chordthmlem2 - Metamath Proof Explorer

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Theorem chordthmlem2 26891

Description: If M is the midpoint of AB, AQ = BQ, and P is on the line AB, then QMP is a right angle. This is proven by reduction to the special case chordthmlem 26890, where P = B, and using angrtmuld 26866 to observe that QMP is right iff QMB is. (Contributed by David Moews, 28-Feb-2017.)

**Hypotheses**
Ref	Expression
chordthmlem2.angdef	⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))
chordthmlem2.A	⊢ (𝜑 → 𝐴 ∈ ℂ)
chordthmlem2.B	⊢ (𝜑 → 𝐵 ∈ ℂ)
chordthmlem2.Q	⊢ (𝜑 → 𝑄 ∈ ℂ)
chordthmlem2.X	⊢ (𝜑 → 𝑋 ∈ ℝ)
chordthmlem2.M	⊢ (𝜑 → 𝑀 = ((𝐴 + 𝐵) / 2))
chordthmlem2.P	⊢ (𝜑 → 𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)))
chordthmlem2.ABequidistQ	⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) = (abs‘(𝐵 − 𝑄)))
chordthmlem2.PneM	⊢ (𝜑 → 𝑃 ≠ 𝑀)
chordthmlem2.QneM	⊢ (𝜑 → 𝑄 ≠ 𝑀)

**Assertion**
Ref	Expression
chordthmlem2	⊢ (𝜑 → ((𝑄 − 𝑀)𝐹(𝑃 − 𝑀)) ∈ {(π / 2), -(π / 2)})

Distinct variable groups: 𝑥,𝑦,𝑄 𝑥,𝑃,𝑦 𝑥,𝑀,𝑦 𝑥,𝐵,𝑦 𝑥,𝐴,𝑦 Allowed substitution hints: 𝜑(𝑥,𝑦) 𝐹(𝑥,𝑦) 𝑋(𝑥,𝑦)

**Proof of Theorem chordthmlem2**
Step	Hyp	Ref	Expression
1		chordthmlem2.angdef	. . 3 ⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))
2		chordthmlem2.A	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ)
3		chordthmlem2.B	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ)
4		chordthmlem2.Q	. . 3 ⊢ (𝜑 → 𝑄 ∈ ℂ)
5		chordthmlem2.M	. . 3 ⊢ (𝜑 → 𝑀 = ((𝐴 + 𝐵) / 2))
6		chordthmlem2.ABequidistQ	. . 3 ⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) = (abs‘(𝐵 − 𝑄)))
7		2re 12338	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
8	7	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℝ)
9		2ne0 12368	. . . . . . . . . 10 ⊢ 2 ≠ 0
10	9	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ≠ 0)
11	8, 10	rereccld 12092	. . . . . . . 8 ⊢ (𝜑 → (1 / 2) ∈ ℝ)
12		chordthmlem2.X	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℝ)
13	11, 12	resubcld 11689	. . . . . . 7 ⊢ (𝜑 → ((1 / 2) − 𝑋) ∈ ℝ)
14	13	recnd 11287	. . . . . 6 ⊢ (𝜑 → ((1 / 2) − 𝑋) ∈ ℂ)
15	3, 2	subcld 11618	. . . . . 6 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℂ)
16	11	recnd 11287	. . . . . . . . 9 ⊢ (𝜑 → (1 / 2) ∈ ℂ)
17	12	recnd 11287	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℂ)
18	16, 17, 15	subdird 11718	. . . . . . . 8 ⊢ (𝜑 → (((1 / 2) − 𝑋) · (𝐵 − 𝐴)) = (((1 / 2) · (𝐵 − 𝐴)) − (𝑋 · (𝐵 − 𝐴))))
19		2cnd 12342	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ∈ ℂ)
20	3, 19, 10	divcan4d 12047	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐵 · 2) / 2) = 𝐵)
21	3	times2d 12508	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 · 2) = (𝐵 + 𝐵))
22	21	oveq1d 7446	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐵 · 2) / 2) = ((𝐵 + 𝐵) / 2))
23	20, 22	eqtr3d 2777	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 = ((𝐵 + 𝐵) / 2))
24	23, 5	oveq12d 7449	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 − 𝑀) = (((𝐵 + 𝐵) / 2) − ((𝐴 + 𝐵) / 2)))
25	3, 3	addcld 11278	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 + 𝐵) ∈ ℂ)
26	2, 3	addcld 11278	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ)
27	25, 26, 19, 10	divsubdird 12080	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐵 + 𝐵) − (𝐴 + 𝐵)) / 2) = (((𝐵 + 𝐵) / 2) − ((𝐴 + 𝐵) / 2)))
28	3, 2, 3	pnpcan2d 11656	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐵 + 𝐵) − (𝐴 + 𝐵)) = (𝐵 − 𝐴))
29	28	oveq1d 7446	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐵 + 𝐵) − (𝐴 + 𝐵)) / 2) = ((𝐵 − 𝐴) / 2))
30	24, 27, 29	3eqtr2d 2781	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 − 𝑀) = ((𝐵 − 𝐴) / 2))
31	15, 19, 10	divrec2d 12045	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐵 − 𝐴) / 2) = ((1 / 2) · (𝐵 − 𝐴)))
32	30, 31	eqtrd 2775	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 − 𝑀) = ((1 / 2) · (𝐵 − 𝐴)))
33		chordthmlem2.P	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)))
34	17, 2	mulcld 11279	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 · 𝐴) ∈ ℂ)
35		1cnd 11254	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℂ)
36	35, 17	subcld 11618	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1 − 𝑋) ∈ ℂ)
37	36, 3	mulcld 11279	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((1 − 𝑋) · 𝐵) ∈ ℂ)
38	34, 37	addcld 11278	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ∈ ℂ)
39	33, 38	eqeltrd 2839	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ ℂ)
40	2, 39, 3, 17	affineequiv 26881	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ↔ (𝐵 − 𝑃) = (𝑋 · (𝐵 − 𝐴))))
41	33, 40	mpbid 232	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 − 𝑃) = (𝑋 · (𝐵 − 𝐴)))
42	32, 41	oveq12d 7449	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 − 𝑀) − (𝐵 − 𝑃)) = (((1 / 2) · (𝐵 − 𝐴)) − (𝑋 · (𝐵 − 𝐴))))
43	26	halfcld 12509	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℂ)
44	5, 43	eqeltrd 2839	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ)
45	3, 44, 39	nnncan1d 11652	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 − 𝑀) − (𝐵 − 𝑃)) = (𝑃 − 𝑀))
46	18, 42, 45	3eqtr2rd 2782	. . . . . . 7 ⊢ (𝜑 → (𝑃 − 𝑀) = (((1 / 2) − 𝑋) · (𝐵 − 𝐴)))
47		chordthmlem2.PneM	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ≠ 𝑀)
48	39, 44, 47	subne0d 11627	. . . . . . 7 ⊢ (𝜑 → (𝑃 − 𝑀) ≠ 0)
49	46, 48	eqnetrrd 3007	. . . . . 6 ⊢ (𝜑 → (((1 / 2) − 𝑋) · (𝐵 − 𝐴)) ≠ 0)
50	14, 15, 49	mulne0bbd 11917	. . . . 5 ⊢ (𝜑 → (𝐵 − 𝐴) ≠ 0)
51	3, 2, 50	subne0ad 11629	. . . 4 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
52	51	necomd 2994	. . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
53		chordthmlem2.QneM	. . 3 ⊢ (𝜑 → 𝑄 ≠ 𝑀)
54	1, 2, 3, 4, 5, 6, 52, 53	chordthmlem 26890	. 2 ⊢ (𝜑 → ((𝑄 − 𝑀)𝐹(𝐵 − 𝑀)) ∈ {(π / 2), -(π / 2)})
55	4, 44	subcld 11618	. . 3 ⊢ (𝜑 → (𝑄 − 𝑀) ∈ ℂ)
56	39, 44	subcld 11618	. . 3 ⊢ (𝜑 → (𝑃 − 𝑀) ∈ ℂ)
57	3, 44	subcld 11618	. . 3 ⊢ (𝜑 → (𝐵 − 𝑀) ∈ ℂ)
58	4, 44, 53	subne0d 11627	. . 3 ⊢ (𝜑 → (𝑄 − 𝑀) ≠ 0)
59	19, 10	recne0d 12035	. . . . 5 ⊢ (𝜑 → (1 / 2) ≠ 0)
60	16, 15, 59, 50	mulne0d 11913	. . . 4 ⊢ (𝜑 → ((1 / 2) · (𝐵 − 𝐴)) ≠ 0)
61	32, 60	eqnetrd 3006	. . 3 ⊢ (𝜑 → (𝐵 − 𝑀) ≠ 0)
62	32, 46	oveq12d 7449	. . . . 5 ⊢ (𝜑 → ((𝐵 − 𝑀) / (𝑃 − 𝑀)) = (((1 / 2) · (𝐵 − 𝐴)) / (((1 / 2) − 𝑋) · (𝐵 − 𝐴))))
63	14, 15, 49	mulne0bad 11916	. . . . . 6 ⊢ (𝜑 → ((1 / 2) − 𝑋) ≠ 0)
64	16, 14, 15, 63, 50	divcan5rd 12068	. . . . 5 ⊢ (𝜑 → (((1 / 2) · (𝐵 − 𝐴)) / (((1 / 2) − 𝑋) · (𝐵 − 𝐴))) = ((1 / 2) / ((1 / 2) − 𝑋)))
65	62, 64	eqtrd 2775	. . . 4 ⊢ (𝜑 → ((𝐵 − 𝑀) / (𝑃 − 𝑀)) = ((1 / 2) / ((1 / 2) − 𝑋)))
66	11, 13, 63	redivcld 12093	. . . 4 ⊢ (𝜑 → ((1 / 2) / ((1 / 2) − 𝑋)) ∈ ℝ)
67	65, 66	eqeltrd 2839	. . 3 ⊢ (𝜑 → ((𝐵 − 𝑀) / (𝑃 − 𝑀)) ∈ ℝ)
68	1, 55, 56, 57, 58, 48, 61, 67	angrtmuld 26866	. 2 ⊢ (𝜑 → (((𝑄 − 𝑀)𝐹(𝑃 − 𝑀)) ∈ {(π / 2), -(π / 2)} ↔ ((𝑄 − 𝑀)𝐹(𝐵 − 𝑀)) ∈ {(π / 2), -(π / 2)}))
69	54, 68	mpbird 257	1 ⊢ (𝜑 → ((𝑄 − 𝑀)𝐹(𝑃 − 𝑀)) ∈ {(π / 2), -(π / 2)})

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∖ cdif 3960 {csn 4631 {cpr 4633 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 − cmin 11490 -cneg 11491 / cdiv 11918 2c2 12319 ℑcim 15134 abscabs 15270 πcpi 16099 logclog 26611

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-ef 16100 df-sin 16102 df-cos 16103 df-pi 16105 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-limc 25916 df-dv 25917 df-log 26613

This theorem is referenced by: chordthmlem3 26892

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