Theorem chordthmlem2 20635

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 20634, where P = B, and using angrtmuld 20611 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 10033	. . . . . . . . . 10 |- 2 e. RR
8	7	a1i 11	. . . . . . . . 9 |- ( ph -> 2 e. RR )
9		2ne0 10047	. . . . . . . . . 10 |- 2 =/= 0
10	9	a1i 11	. . . . . . . . 9 |- ( ph -> 2 =/= 0 )
11	8, 10	rereccld 9805	. . . . . . . 8 |- ( ph -> ( 1 / 2 ) e. RR )
12		chordthmlem2.X	. . . . . . . 8 |- ( ph -> X e. RR )
13	11, 12	resubcld 9429	. . . . . . 7 |- ( ph -> ( ( 1 / 2 ) - X ) e. RR )
14	13	recnd 9078	. . . . . 6 |- ( ph -> ( ( 1 / 2 ) - X ) e. CC )
15	3, 2	subcld 9375	. . . . . 6 |- ( ph -> ( B - A ) e. CC )
16	11	recnd 9078	. . . . . . . . 9 |- ( ph -> ( 1 / 2 ) e. CC )
17	12	recnd 9078	. . . . . . . . 9 |- ( ph -> X e. CC )
18	16, 17, 15	subdird 9454	. . . . . . . 8 |- ( ph -> ( ( ( 1 / 2 ) - X ) x. ( B - A ) ) = ( ( ( 1 / 2 ) x. ( B - A ) ) - ( X x. ( B - A ) ) ) )
19		2cn 10034	. . . . . . . . . . . . . . 15 |- 2 e. CC
20	19	a1i 11	. . . . . . . . . . . . . 14 |- ( ph -> 2 e. CC )
21	3, 20, 10	divcan4d 9760	. . . . . . . . . . . . 13 |- ( ph -> ( ( B x. 2 ) / 2 ) = B )
22	3	times2d 10175	. . . . . . . . . . . . . 14 |- ( ph -> ( B x. 2 ) = ( B + B ) )
23	22	oveq1d 6063	. . . . . . . . . . . . 13 |- ( ph -> ( ( B x. 2 ) / 2 ) = ( ( B + B ) / 2 ) )
24	21, 23	eqtr3d 2446	. . . . . . . . . . . 12 |- ( ph -> B = ( ( B + B ) / 2 ) )
25	24, 5	oveq12d 6066	. . . . . . . . . . 11 |- ( ph -> ( B - M ) = ( ( ( B + B ) / 2 ) - ( ( A + B ) / 2 ) ) )
26	3, 3	addcld 9071	. . . . . . . . . . . 12 |- ( ph -> ( B + B ) e. CC )
27	2, 3	addcld 9071	. . . . . . . . . . . 12 |- ( ph -> ( A + B ) e. CC )
28	26, 27, 20, 10	divsubdird 9793	. . . . . . . . . . 11 |- ( ph -> ( ( ( B + B ) - ( A + B ) ) / 2 ) = ( ( ( B + B ) / 2 ) - ( ( A + B ) / 2 ) ) )
29	3, 2, 3	pnpcan2d 9413	. . . . . . . . . . . 12 |- ( ph -> ( ( B + B ) - ( A + B ) ) = ( B - A ) )
30	29	oveq1d 6063	. . . . . . . . . . 11 |- ( ph -> ( ( ( B + B ) - ( A + B ) ) / 2 ) = ( ( B - A ) / 2 ) )
31	25, 28, 30	3eqtr2d 2450	. . . . . . . . . 10 |- ( ph -> ( B - M ) = ( ( B - A ) / 2 ) )
32	15, 20, 10	divrec2d 9758	. . . . . . . . . 10 |- ( ph -> ( ( B - A ) / 2 ) = ( ( 1 / 2 ) x. ( B - A ) ) )
33	31, 32	eqtrd 2444	. . . . . . . . 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 9072	. . . . . . . . . . . . 13 |- ( ph -> ( X x. A ) e. CC )
36		ax-1cn 9012	. . . . . . . . . . . . . . . 16 |- 1 e. CC
37	36	a1i 11	. . . . . . . . . . . . . . 15 |- ( ph -> 1 e. CC )
38	37, 17	subcld 9375	. . . . . . . . . . . . . 14 |- ( ph -> ( 1 - X ) e. CC )
39	38, 3	mulcld 9072	. . . . . . . . . . . . 13 |- ( ph -> ( ( 1 - X ) x. B ) e. CC )
40	35, 39	addcld 9071	. . . . . . . . . . . 12 |- ( ph -> ( ( X x. A ) + ( ( 1 - X ) x. B ) ) e. CC )
41	34, 40	eqeltrd 2486	. . . . . . . . . . 11 |- ( ph -> P e. CC )
42	2, 41, 3, 17	affineequiv 20628	. . . . . . . . . 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 6066	. . . . . . . 8 |- ( ph -> ( ( B - M ) - ( B - P ) ) = ( ( ( 1 / 2 ) x. ( B - A ) ) - ( X x. ( B - A ) ) ) )
45	27	halfcld 10176	. . . . . . . . . 10 |- ( ph -> ( ( A + B ) / 2 ) e. CC )
46	5, 45	eqeltrd 2486	. . . . . . . . 9 |- ( ph -> M e. CC )
47	3, 46, 41	nnncan1d 9409	. . . . . . . 8 |- ( ph -> ( ( B - M ) - ( B - P ) ) = ( P - M ) )
48	18, 44, 47	3eqtr2rd 2451	. . . . . . 7 |- ( ph -> ( P - M ) = ( ( ( 1 / 2 ) - X ) x. ( B - A ) ) )
49		chordthmlem2.PneM	. . . . . . . 8 |- ( ph -> P =/= M )
50	41, 46, 49	subne0d 9384	. . . . . . 7 |- ( ph -> ( P - M ) =/= 0 )
51	48, 50	eqnetrrd 2595	. . . . . 6 |- ( ph -> ( ( ( 1 / 2 ) - X ) x. ( B - A ) ) =/= 0 )
52	14, 15, 51	mulne0bbd 9640	. . . . 5 |- ( ph -> ( B - A ) =/= 0 )
53	3, 2, 52	subne0ad 9386	. . . 4 |- ( ph -> B =/= A )
54	53	necomd 2658	. . 3 |- ( ph -> A =/= B )
55		chordthmlem2.QneM	. . 3 |- ( ph -> Q =/= M )
56	1, 2, 3, 4, 5, 6, 54, 55	chordthmlem 20634	. 2 |- ( ph -> ( ( Q - M ) F ( B - M ) ) e. { ( pi / 2 ) , -u ( pi / 2 ) } )
57	4, 46	subcld 9375	. . 3 |- ( ph -> ( Q - M ) e. CC )
58	41, 46	subcld 9375	. . 3 |- ( ph -> ( P - M ) e. CC )
59	3, 46	subcld 9375	. . 3 |- ( ph -> ( B - M ) e. CC )
60	4, 46, 55	subne0d 9384	. . 3 |- ( ph -> ( Q - M ) =/= 0 )
61	20, 10	recne0d 9748	. . . . 5 |- ( ph -> ( 1 / 2 ) =/= 0 )
62	16, 15, 61, 52	mulne0d 9638	. . . 4 |- ( ph -> ( ( 1 / 2 ) x. ( B - A ) ) =/= 0 )
63	33, 62	eqnetrd 2593	. . 3 |- ( ph -> ( B - M ) =/= 0 )
64	33, 48	oveq12d 6066	. . . . 5 |- ( ph -> ( ( B - M ) / ( P - M ) ) = ( ( ( 1 / 2 ) x. ( B - A ) ) / ( ( ( 1 / 2 ) - X ) x. ( B - A ) ) ) )
65	14, 15, 51	mulne0bad 9639	. . . . . 6 |- ( ph -> ( ( 1 / 2 ) - X ) =/= 0 )
66	16, 14, 15, 65, 52	divcan5rd 9781	. . . . 5 |- ( ph -> ( ( ( 1 / 2 ) x. ( B - A ) ) / ( ( ( 1 / 2 ) - X ) x. ( B - A ) ) ) = ( ( 1 / 2 ) / ( ( 1 / 2 ) - X ) ) )
67	64, 66	eqtrd 2444	. . . 4 |- ( ph -> ( ( B - M ) / ( P - M ) ) = ( ( 1 / 2 ) / ( ( 1 / 2 ) - X ) ) )
68	11, 13, 65	redivcld 9806	. . . 4 |- ( ph -> ( ( 1 / 2 ) / ( ( 1 / 2 ) - X ) ) e. RR )
69	67, 68	eqeltrd 2486	. . 3 |- ( ph -> ( ( B - M ) / ( P - M ) ) e. RR )
70	1, 57, 58, 59, 60, 50, 63, 69	angrtmuld 20611	. 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 1649   e. wcel 1721   =/= wne 2575   \ cdif 3285   {csn 3782   {cpr 3783   ` cfv 5421  (class class class)co 6048   e. cmpt2 6050   CCcc 8952   RRcr 8953   0cc0 8954   1c1 8955   + caddc 8957   x. cmul 8959   - cmin 9255   -ucneg 9256   / cdiv 9641   2c2 10013   Imcim 11866   abscabs 12002   picpi 12632   logclog 20413

This theorem is referenced by:  chordthmlem3  20636

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032  ax-addf 9033  ax-mulf 9034

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6872  df-map 6987  df-pm 6988  df-ixp 7031  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-fi 7382  df-sup 7412  df-oi 7443  df-card 7790  df-cda 8012  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-10 10030  df-n0 10186  df-z 10247  df-dec 10347  df-uz 10453  df-q 10539  df-rp 10577  df-xneg 10674  df-xadd 10675  df-xmul 10676  df-ioo 10884  df-ioc 10885  df-ico 10886  df-icc 10887  df-fz 11008  df-fzo 11099  df-fl 11165  df-mod 11214  df-seq 11287  df-exp 11346  df-fac 11530  df-bc 11557  df-hash 11582  df-shft 11845  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-limsup 12228  df-clim 12245  df-rlim 12246  df-sum 12443  df-ef 12633  df-sin 12635  df-cos 12636  df-pi 12638  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-mulr 13506  df-starv 13507  df-sca 13508  df-vsca 13509  df-tset 13511  df-ple 13512  df-ds 13514  df-unif 13515  df-hom 13516  df-cco 13517  df-rest 13613  df-topn 13614  df-topgen 13630  df-pt 13631  df-prds 13634  df-xrs 13689  df-0g 13690  df-gsum 13691  df-qtop 13696  df-imas 13697  df-xps 13699  df-mre 13774  df-mrc 13775  df-acs 13777  df-mnd 14653  df-submnd 14702  df-mulg 14778  df-cntz 15079  df-cmn 15377  df-psmet 16657  df-xmet 16658  df-met 16659  df-bl 16660  df-mopn 16661  df-fbas 16662  df-fg 16663  df-cnfld 16667  df-top 16926  df-bases 16928  df-topon 16929  df-topsp 16930  df-cld 17046  df-ntr 17047  df-cls 17048  df-nei 17125  df-lp 17163  df-perf 17164  df-cn 17253  df-cnp 17254  df-haus 17341  df-tx 17555  df-hmeo 17748  df-fil 17839  df-fm 17931  df-flim 17932  df-flf 17933  df-xms 18311  df-ms 18312  df-tms 18313  df-cncf 18869  df-limc 19714  df-dv 19715  df-log 20415