Theorem chordthmlem3 24561

Description: If M is the midpoint of AB, AQ = BQ, and P is on the line AB, then PQ ² = QM ² + PM ² . This follows from chordthmlem2 24560 and the Pythagorean theorem (pythag 24547) in the case where P and Q are unequal to M. If either P or Q equals M, the result is trivial. (Contributed by David Moews, 28-Feb-2017.)

Hypotheses

Ref	Expression
chordthmlem3.A	|- ( ph -> A e. CC )
chordthmlem3.B	|- ( ph -> B e. CC )
chordthmlem3.Q	|- ( ph -> Q e. CC )
chordthmlem3.X	|- ( ph -> X e. RR )
chordthmlem3.M	|- ( ph -> M = ( ( A + B ) / 2 ) )
chordthmlem3.P	|- ( ph -> P = ( ( X x. A ) + ( ( 1 - X ) x. B ) ) )
chordthmlem3.ABequidistQ	|- ( ph -> ( abs ` ( A - Q ) ) = ( abs ` ( B - Q ) ) )

Assertion

Ref	Expression
chordthmlem3	|- ( ph -> ( ( abs ` ( P - Q ) ) ^ 2 ) = ( ( ( abs ` ( Q - M ) ) ^ 2 ) + ( ( abs ` ( P - M ) ) ^ 2 ) ) )

Proof of Theorem chordthmlem3

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		chordthmlem3.Q	. . . . . . . . 9 |- ( ph -> Q e. CC )
2		chordthmlem3.M	. . . . . . . . . 10 |- ( ph -> M = ( ( A + B ) / 2 ) )
3		chordthmlem3.A	. . . . . . . . . . . 12 |- ( ph -> A e. CC )
4		chordthmlem3.B	. . . . . . . . . . . 12 |- ( ph -> B e. CC )
5	3, 4	addcld 10059	. . . . . . . . . . 11 |- ( ph -> ( A + B ) e. CC )
6	5	halfcld 11277	. . . . . . . . . 10 |- ( ph -> ( ( A + B ) / 2 ) e. CC )
7	2, 6	eqeltrd 2701	. . . . . . . . 9 |- ( ph -> M e. CC )
8	1, 7	subcld 10392	. . . . . . . 8 |- ( ph -> ( Q - M ) e. CC )
9	8	abscld 14175	. . . . . . 7 |- ( ph -> ( abs ` ( Q - M ) ) e. RR )
10	9	recnd 10068	. . . . . 6 |- ( ph -> ( abs ` ( Q - M ) ) e. CC )
11	10	sqcld 13006	. . . . 5 |- ( ph -> ( ( abs ` ( Q - M ) ) ^ 2 ) e. CC )
12	11	adantr 481	. . . 4 |- ( ( ph /\ P = M ) -> ( ( abs ` ( Q - M ) ) ^ 2 ) e. CC )
13	12	addid1d 10236	. . 3 |- ( ( ph /\ P = M ) -> ( ( ( abs ` ( Q - M ) ) ^ 2 ) + 0 ) = ( ( abs ` ( Q - M ) ) ^ 2 ) )
14		chordthmlem3.P	. . . . . . . . 9 |- ( ph -> P = ( ( X x. A ) + ( ( 1 - X ) x. B ) ) )
15		chordthmlem3.X	. . . . . . . . . . . 12 |- ( ph -> X e. RR )
16	15	recnd 10068	. . . . . . . . . . 11 |- ( ph -> X e. CC )
17	16, 3	mulcld 10060	. . . . . . . . . 10 |- ( ph -> ( X x. A ) e. CC )
18		1cnd 10056	. . . . . . . . . . . 12 |- ( ph -> 1 e. CC )
19	18, 16	subcld 10392	. . . . . . . . . . 11 |- ( ph -> ( 1 - X ) e. CC )
20	19, 4	mulcld 10060	. . . . . . . . . 10 |- ( ph -> ( ( 1 - X ) x. B ) e. CC )
21	17, 20	addcld 10059	. . . . . . . . 9 |- ( ph -> ( ( X x. A ) + ( ( 1 - X ) x. B ) ) e. CC )
22	14, 21	eqeltrd 2701	. . . . . . . 8 |- ( ph -> P e. CC )
23	22	adantr 481	. . . . . . 7 |- ( ( ph /\ P = M ) -> P e. CC )
24		simpr 477	. . . . . . 7 |- ( ( ph /\ P = M ) -> P = M )
25	23, 24	subeq0bd 10456	. . . . . 6 |- ( ( ph /\ P = M ) -> ( P - M ) = 0 )
26	25	abs00bd 14031	. . . . 5 |- ( ( ph /\ P = M ) -> ( abs ` ( P - M ) ) = 0 )
27	26	sq0id 12957	. . . 4 |- ( ( ph /\ P = M ) -> ( ( abs ` ( P - M ) ) ^ 2 ) = 0 )
28	27	oveq2d 6666	. . 3 |- ( ( ph /\ P = M ) -> ( ( ( abs ` ( Q - M ) ) ^ 2 ) + ( ( abs ` ( P - M ) ) ^ 2 ) ) = ( ( ( abs ` ( Q - M ) ) ^ 2 ) + 0 ) )
29	1	adantr 481	. . . . . 6 |- ( ( ph /\ P = M ) -> Q e. CC )
30	29, 23	abssubd 14192	. . . . 5 |- ( ( ph /\ P = M ) -> ( abs ` ( Q - P ) ) = ( abs ` ( P - Q ) ) )
31	24	oveq2d 6666	. . . . . 6 |- ( ( ph /\ P = M ) -> ( Q - P ) = ( Q - M ) )
32	31	fveq2d 6195	. . . . 5 |- ( ( ph /\ P = M ) -> ( abs ` ( Q - P ) ) = ( abs ` ( Q - M ) ) )
33	30, 32	eqtr3d 2658	. . . 4 |- ( ( ph /\ P = M ) -> ( abs ` ( P - Q ) ) = ( abs ` ( Q - M ) ) )
34	33	oveq1d 6665	. . 3 |- ( ( ph /\ P = M ) -> ( ( abs ` ( P - Q ) ) ^ 2 ) = ( ( abs ` ( Q - M ) ) ^ 2 ) )
35	13, 28, 34	3eqtr4rd 2667	. 2 |- ( ( ph /\ P = M ) -> ( ( abs ` ( P - Q ) ) ^ 2 ) = ( ( ( abs ` ( Q - M ) ) ^ 2 ) + ( ( abs ` ( P - M ) ) ^ 2 ) ) )
36	22, 7	subcld 10392	. . . . . . . 8 |- ( ph -> ( P - M ) e. CC )
37	36	abscld 14175	. . . . . . 7 |- ( ph -> ( abs ` ( P - M ) ) e. RR )
38	37	recnd 10068	. . . . . 6 |- ( ph -> ( abs ` ( P - M ) ) e. CC )
39	38	sqcld 13006	. . . . 5 |- ( ph -> ( ( abs ` ( P - M ) ) ^ 2 ) e. CC )
40	39	adantr 481	. . . 4 |- ( ( ph /\ Q = M ) -> ( ( abs ` ( P - M ) ) ^ 2 ) e. CC )
41	40	addid2d 10237	. . 3 |- ( ( ph /\ Q = M ) -> ( 0 + ( ( abs ` ( P - M ) ) ^ 2 ) ) = ( ( abs ` ( P - M ) ) ^ 2 ) )
42	1	adantr 481	. . . . . . 7 |- ( ( ph /\ Q = M ) -> Q e. CC )
43		simpr 477	. . . . . . 7 |- ( ( ph /\ Q = M ) -> Q = M )
44	42, 43	subeq0bd 10456	. . . . . 6 |- ( ( ph /\ Q = M ) -> ( Q - M ) = 0 )
45	44	abs00bd 14031	. . . . 5 |- ( ( ph /\ Q = M ) -> ( abs ` ( Q - M ) ) = 0 )
46	45	sq0id 12957	. . . 4 |- ( ( ph /\ Q = M ) -> ( ( abs ` ( Q - M ) ) ^ 2 ) = 0 )
47	46	oveq1d 6665	. . 3 |- ( ( ph /\ Q = M ) -> ( ( ( abs ` ( Q - M ) ) ^ 2 ) + ( ( abs ` ( P - M ) ) ^ 2 ) ) = ( 0 + ( ( abs ` ( P - M ) ) ^ 2 ) ) )
48	43	oveq2d 6666	. . . . 5 |- ( ( ph /\ Q = M ) -> ( P - Q ) = ( P - M ) )
49	48	fveq2d 6195	. . . 4 |- ( ( ph /\ Q = M ) -> ( abs ` ( P - Q ) ) = ( abs ` ( P - M ) ) )
50	49	oveq1d 6665	. . 3 |- ( ( ph /\ Q = M ) -> ( ( abs ` ( P - Q ) ) ^ 2 ) = ( ( abs ` ( P - M ) ) ^ 2 ) )
51	41, 47, 50	3eqtr4rd 2667	. 2 |- ( ( ph /\ Q = M ) -> ( ( abs ` ( P - Q ) ) ^ 2 ) = ( ( ( abs ` ( Q - M ) ) ^ 2 ) + ( ( abs ` ( P - M ) ) ^ 2 ) ) )
52	22	adantr 481	. . 3 |- ( ( ph /\ ( P =/= M /\ Q =/= M ) ) -> P e. CC )
53	1	adantr 481	. . 3 |- ( ( ph /\ ( P =/= M /\ Q =/= M ) ) -> Q e. CC )
54	7	adantr 481	. . 3 |- ( ( ph /\ ( P =/= M /\ Q =/= M ) ) -> M e. CC )
55		simprl 794	. . 3 |- ( ( ph /\ ( P =/= M /\ Q =/= M ) ) -> P =/= M )
56		simprr 796	. . 3 |- ( ( ph /\ ( P =/= M /\ Q =/= M ) ) -> Q =/= M )
57		eqid 2622	. . . 4 |- ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) ) = ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) )
58	3	adantr 481	. . . 4 |- ( ( ph /\ ( P =/= M /\ Q =/= M ) ) -> A e. CC )
59	4	adantr 481	. . . 4 |- ( ( ph /\ ( P =/= M /\ Q =/= M ) ) -> B e. CC )
60	15	adantr 481	. . . 4 |- ( ( ph /\ ( P =/= M /\ Q =/= M ) ) -> X e. RR )
61	2	adantr 481	. . . 4 |- ( ( ph /\ ( P =/= M /\ Q =/= M ) ) -> M = ( ( A + B ) / 2 ) )
62	14	adantr 481	. . . 4 |- ( ( ph /\ ( P =/= M /\ Q =/= M ) ) -> P = ( ( X x. A ) + ( ( 1 - X ) x. B ) ) )
63		chordthmlem3.ABequidistQ	. . . . 5 |- ( ph -> ( abs ` ( A - Q ) ) = ( abs ` ( B - Q ) ) )
64	63	adantr 481	. . . 4 |- ( ( ph /\ ( P =/= M /\ Q =/= M ) ) -> ( abs ` ( A - Q ) ) = ( abs ` ( B - Q ) ) )
65	57, 58, 59, 53, 60, 61, 62, 64, 55, 56	chordthmlem2 24560	. . 3 |- ( ( ph /\ ( P =/= M /\ Q =/= M ) ) -> ( ( Q - M ) ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) ) ( P - M ) ) e. { ( pi / 2 ) , -u ( pi / 2 ) } )
66		eqid 2622	. . . 4 |- ( abs ` ( Q - M ) ) = ( abs ` ( Q - M ) )
67		eqid 2622	. . . 4 |- ( abs ` ( P - M ) ) = ( abs ` ( P - M ) )
68		eqid 2622	. . . 4 |- ( abs ` ( P - Q ) ) = ( abs ` ( P - Q ) )
69		eqid 2622	. . . 4 |- ( ( Q - M ) ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) ) ( P - M ) ) = ( ( Q - M ) ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) ) ( P - M ) )
70	57, 66, 67, 68, 69	pythag 24547	. . 3 |- ( ( ( P e. CC /\ Q e. CC /\ M e. CC ) /\ ( P =/= M /\ Q =/= M ) /\ ( ( Q - M ) ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) ) ( P - M ) ) e. { ( pi / 2 ) , -u ( pi / 2 ) } ) -> ( ( abs ` ( P - Q ) ) ^ 2 ) = ( ( ( abs ` ( Q - M ) ) ^ 2 ) + ( ( abs ` ( P - M ) ) ^ 2 ) ) )
71	52, 53, 54, 55, 56, 65, 70	syl321anc 1348	. 2 |- ( ( ph /\ ( P =/= M /\ Q =/= M ) ) -> ( ( abs ` ( P - Q ) ) ^ 2 ) = ( ( ( abs ` ( Q - M ) ) ^ 2 ) + ( ( abs ` ( P - M ) ) ^ 2 ) ) )
72	35, 51, 71	pm2.61da2ne 2882	1 |- ( ph -> ( ( abs ` ( P - Q ) ) ^ 2 ) = ( ( ( abs ` ( Q - M ) ) ^ 2 ) + ( ( abs ` ( P - M ) ) ^ 2 ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   {csn 4177   {cpr 4179   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   / cdiv 10684   2c2 11070   ^cexp 12860   Imcim 13838   abscabs 13974   picpi 14797   logclog 24301

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  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  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  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-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303

This theorem is referenced by:  chordthmlem5  24563