chordthmlem5 - Metamath Proof Explorer

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Theorem chordthmlem5 26781

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 26779 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 26780. (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 11258	. . . . . . . 8 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ)
5	4	halfcld 12482	. . . . . . 7 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℂ)
6	1, 5	subcld 11596	. . . . . 6 ⊢ (𝜑 → (𝑄 − ((𝐴 + 𝐵) / 2)) ∈ ℂ)
7	6	abscld 15410	. . . . 5 ⊢ (𝜑 → (abs‘(𝑄 − ((𝐴 + 𝐵) / 2))) ∈ ℝ)
8	7	recnd 11267	. . . 4 ⊢ (𝜑 → (abs‘(𝑄 − ((𝐴 + 𝐵) / 2))) ∈ ℂ)
9	8	sqcld 14135	. . 3 ⊢ (𝜑 → ((abs‘(𝑄 − ((𝐴 + 𝐵) / 2)))↑2) ∈ ℂ)
10	3, 5	subcld 11596	. . . . . 6 ⊢ (𝜑 → (𝐵 − ((𝐴 + 𝐵) / 2)) ∈ ℂ)
11	10	abscld 15410	. . . . 5 ⊢ (𝜑 → (abs‘(𝐵 − ((𝐴 + 𝐵) / 2))) ∈ ℝ)
12	11	recnd 11267	. . . 4 ⊢ (𝜑 → (abs‘(𝐵 − ((𝐴 + 𝐵) / 2))) ∈ ℂ)
13	12	sqcld 14135	. . 3 ⊢ (𝜑 → ((abs‘(𝐵 − ((𝐴 + 𝐵) / 2)))↑2) ∈ ℂ)
14		chordthmlem5.P	. . . . . . . 8 ⊢ (𝜑 → 𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)))
15		unitssre 13503	. . . . . . . . . . . 12 ⊢ (0[,]1) ⊆ ℝ
16		chordthmlem5.X	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ (0[,]1))
17	15, 16	sselid 3971	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ ℝ)
18	17	recnd 11267	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ℂ)
19	18, 2	mulcld 11259	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 · 𝐴) ∈ ℂ)
20		1cnd 11234	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℂ)
21	20, 18	subcld 11596	. . . . . . . . . 10 ⊢ (𝜑 → (1 − 𝑋) ∈ ℂ)
22	21, 3	mulcld 11259	. . . . . . . . 9 ⊢ (𝜑 → ((1 − 𝑋) · 𝐵) ∈ ℂ)
23	19, 22	addcld 11258	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ∈ ℂ)
24	14, 23	eqeltrd 2825	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℂ)
25	24, 5	subcld 11596	. . . . . 6 ⊢ (𝜑 → (𝑃 − ((𝐴 + 𝐵) / 2)) ∈ ℂ)
26	25	abscld 15410	. . . . 5 ⊢ (𝜑 → (abs‘(𝑃 − ((𝐴 + 𝐵) / 2))) ∈ ℝ)
27	26	recnd 11267	. . . 4 ⊢ (𝜑 → (abs‘(𝑃 − ((𝐴 + 𝐵) / 2))) ∈ ℂ)
28	27	sqcld 14135	. . 3 ⊢ (𝜑 → ((abs‘(𝑃 − ((𝐴 + 𝐵) / 2)))↑2) ∈ ℂ)
29	9, 13, 28	pnpcand 11633	. 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 11242	. . . 4 ⊢ (𝜑 → 0 ∈ ℝ)
31		eqidd 2726	. . . 4 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) = ((𝐴 + 𝐵) / 2))
32	2	mul02d 11437	. . . . . 6 ⊢ (𝜑 → (0 · 𝐴) = 0)
33	20	subid1d 11585	. . . . . . . 8 ⊢ (𝜑 → (1 − 0) = 1)
34	33	oveq1d 7428	. . . . . . 7 ⊢ (𝜑 → ((1 − 0) · 𝐵) = (1 · 𝐵))
35	3	mullidd 11257	. . . . . . 7 ⊢ (𝜑 → (1 · 𝐵) = 𝐵)
36	34, 35	eqtrd 2765	. . . . . 6 ⊢ (𝜑 → ((1 − 0) · 𝐵) = 𝐵)
37	32, 36	oveq12d 7431	. . . . 5 ⊢ (𝜑 → ((0 · 𝐴) + ((1 − 0) · 𝐵)) = (0 + 𝐵))
38	3	addlidd 11440	. . . . 5 ⊢ (𝜑 → (0 + 𝐵) = 𝐵)
39	37, 38	eqtr2d 2766	. . . 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 26779	. . 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 26779	. . 3 ⊢ (𝜑 → ((abs‘(𝑃 − 𝑄))↑2) = (((abs‘(𝑄 − ((𝐴 + 𝐵) / 2)))↑2) + ((abs‘(𝑃 − ((𝐴 + 𝐵) / 2)))↑2)))
43	41, 42	oveq12d 7431	. 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 26780	. 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 2776	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 1533 ∈ wcel 2098 ‘cfv 6543 (class class class)co 7413 ℂcc 11131 ℝcr 11132 0cc0 11133 1c1 11134 + caddc 11136 · cmul 11138 − cmin 11469 / cdiv 11896 2c2 12292 [,]cicc 13354 ↑cexp 14053 abscabs 15208

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-inf2 9659 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 ax-addf 11212

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-om 7866 df-1st 7987 df-2nd 7988 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9381 df-fi 9429 df-sup 9460 df-inf 9461 df-oi 9528 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-q 12958 df-rp 13002 df-xneg 13119 df-xadd 13120 df-xmul 13121 df-ioo 13355 df-ioc 13356 df-ico 13357 df-icc 13358 df-fz 13512 df-fzo 13655 df-fl 13784 df-mod 13862 df-seq 13994 df-exp 14054 df-fac 14260 df-bc 14289 df-hash 14317 df-shft 15041 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-limsup 15442 df-clim 15459 df-rlim 15460 df-sum 15660 df-ef 16038 df-sin 16040 df-cos 16041 df-pi 16043 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17398 df-topn 17399 df-0g 17417 df-gsum 17418 df-topgen 17419 df-pt 17420 df-prds 17423 df-xrs 17478 df-qtop 17483 df-imas 17484 df-xps 17486 df-mre 17560 df-mrc 17561 df-acs 17563 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-submnd 18735 df-mulg 19023 df-cntz 19267 df-cmn 19736 df-psmet 21270 df-xmet 21271 df-met 21272 df-bl 21273 df-mopn 21274 df-fbas 21275 df-fg 21276 df-cnfld 21279 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22862 df-cld 22936 df-ntr 22937 df-cls 22938 df-nei 23015 df-lp 23053 df-perf 23054 df-cn 23144 df-cnp 23145 df-haus 23232 df-tx 23479 df-hmeo 23672 df-fil 23763 df-fm 23855 df-flim 23856 df-flf 23857 df-xms 24239 df-ms 24240 df-tms 24241 df-cncf 24811 df-limc 25808 df-dv 25809 df-log 26503

This theorem is referenced by: chordthm 26782 chordthmALT 44433

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