chordthmlem5 - Metamath Proof Explorer

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Theorem chordthmlem5 26742

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 26740 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 26741. (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 11249	. . . . . . . 8 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ)
5	4	halfcld 12473	. . . . . . 7 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℂ)
6	1, 5	subcld 11587	. . . . . 6 ⊢ (𝜑 → (𝑄 − ((𝐴 + 𝐵) / 2)) ∈ ℂ)
7	6	abscld 15401	. . . . 5 ⊢ (𝜑 → (abs‘(𝑄 − ((𝐴 + 𝐵) / 2))) ∈ ℝ)
8	7	recnd 11258	. . . 4 ⊢ (𝜑 → (abs‘(𝑄 − ((𝐴 + 𝐵) / 2))) ∈ ℂ)
9	8	sqcld 14126	. . 3 ⊢ (𝜑 → ((abs‘(𝑄 − ((𝐴 + 𝐵) / 2)))↑2) ∈ ℂ)
10	3, 5	subcld 11587	. . . . . 6 ⊢ (𝜑 → (𝐵 − ((𝐴 + 𝐵) / 2)) ∈ ℂ)
11	10	abscld 15401	. . . . 5 ⊢ (𝜑 → (abs‘(𝐵 − ((𝐴 + 𝐵) / 2))) ∈ ℝ)
12	11	recnd 11258	. . . 4 ⊢ (𝜑 → (abs‘(𝐵 − ((𝐴 + 𝐵) / 2))) ∈ ℂ)
13	12	sqcld 14126	. . 3 ⊢ (𝜑 → ((abs‘(𝐵 − ((𝐴 + 𝐵) / 2)))↑2) ∈ ℂ)
14		chordthmlem5.P	. . . . . . . 8 ⊢ (𝜑 → 𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)))
15		unitssre 13494	. . . . . . . . . . . 12 ⊢ (0[,]1) ⊆ ℝ
16		chordthmlem5.X	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ (0[,]1))
17	15, 16	sselid 3976	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ ℝ)
18	17	recnd 11258	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ℂ)
19	18, 2	mulcld 11250	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 · 𝐴) ∈ ℂ)
20		1cnd 11225	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℂ)
21	20, 18	subcld 11587	. . . . . . . . . 10 ⊢ (𝜑 → (1 − 𝑋) ∈ ℂ)
22	21, 3	mulcld 11250	. . . . . . . . 9 ⊢ (𝜑 → ((1 − 𝑋) · 𝐵) ∈ ℂ)
23	19, 22	addcld 11249	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ∈ ℂ)
24	14, 23	eqeltrd 2828	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℂ)
25	24, 5	subcld 11587	. . . . . 6 ⊢ (𝜑 → (𝑃 − ((𝐴 + 𝐵) / 2)) ∈ ℂ)
26	25	abscld 15401	. . . . 5 ⊢ (𝜑 → (abs‘(𝑃 − ((𝐴 + 𝐵) / 2))) ∈ ℝ)
27	26	recnd 11258	. . . 4 ⊢ (𝜑 → (abs‘(𝑃 − ((𝐴 + 𝐵) / 2))) ∈ ℂ)
28	27	sqcld 14126	. . 3 ⊢ (𝜑 → ((abs‘(𝑃 − ((𝐴 + 𝐵) / 2)))↑2) ∈ ℂ)
29	9, 13, 28	pnpcand 11624	. 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 11233	. . . 4 ⊢ (𝜑 → 0 ∈ ℝ)
31		eqidd 2728	. . . 4 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) = ((𝐴 + 𝐵) / 2))
32	2	mul02d 11428	. . . . . 6 ⊢ (𝜑 → (0 · 𝐴) = 0)
33	20	subid1d 11576	. . . . . . . 8 ⊢ (𝜑 → (1 − 0) = 1)
34	33	oveq1d 7429	. . . . . . 7 ⊢ (𝜑 → ((1 − 0) · 𝐵) = (1 · 𝐵))
35	3	mullidd 11248	. . . . . . 7 ⊢ (𝜑 → (1 · 𝐵) = 𝐵)
36	34, 35	eqtrd 2767	. . . . . 6 ⊢ (𝜑 → ((1 − 0) · 𝐵) = 𝐵)
37	32, 36	oveq12d 7432	. . . . 5 ⊢ (𝜑 → ((0 · 𝐴) + ((1 − 0) · 𝐵)) = (0 + 𝐵))
38	3	addlidd 11431	. . . . 5 ⊢ (𝜑 → (0 + 𝐵) = 𝐵)
39	37, 38	eqtr2d 2768	. . . 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 26740	. . 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 26740	. . 3 ⊢ (𝜑 → ((abs‘(𝑃 − 𝑄))↑2) = (((abs‘(𝑄 − ((𝐴 + 𝐵) / 2)))↑2) + ((abs‘(𝑃 − ((𝐴 + 𝐵) / 2)))↑2)))
43	41, 42	oveq12d 7432	. 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 26741	. 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 2778	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 1534 ∈ wcel 2099 ‘cfv 6542 (class class class)co 7414 ℂcc 11122 ℝcr 11123 0cc0 11124 1c1 11125 + caddc 11127 · cmul 11129 − cmin 11460 / cdiv 11887 2c2 12283 [,]cicc 13345 ↑cexp 14044 abscabs 15199

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-inf2 9650 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 ax-addf 11203

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7677 df-om 7863 df-1st 7985 df-2nd 7986 df-supp 8158 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-er 8716 df-map 8836 df-pm 8837 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9376 df-fi 9420 df-sup 9451 df-inf 9452 df-oi 9519 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-z 12575 df-dec 12694 df-uz 12839 df-q 12949 df-rp 12993 df-xneg 13110 df-xadd 13111 df-xmul 13112 df-ioo 13346 df-ioc 13347 df-ico 13348 df-icc 13349 df-fz 13503 df-fzo 13646 df-fl 13775 df-mod 13853 df-seq 13985 df-exp 14045 df-fac 14251 df-bc 14280 df-hash 14308 df-shft 15032 df-cj 15064 df-re 15065 df-im 15066 df-sqrt 15200 df-abs 15201 df-limsup 15433 df-clim 15450 df-rlim 15451 df-sum 15651 df-ef 16029 df-sin 16031 df-cos 16032 df-pi 16034 df-struct 17101 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ress 17195 df-plusg 17231 df-mulr 17232 df-starv 17233 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-unif 17241 df-hom 17242 df-cco 17243 df-rest 17389 df-topn 17390 df-0g 17408 df-gsum 17409 df-topgen 17410 df-pt 17411 df-prds 17414 df-xrs 17469 df-qtop 17474 df-imas 17475 df-xps 17477 df-mre 17551 df-mrc 17552 df-acs 17554 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-submnd 18726 df-mulg 19008 df-cntz 19252 df-cmn 19721 df-psmet 21251 df-xmet 21252 df-met 21253 df-bl 21254 df-mopn 21255 df-fbas 21256 df-fg 21257 df-cnfld 21260 df-top 22770 df-topon 22787 df-topsp 22809 df-bases 22823 df-cld 22897 df-ntr 22898 df-cls 22899 df-nei 22976 df-lp 23014 df-perf 23015 df-cn 23105 df-cnp 23106 df-haus 23193 df-tx 23440 df-hmeo 23633 df-fil 23724 df-fm 23816 df-flim 23817 df-flf 23818 df-xms 24200 df-ms 24201 df-tms 24202 df-cncf 24772 df-limc 25769 df-dv 25770 df-log 26464

This theorem is referenced by: chordthm 26743 chordthmALT 44285

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