d 6088	. . . . . . . . . . 11 |- ( ph -> ( ( ( B + B ) - ( A + B ) ) / 2 ) = ( ( B - A ) / 2 ) )
31	25, 28, 30	3eqtr2d 2473	. . . . . . . . . 10 |- ( ph -> ( B - M ) = ( ( B - A ) / 2 ) )
32	15, 20, 10	divrec2d 9786	. . . . . . . . . 10 |- ( ph -> ( ( B - A ) / 2 ) = ( ( 1 / 2 ) x. ( B - A ) ) )
33	31, 32	eqtrd 2467	. . . . . . . . 9 |- ( ph -> ( B - M ) = ( ( 1 / 2 ) x. ( B - A ) ) )
34		chordthmlem2.P	. . . . . . . . . 10 |- ( ph -> P = ( ( X x. A ) + ( ( 1 - X ) x. B ) ) )
35	17, 2	mulcld 9100	. . . . . . . . . . . . 13 |- ( ph -> ( X x. A ) e. CC )
36		ax-1cn 9040	. . . . . . . . . . . . . . . 16 |- 1 e. CC
37	36	a1i 11	. . . . . . . . . . . . . . 15 |- ( ph -> 1 e. CC )
38	37, 17	subcld 9403	. . . . . . . . . . . . . 14 |- ( ph -> ( 1 - X ) e. CC )
39	38, 3	mulcld 9100	. . . . . . . . . . . . 13 |- ( ph -> ( ( 1 - X ) x. B ) e. CC )
40	35, 39	addcld 9099	. . . . . . . . . . . 12 |- ( ph -> ( ( X x. A ) + ( ( 1 - X ) x. B ) ) e. CC )
41	34, 40	eqeltrd 2509	. . . . . . . . . . 11 |- ( ph -> P e. CC )
42	2, 41, 3, 17	affineequiv 20659	. . . . . . . . . 10 |- ( ph -> ( P = ( ( X x. A ) + ( ( 1 - X ) x. B ) ) <-> ( B - P ) = ( X x. ( B - A ) ) ) )
43	34, 42	mpbid 202	. . . . . . . . 9 |- ( ph -> ( B - P ) = ( X x. ( B - A ) ) )
44	33, 43	oveq12d 6091	. . . . . . . 8 |- ( ph -> ( ( B - M ) - ( B - P ) ) = ( ( ( 1 / 2 ) x. ( B - A ) ) - ( X x. ( B - A ) ) ) )
45	27	halfcld 10204	. . . . . . . . . 10 |- ( ph -> ( ( A + B ) / 2 ) e. CC )
46	5, 45	eqeltrd 2509	. . . . . . . . 9 |- ( ph -> M e. CC )
47	3, 46, 41	nnncan1d 9437	. . . . . . . 8 |- ( ph -> ( ( B - M ) - ( B - P ) ) = ( P - M ) )
48	18, 44, 47	3eqtr2rd 2474	. . . . . . 7 |- ( ph -> ( P - M ) = ( ( ( 1 / 2 ) - X ) x. ( B - A ) ) )
49		chordthmlem2.PneM	. . . . . . . 8 |- ( ph -> P =/= M )
50	41, 46, 49	subne0d 9412	. . . . . . 7 |- ( ph -> ( P - M ) =/= 0 )
51	48, 50	eqnetrrd 2618	. . . . . 6 |- ( ph -> ( ( ( 1 / 2 ) - X ) x. ( B - A ) ) =/= 0 )
52	14, 15, 51	mulne0bbd 9668	. . . . 5 |- ( ph -> ( B - A ) =/= 0 )
53	3, 2, 52	subne0ad 9414	. . . 4 |- ( ph -> B =/= A )
54	53	necomd 2681	. . 3 |- ( ph -> A =/= B )
55		chordthmlem2.QneM	. . 3 |- ( ph -> Q =/= M )
56	1, 2, 3, 4, 5, 6, 54, 55	chordthmlem 20665	. 2 |- ( ph -> ( ( Q - M ) F ( B - M ) ) e. { ( pi / 2 ) , -u ( pi / 2 ) } )
57	4, 46	subcld 9403	. . 3 |- ( ph -> ( Q - M ) e. CC )
58	41, 46	subcld 9403	. . 3 |- ( ph -> ( P - M ) e. CC )
59	3, 46	subcld 9403	. . 3 |- ( ph -> ( B - M ) e. CC )
60	4, 46, 55	subne0d 9412	. . 3 |- ( ph -> ( Q - M ) =/= 0 )
61	20, 10	recne0d 9776	. . . . 5 |- ( ph -> ( 1 / 2 ) =/= 0 )
62	16, 15, 61, 52	mulne0d 9666	. . . 4 |- ( ph -> ( ( 1 / 2 ) x. ( B - A ) ) =/= 0 )
63	33, 62	eqnetrd 2616	. . 3 |- ( ph -> ( B - M ) =/= 0 )
64	33, 48	oveq12d 6091	. . . . 5 |- ( ph -> ( ( B - M ) / ( P - M ) ) = ( ( ( 1 / 2 ) x. ( B - A ) ) / ( ( ( 1 / 2 ) - X ) x. ( B - A ) ) ) )
65	14, 15, 51	mulne0bad 9667	. . . . . 6 |- ( ph -> ( ( 1 / 2 ) - X ) =/= 0 )
66	16, 14, 15, 65, 52	divcan5rd 9809	. . . . 5 |- ( ph -> ( ( ( 1 / 2 ) x. ( B - A ) ) / ( ( ( 1 / 2 ) - X ) x. ( B - A ) ) ) = ( ( 1 / 2 ) / ( ( 1 / 2 ) - X ) ) )
67	64, 66	eqtrd 2467	. . . 4 |- ( ph -> ( ( B - M ) / ( P - M ) ) = ( ( 1 / 2 ) / ( ( 1 / 2 ) - X ) ) )
68	11, 13, 65	redivcld 9834	. . . 4 |- ( ph -> ( ( 1 / 2 ) / ( ( 1 / 2 ) - X ) ) e. RR )
69	67, 68	eqeltrd 2509	. . 3 |- ( ph -> ( ( B - M ) / ( P - M ) ) e. RR )
70	1, 57, 58, 59, 60, 50, 63, 69	angrtmuld 20642	. 2 |- ( ph -> ( ( ( Q - M ) F ( P - M ) ) e. { ( pi / 2 ) , -u ( pi / 2 ) } <-> ( ( Q - M ) F ( B - M ) ) e. { ( pi / 2 ) , -u ( pi / 2 ) } ) )
71	56, 70	mpbird 224	1 |- ( ph -> ( ( Q - M ) F ( P - M ) ) e. { ( pi / 2 ) , -u ( pi / 2 ) } )

Syntax hints:   -> wi 4   = wceq 1652   e. wcel 1725   =/= wne 2598   \ cdif 3309   {csn 3806   {cpr 3807   ` cfv 5446  (class class class)co 6073   e. cmpt2 6075   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983   + caddc 8985   x. cmul 8987   - cmin 9283   -ucneg 9284   / cdiv 9669   2c2 10041   Imcim 11895   abscabs 12031   picpi 12661   logclog 20444

This theorem is referenced by:  chordthmlem3  20667

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-ioo 10912  df-ioc 10913  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-fac 11559  df-bc 11586  df-hash 11611  df-shft 11874  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-limsup 12257  df-clim 12274  df-rlim 12275  df-sum 12472  df-ef 12662  df-sin 12664  df-cos 12665  df-pi 12667  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-hom 13545  df-cco 13546  df-rest 13642  df-topn 13643  df-topgen 13659  df-pt 13660  df-prds 13663  df-xrs 13718  df-0g 13719  df-gsum 13720  df-qtop 13725  df-imas 13726  df-xps 13728  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-submnd 14731  df-mulg 14807  df-cntz 15108  df-cmn 15406  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-fbas 16691  df-fg 16692  df-cnfld 16696  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cld 17075  df-ntr 17076  df-cls 17077  df-nei 17154  df-lp 17192  df-perf 17193  df-cn 17283  df-cnp 17284  df-haus 17371  df-tx 17586  df-hmeo 17779  df-fil 17870  df-fm 17962  df-flim 17963  df-flf 17964  df-xms 18342  df-ms 18343  df-tms 18344  df-cncf 18900  df-limc 19745  df-dv 19746  df-log 20446