chordthmlem5 - Metamath Proof Explorer

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Theorem chordthmlem5 24563

Description: If P is on the segment AB and AQ = BQ, then PA · PB = BQ ² − PQ ² . This follows from two uses of chordthmlem3 24561 to show that PQ ² = QM ² + PM ² and BQ ² = QM ² + BM ² , so BQ ² − PQ ² = (QM ² + BM ² ) − (QM ² + PM ² ) = BM ² − PM ² , which equals PA · PB by chordthmlem4 24562. (Contributed by David Moews, 28-Feb-2017.)

**Hypotheses**
Ref	Expression
chordthmlem5.A	⊢ (𝜑 → 𝐴 ∈ ℂ)
chordthmlem5.B	⊢ (𝜑 → 𝐵 ∈ ℂ)
chordthmlem5.Q	⊢ (𝜑 → 𝑄 ∈ ℂ)
chordthmlem5.X	⊢ (𝜑 → 𝑋 ∈ (0[,]1))
chordthmlem5.P	⊢ (𝜑 → 𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)))
chordthmlem5.ABequidistQ	⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) = (abs‘(𝐵 − 𝑄)))

**Assertion**
Ref	Expression
chordthmlem5	⊢ (𝜑 → ((abs‘(𝑃 − 𝐴)) · (abs‘(𝑃 − 𝐵))) = (((abs‘(𝐵 − 𝑄))↑2) − ((abs‘(𝑃 − 𝑄))↑2)))

**Proof of Theorem chordthmlem5**
Step	Hyp	Ref	Expression
1		chordthmlem5.Q	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℂ)
2		chordthmlem5.A	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
3		chordthmlem5.B	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℂ)
4	2, 3	addcld 10059	. . . . . . . 8 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ)
5	4	halfcld 11277	. . . . . . 7 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℂ)
6	1, 5	subcld 10392	. . . . . 6 ⊢ (𝜑 → (𝑄 − ((𝐴 + 𝐵) / 2)) ∈ ℂ)
7	6	abscld 14175	. . . . 5 ⊢ (𝜑 → (abs‘(𝑄 − ((𝐴 + 𝐵) / 2))) ∈ ℝ)
8	7	recnd 10068	. . . 4 ⊢ (𝜑 → (abs‘(𝑄 − ((𝐴 + 𝐵) / 2))) ∈ ℂ)
9	8	sqcld 13006	. . 3 ⊢ (𝜑 → ((abs‘(𝑄 − ((𝐴 + 𝐵) / 2)))↑2) ∈ ℂ)
10	3, 5	subcld 10392	. . . . . 6 ⊢ (𝜑 → (𝐵 − ((𝐴 + 𝐵) / 2)) ∈ ℂ)
11	10	abscld 14175	. . . . 5 ⊢ (𝜑 → (abs‘(𝐵 − ((𝐴 + 𝐵) / 2))) ∈ ℝ)
12	11	recnd 10068	. . . 4 ⊢ (𝜑 → (abs‘(𝐵 − ((𝐴 + 𝐵) / 2))) ∈ ℂ)
13	12	sqcld 13006	. . 3 ⊢ (𝜑 → ((abs‘(𝐵 − ((𝐴 + 𝐵) / 2)))↑2) ∈ ℂ)
14		chordthmlem5.P	. . . . . . . 8 ⊢ (𝜑 → 𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)))
15		unitssre 12319	. . . . . . . . . . . 12 ⊢ (0[,]1) ⊆ ℝ
16		chordthmlem5.X	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ (0[,]1))
17	15, 16	sseldi 3601	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ ℝ)
18	17	recnd 10068	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ℂ)
19	18, 2	mulcld 10060	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 · 𝐴) ∈ ℂ)
20		1cnd 10056	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℂ)
21	20, 18	subcld 10392	. . . . . . . . . 10 ⊢ (𝜑 → (1 − 𝑋) ∈ ℂ)
22	21, 3	mulcld 10060	. . . . . . . . 9 ⊢ (𝜑 → ((1 − 𝑋) · 𝐵) ∈ ℂ)
23	19, 22	addcld 10059	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ∈ ℂ)
24	14, 23	eqeltrd 2701	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℂ)
25	24, 5	subcld 10392	. . . . . 6 ⊢ (𝜑 → (𝑃 − ((𝐴 + 𝐵) / 2)) ∈ ℂ)
26	25	abscld 14175	. . . . 5 ⊢ (𝜑 → (abs‘(𝑃 − ((𝐴 + 𝐵) / 2))) ∈ ℝ)
27	26	recnd 10068	. . . 4 ⊢ (𝜑 → (abs‘(𝑃 − ((𝐴 + 𝐵) / 2))) ∈ ℂ)
28	27	sqcld 13006	. . 3 ⊢ (𝜑 → ((abs‘(𝑃 − ((𝐴 + 𝐵) / 2)))↑2) ∈ ℂ)
29	9, 13, 28	pnpcand 10429	. 2 ⊢ (𝜑 → ((((abs‘(𝑄 − ((𝐴 + 𝐵) / 2)))↑2) + ((abs‘(𝐵 − ((𝐴 + 𝐵) / 2)))↑2)) − (((abs‘(𝑄 − ((𝐴 + 𝐵) / 2)))↑2) + ((abs‘(𝑃 − ((𝐴 + 𝐵) / 2)))↑2))) = (((abs‘(𝐵 − ((𝐴 + 𝐵) / 2)))↑2) − ((abs‘(𝑃 − ((𝐴 + 𝐵) / 2)))↑2)))
30		0red 10041	. . . 4 ⊢ (𝜑 → 0 ∈ ℝ)
31		eqidd 2623	. . . 4 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) = ((𝐴 + 𝐵) / 2))
32	2	mul02d 10234	. . . . . 6 ⊢ (𝜑 → (0 · 𝐴) = 0)
33	20	subid1d 10381	. . . . . . . 8 ⊢ (𝜑 → (1 − 0) = 1)
34	33	oveq1d 6665	. . . . . . 7 ⊢ (𝜑 → ((1 − 0) · 𝐵) = (1 · 𝐵))
35	3	mulid2d 10058	. . . . . . 7 ⊢ (𝜑 → (1 · 𝐵) = 𝐵)
36	34, 35	eqtrd 2656	. . . . . 6 ⊢ (𝜑 → ((1 − 0) · 𝐵) = 𝐵)
37	32, 36	oveq12d 6668	. . . . 5 ⊢ (𝜑 → ((0 · 𝐴) + ((1 − 0) · 𝐵)) = (0 + 𝐵))
38	3	addid2d 10237	. . . . 5 ⊢ (𝜑 → (0 + 𝐵) = 𝐵)
39	37, 38	eqtr2d 2657	. . . 4 ⊢ (𝜑 → 𝐵 = ((0 · 𝐴) + ((1 − 0) · 𝐵)))
40		chordthmlem5.ABequidistQ	. . . 4 ⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) = (abs‘(𝐵 − 𝑄)))
41	2, 3, 1, 30, 31, 39, 40	chordthmlem3 24561	. . 3 ⊢ (𝜑 → ((abs‘(𝐵 − 𝑄))↑2) = (((abs‘(𝑄 − ((𝐴 + 𝐵) / 2)))↑2) + ((abs‘(𝐵 − ((𝐴 + 𝐵) / 2)))↑2)))
42	2, 3, 1, 17, 31, 14, 40	chordthmlem3 24561	. . 3 ⊢ (𝜑 → ((abs‘(𝑃 − 𝑄))↑2) = (((abs‘(𝑄 − ((𝐴 + 𝐵) / 2)))↑2) + ((abs‘(𝑃 − ((𝐴 + 𝐵) / 2)))↑2)))
43	41, 42	oveq12d 6668	. 2 ⊢ (𝜑 → (((abs‘(𝐵 − 𝑄))↑2) − ((abs‘(𝑃 − 𝑄))↑2)) = ((((abs‘(𝑄 − ((𝐴 + 𝐵) / 2)))↑2) + ((abs‘(𝐵 − ((𝐴 + 𝐵) / 2)))↑2)) − (((abs‘(𝑄 − ((𝐴 + 𝐵) / 2)))↑2) + ((abs‘(𝑃 − ((𝐴 + 𝐵) / 2)))↑2))))
44	2, 3, 16, 31, 14	chordthmlem4 24562	. 2 ⊢ (𝜑 → ((abs‘(𝑃 − 𝐴)) · (abs‘(𝑃 − 𝐵))) = (((abs‘(𝐵 − ((𝐴 + 𝐵) / 2)))↑2) − ((abs‘(𝑃 − ((𝐴 + 𝐵) / 2)))↑2)))
45	29, 43, 44	3eqtr4rd 2667	1 ⊢ (𝜑 → ((abs‘(𝑃 − 𝐴)) · (abs‘(𝑃 − 𝐵))) = (((abs‘(𝐵 − 𝑄))↑2) − ((abs‘(𝑃 − 𝑄))↑2)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 − cmin 10266 / cdiv 10684 2c2 11070 [,]cicc 12178 ↑cexp 12860 abscabs 13974

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: chordthm 24564 chordthmALT 39169

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