chordthmlem5 - Metamath Proof Explorer

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Theorem chordthmlem5 26773

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 26771 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 26772. (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 11131	. . . . . . . 8 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ)
5	4	halfcld 12366	. . . . . . 7 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℂ)
6	1, 5	subcld 11472	. . . . . 6 ⊢ (𝜑 → (𝑄 − ((𝐴 + 𝐵) / 2)) ∈ ℂ)
7	6	abscld 15346	. . . . 5 ⊢ (𝜑 → (abs‘(𝑄 − ((𝐴 + 𝐵) / 2))) ∈ ℝ)
8	7	recnd 11140	. . . 4 ⊢ (𝜑 → (abs‘(𝑄 − ((𝐴 + 𝐵) / 2))) ∈ ℂ)
9	8	sqcld 14051	. . 3 ⊢ (𝜑 → ((abs‘(𝑄 − ((𝐴 + 𝐵) / 2)))↑2) ∈ ℂ)
10	3, 5	subcld 11472	. . . . . 6 ⊢ (𝜑 → (𝐵 − ((𝐴 + 𝐵) / 2)) ∈ ℂ)
11	10	abscld 15346	. . . . 5 ⊢ (𝜑 → (abs‘(𝐵 − ((𝐴 + 𝐵) / 2))) ∈ ℝ)
12	11	recnd 11140	. . . 4 ⊢ (𝜑 → (abs‘(𝐵 − ((𝐴 + 𝐵) / 2))) ∈ ℂ)
13	12	sqcld 14051	. . 3 ⊢ (𝜑 → ((abs‘(𝐵 − ((𝐴 + 𝐵) / 2)))↑2) ∈ ℂ)
14		chordthmlem5.P	. . . . . . . 8 ⊢ (𝜑 → 𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)))
15		unitssre 13399	. . . . . . . . . . . 12 ⊢ (0[,]1) ⊆ ℝ
16		chordthmlem5.X	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ (0[,]1))
17	15, 16	sselid 3927	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ ℝ)
18	17	recnd 11140	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ℂ)
19	18, 2	mulcld 11132	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 · 𝐴) ∈ ℂ)
20		1cnd 11107	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℂ)
21	20, 18	subcld 11472	. . . . . . . . . 10 ⊢ (𝜑 → (1 − 𝑋) ∈ ℂ)
22	21, 3	mulcld 11132	. . . . . . . . 9 ⊢ (𝜑 → ((1 − 𝑋) · 𝐵) ∈ ℂ)
23	19, 22	addcld 11131	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ∈ ℂ)
24	14, 23	eqeltrd 2831	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℂ)
25	24, 5	subcld 11472	. . . . . 6 ⊢ (𝜑 → (𝑃 − ((𝐴 + 𝐵) / 2)) ∈ ℂ)
26	25	abscld 15346	. . . . 5 ⊢ (𝜑 → (abs‘(𝑃 − ((𝐴 + 𝐵) / 2))) ∈ ℝ)
27	26	recnd 11140	. . . 4 ⊢ (𝜑 → (abs‘(𝑃 − ((𝐴 + 𝐵) / 2))) ∈ ℂ)
28	27	sqcld 14051	. . 3 ⊢ (𝜑 → ((abs‘(𝑃 − ((𝐴 + 𝐵) / 2)))↑2) ∈ ℂ)
29	9, 13, 28	pnpcand 11509	. 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 11115	. . . 4 ⊢ (𝜑 → 0 ∈ ℝ)
31		eqidd 2732	. . . 4 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) = ((𝐴 + 𝐵) / 2))
32	2	mul02d 11311	. . . . . 6 ⊢ (𝜑 → (0 · 𝐴) = 0)
33	20	subid1d 11461	. . . . . . . 8 ⊢ (𝜑 → (1 − 0) = 1)
34	33	oveq1d 7361	. . . . . . 7 ⊢ (𝜑 → ((1 − 0) · 𝐵) = (1 · 𝐵))
35	3	mullidd 11130	. . . . . . 7 ⊢ (𝜑 → (1 · 𝐵) = 𝐵)
36	34, 35	eqtrd 2766	. . . . . 6 ⊢ (𝜑 → ((1 − 0) · 𝐵) = 𝐵)
37	32, 36	oveq12d 7364	. . . . 5 ⊢ (𝜑 → ((0 · 𝐴) + ((1 − 0) · 𝐵)) = (0 + 𝐵))
38	3	addlidd 11314	. . . . 5 ⊢ (𝜑 → (0 + 𝐵) = 𝐵)
39	37, 38	eqtr2d 2767	. . . 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 26771	. . 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 26771	. . 3 ⊢ (𝜑 → ((abs‘(𝑃 − 𝑄))↑2) = (((abs‘(𝑄 − ((𝐴 + 𝐵) / 2)))↑2) + ((abs‘(𝑃 − ((𝐴 + 𝐵) / 2)))↑2)))
43	41, 42	oveq12d 7364	. 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 26772	. 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 2777	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 1541 ∈ wcel 2111 ‘cfv 6481 (class class class)co 7346 ℂcc 11004 ℝcr 11005 0cc0 11006 1c1 11007 + caddc 11009 · cmul 11011 − cmin 11344 / cdiv 11774 2c2 12180 [,]cicc 13248 ↑cexp 13968 abscabs 15141

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-inf2 9531 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-ioo 13249 df-ioc 13250 df-ico 13251 df-icc 13252 df-fz 13408 df-fzo 13555 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14974 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396 df-sum 15594 df-ef 15974 df-sin 15976 df-cos 15977 df-pi 15979 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-mulg 18981 df-cntz 19229 df-cmn 19694 df-psmet 21283 df-xmet 21284 df-met 21285 df-bl 21286 df-mopn 21287 df-fbas 21288 df-fg 21289 df-cnfld 21292 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22861 df-cld 22934 df-ntr 22935 df-cls 22936 df-nei 23013 df-lp 23051 df-perf 23052 df-cn 23142 df-cnp 23143 df-haus 23230 df-tx 23477 df-hmeo 23670 df-fil 23761 df-fm 23853 df-flim 23854 df-flf 23855 df-xms 24235 df-ms 24236 df-tms 24237 df-cncf 24798 df-limc 25794 df-dv 25795 df-log 26492

This theorem is referenced by: chordthm 26774 chordthmALT 45035

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