Theorem chordthmlem2 20078

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 20077, where P = B, and using angrtmuld 20054 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 9769	. . . . . . . . . 10 |- 2 e. RR
8	7	a1i 12	. . . . . . . . 9 |- ( ph -> 2 e. RR )
9		2ne0 9783	. . . . . . . . . 10 |- 2 =/= 0
10	9	a1i 12	. . . . . . . . 9 |- ( ph -> 2 =/= 0 )
11	8, 10	rereccld 9541	. . . . . . . 8 |- ( ph -> ( 1 / 2 ) e. RR )
12		chordthmlem2.X	. . . . . . . 8 |- ( ph -> X e. RR )
13	11, 12	resubcld 9165	. . . . . . 7 |- ( ph -> ( ( 1 / 2 ) - X ) e. RR )
14	13	recnd 8815	. . . . . 6 |- ( ph -> ( ( 1 / 2 ) - X ) e. CC )
15	3, 2	subcld 9111	. . . . . 6 |- ( ph -> ( B - A ) e. CC )
16	11	recnd 8815	. . . . . . . . 9 |- ( ph -> ( 1 / 2 ) e. CC )
17	12	recnd 8815	. . . . . . . . 9 |- ( ph -> X e. CC )
18	16, 17, 15	subdird 9190	. . . . . . . 8 |- ( ph -> ( ( ( 1 / 2 ) - X ) x. ( B - A ) ) = ( ( ( 1 / 2 ) x. ( B - A ) ) - ( X x. ( B - A ) ) ) )
19		2cn 9770	. . . . . . . . . . . . . . 15 |- 2 e. CC
20	19	a1i 12	. . . . . . . . . . . . . 14 |- ( ph -> 2 e. CC )
21	3, 20, 10	divcan4d 9496	. . . . . . . . . . . . 13 |- ( ph -> ( ( B x. 2 ) / 2 ) = B )
22	3	times2d 9908	. . . . . . . . . . . . . 14 |- ( ph -> ( B x. 2 ) = ( B + B ) )
23	22	oveq1d 5793	. . . . . . . . . . . . 13 |- ( ph -> ( ( B x. 2 ) / 2 ) = ( ( B + B ) / 2 ) )
24	21, 23	eqtr3d 2290	. . . . . . . . . . . 12 |- ( ph -> B = ( ( B + B ) / 2 ) )
25	24, 5	oveq12d 5796	. . . . . . . . . . 11 |- ( ph -> ( B - M ) = ( ( ( B + B ) / 2 ) - ( ( A + B ) / 2 ) ) )
26	3, 3	addcld 8808	. . . . . . . . . . . 12 |- ( ph -> ( B + B ) e. CC )
27	2, 3	addcld 8808	. . . . . . . . . . . 12 |- ( ph -> ( A + B ) e. CC )
28	26, 27, 20, 10	divsubdird 9529	. . . . . . . . . . 11 |- ( ph -> ( ( ( B + B ) - ( A + B ) ) / 2 ) = ( ( ( B + B ) / 2 ) - ( ( A + B ) / 2 ) ) )
29	3, 2, 3	pnpcan2d 9149	. . . . . . . . . . . 12 |- ( ph -> ( ( B + B ) - ( A + B ) ) = ( B - A ) )
30	29	oveq1d 5793	. . . . . . . . . . 11 |- ( ph -> ( ( ( B + B ) - ( A + B ) ) / 2 ) = ( ( B - A ) / 2 ) )
31	25, 28, 30	3eqtr2d 2294	. . . . . . . . . 10 |- ( ph -> ( B - M ) = ( ( B - A ) / 2 ) )
32	15, 20, 10	divrec2d 9494	. . . . . . . . . 10 |- ( ph -> ( ( B - A ) / 2 ) = ( ( 1 / 2 ) x. ( B - A ) ) )
33	31, 32	eqtrd 2288	. . . . . . . . 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 8809	. . . . . . . . . . . . 13 |- ( ph -> ( X x. A ) e. CC )
36		ax-1cn 8749	. . . . . . . . . . . . . . . 16 |- 1 e. CC
37	36	a1i 12	. . . . . . . . . . . . . . 15 |- ( ph -> 1 e. CC )
38	37, 17	subcld 9111	. . . . . . . . . . . . . 14 |- ( ph -> ( 1 - X ) e. CC )
39	38, 3	mulcld 8809	. . . . . . . . . . . . 13 |- ( ph -> ( ( 1 - X ) x. B ) e. CC )
40	35, 39	addcld 8808	. . . . . . . . . . . 12 |- ( ph -> ( ( X x. A ) + ( ( 1 - X ) x. B ) ) e. CC )
41	34, 40	eqeltrd 2330	. . . . . . . . . . 11 |- ( ph -> P e. CC )
42	2, 41, 3, 17	affineequiv 20071	. . . . . . . . . 10 |- ( ph -> ( P = ( ( X x. A ) + ( ( 1 - X ) x. B ) ) <-> ( B - P ) = ( X x. ( B - A ) ) ) )
43	34, 42	mpbid 203	. . . . . . . . 9 |- ( ph -> ( B - P ) = ( X x. ( B - A ) ) )
44	33, 43	oveq12d 5796	. . . . . . . 8 |- ( ph -> ( ( B - M ) - ( B - P ) ) = ( ( ( 1 / 2 ) x. ( B - A ) ) - ( X x. ( B - A ) ) ) )
45	27	halfcld 9909	. . . . . . . . . 10 |- ( ph -> ( ( A + B ) / 2 ) e. CC )
46	5, 45	eqeltrd 2330	. . . . . . . . 9 |- ( ph -> M e. CC )
47	3, 46, 41	nnncan1d 9145	. . . . . . . 8 |- ( ph -> ( ( B - M ) - ( B - P ) ) = ( P - M ) )
48	18, 44, 47	3eqtr2rd 2295	. . . . . . 7 |- ( ph -> ( P - M ) = ( ( ( 1 / 2 ) - X ) x. ( B - A ) ) )
49		chordthmlem2.PneM	. . . . . . . 8 |- ( ph -> P =/= M )
50	41, 46, 49	subne0d 9120	. . . . . . 7 |- ( ph -> ( P - M ) =/= 0 )
51	48, 50	eqnetrrd 2439	. . . . . 6 |- ( ph -> ( ( ( 1 / 2 ) - X ) x. ( B - A ) ) =/= 0 )
52	14, 15, 51	mulne0bbd 9376	. . . . 5 |- ( ph -> ( B - A ) =/= 0 )
53	3, 2, 52	subne0ad 9122	. . . 4 |- ( ph -> B =/= A )
54	53	necomd 2502	. . 3 |- ( ph -> A =/= B )
55		chordthmlem2.QneM	. . 3 |- ( ph -> Q =/= M )
56	1, 2, 3, 4, 5, 6, 54, 55	chordthmlem 20077	. 2 |- ( ph -> ( ( Q - M ) F ( B - M ) ) e. { ( pi / 2 ) , -u ( pi / 2 ) } )
57	4, 46	subcld 9111	. . 3 |- ( ph -> ( Q - M ) e. CC )
58	41, 46	subcld 9111	. . 3 |- ( ph -> ( P - M ) e. CC )
59	3, 46	subcld 9111	. . 3 |- ( ph -> ( B - M ) e. CC )
60	4, 46, 55	subne0d 9120	. . 3 |- ( ph -> ( Q - M ) =/= 0 )
61	20, 10	recne0d 9484	. . . . 5 |- ( ph -> ( 1 / 2 ) =/= 0 )
62	16, 15, 61, 52	mulne0d 9374	. . . 4 |- ( ph -> ( ( 1 / 2 ) x. ( B - A ) ) =/= 0 )
63	33, 62	eqnetrd 2437	. . 3 |- ( ph -> ( B - M ) =/= 0 )
64	33, 48	oveq12d 5796	. . . . 5 |- ( ph -> ( ( B - M ) / ( P - M ) ) = ( ( ( 1 / 2 ) x. ( B - A ) ) / ( ( ( 1 / 2 ) - X ) x. ( B - A ) ) ) )
65	14, 15, 51	mulne0bad 9375	. . . . . 6 |- ( ph -> ( ( 1 / 2 ) - X ) =/= 0 )
66	16, 14, 15, 65, 52	divcan5rd 9517	. . . . 5 |- ( ph -> ( ( ( 1 / 2 ) x. ( B - A ) ) / ( ( ( 1 / 2 ) - X ) x. ( B - A ) ) ) = ( ( 1 / 2 ) / ( ( 1 / 2 ) - X ) ) )
67	64, 66	eqtrd 2288	. . . 4 |- ( ph -> ( ( B - M ) / ( P - M ) ) = ( ( 1 / 2 ) / ( ( 1 / 2 ) - X ) ) )
68	11, 13, 65	redivcld 9542	. . . 4 |- ( ph -> ( ( 1 / 2 ) / ( ( 1 / 2 ) - X ) ) e. RR )
69	67, 68	eqeltrd 2330	. . 3 |- ( ph -> ( ( B - M ) / ( P - M ) ) e. RR )
70	1, 57, 58, 59, 60, 50, 63, 69	angrtmuld 20054	. 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 225	1 |- ( ph -> ( ( Q - M ) F ( P - M ) ) e. { ( pi / 2 ) , -u ( pi / 2 ) } )

Syntax hints:   -> wi 6   = wceq 1619   e. wcel 1621   =/= wne 2419   \ cdif 3110   {csn 3600   {cpr 3601   ` cfv 4659  (class class class)co 5778   e. cmpt2 5780   CCcc 8689   RRcr 8690   0cc0 8691   1c1 8692   + caddc 8694   x. cmul 8696   - cmin 8991   -ucneg 8992   / cdiv 9377   2c2 9749   Imcim 11534   abscabs 11670   picpi 12296   logclog 19860

This theorem is referenced by:  chordthmlem3  20079

This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-inf2 7296  ax-cnex 8747  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768  ax-pre-sup 8769  ax-addf 8770  ax-mulf 8771

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-iin 3868  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-se 4311  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-isom 4676  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-of 5998  df-1st 6042  df-2nd 6043  df-iota 6211  df-riota 6258  df-recs 6342  df-rdg 6377  df-1o 6433  df-2o 6434  df-oadd 6437  df-er 6614  df-map 6728  df-pm 6729  df-ixp 6772  df-en 6818  df-dom 6819  df-sdom 6820  df-fin 6821  df-fi 7119  df-sup 7148  df-oi 7179  df-card 7526  df-cda 7748  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-div 9378  df-n 9701  df-2 9758  df-3 9759  df-4 9760  df-5 9761  df-6 9762  df-7 9763  df-8 9764  df-9 9765  df-10 9766  df-n0 9919  df-z 9978  df-dec 10078  df-uz 10184  df-q 10270  df-rp 10308  df-xneg 10405  df-xadd 10406  df-xmul 10407  df-ioo 10612  df-ioc 10613  df-ico 10614  df-icc 10615  df-fz 10735  df-fzo 10823  df-fl 10877  df-mod 10926  df-seq 10999  df-exp 11057  df-fac 11241  df-bc 11268  df-hash 11290  df-shft 11513  df-cj 11535  df-re 11536  df-im 11537  df-sqr 11671  df-abs 11672  df-limsup 11896  df-clim 11913  df-rlim 11914  df-sum 12110  df-ef 12297  df-sin 12299  df-cos 12300  df-pi 12302  df-struct 13098  df-ndx 13099  df-slot 13100  df-base 13101  df-sets 13102  df-ress 13103  df-plusg 13169  df-mulr 13170  df-starv 13171  df-sca 13172  df-vsca 13173  df-tset 13175  df-ple 13176  df-ds 13178  df-hom 13180  df-cco 13181  df-rest 13275  df-topn 13276  df-topgen 13292  df-pt 13293  df-prds 13296  df-xrs 13351  df-0g 13352  df-gsum 13353  df-qtop 13358  df-imas 13359  df-xps 13361  df-mre 13436  df-mrc 13437  df-acs 13439  df-mnd 14315  df-submnd 14364  df-mulg 14440  df-cntz 14741  df-cmn 15039  df-xmet 16321  df-met 16322  df-bl 16323  df-mopn 16324  df-cnfld 16326  df-top 16584  df-bases 16586  df-topon 16587  df-topsp 16588  df-cld 16704  df-ntr 16705  df-cls 16706  df-nei 16783  df-lp 16816  df-perf 16817  df-cn 16905  df-cnp 16906  df-haus 16991  df-tx 17205  df-hmeo 17394  df-fbas 17468  df-fg 17469  df-fil 17489  df-fm 17581  df-flim 17582  df-flf 17583  df-xms 17833  df-ms 17834  df-tms 17835  df-cncf 18330  df-limc 19164  df-dv 19165  df-log 19862