chordthmlem2 - Metamath Proof Explorer

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Theorem chordthmlem2 26797

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 26796, where P = B, and using angrtmuld 26772 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 12217	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
8	7	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℝ)
9		2ne0 12247	. . . . . . . . . 10 ⊢ 2 ≠ 0
10	9	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ≠ 0)
11	8, 10	rereccld 11966	. . . . . . . 8 ⊢ (𝜑 → (1 / 2) ∈ ℝ)
12		chordthmlem2.X	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℝ)
13	11, 12	resubcld 11563	. . . . . . 7 ⊢ (𝜑 → ((1 / 2) − 𝑋) ∈ ℝ)
14	13	recnd 11158	. . . . . 6 ⊢ (𝜑 → ((1 / 2) − 𝑋) ∈ ℂ)
15	3, 2	subcld 11490	. . . . . 6 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℂ)
16	11	recnd 11158	. . . . . . . . 9 ⊢ (𝜑 → (1 / 2) ∈ ℂ)
17	12	recnd 11158	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℂ)
18	16, 17, 15	subdird 11592	. . . . . . . 8 ⊢ (𝜑 → (((1 / 2) − 𝑋) · (𝐵 − 𝐴)) = (((1 / 2) · (𝐵 − 𝐴)) − (𝑋 · (𝐵 − 𝐴))))
19		2cnd 12221	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ∈ ℂ)
20	3, 19, 10	divcan4d 11921	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐵 · 2) / 2) = 𝐵)
21	3	times2d 12383	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 · 2) = (𝐵 + 𝐵))
22	21	oveq1d 7371	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐵 · 2) / 2) = ((𝐵 + 𝐵) / 2))
23	20, 22	eqtr3d 2771	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 = ((𝐵 + 𝐵) / 2))
24	23, 5	oveq12d 7374	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 − 𝑀) = (((𝐵 + 𝐵) / 2) − ((𝐴 + 𝐵) / 2)))
25	3, 3	addcld 11149	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 + 𝐵) ∈ ℂ)
26	2, 3	addcld 11149	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ)
27	25, 26, 19, 10	divsubdird 11954	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐵 + 𝐵) − (𝐴 + 𝐵)) / 2) = (((𝐵 + 𝐵) / 2) − ((𝐴 + 𝐵) / 2)))
28	3, 2, 3	pnpcan2d 11528	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐵 + 𝐵) − (𝐴 + 𝐵)) = (𝐵 − 𝐴))
29	28	oveq1d 7371	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐵 + 𝐵) − (𝐴 + 𝐵)) / 2) = ((𝐵 − 𝐴) / 2))
30	24, 27, 29	3eqtr2d 2775	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 − 𝑀) = ((𝐵 − 𝐴) / 2))
31	15, 19, 10	divrec2d 11919	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐵 − 𝐴) / 2) = ((1 / 2) · (𝐵 − 𝐴)))
32	30, 31	eqtrd 2769	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 − 𝑀) = ((1 / 2) · (𝐵 − 𝐴)))
33		chordthmlem2.P	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)))
34	17, 2	mulcld 11150	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 · 𝐴) ∈ ℂ)
35		1cnd 11125	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℂ)
36	35, 17	subcld 11490	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1 − 𝑋) ∈ ℂ)
37	36, 3	mulcld 11150	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((1 − 𝑋) · 𝐵) ∈ ℂ)
38	34, 37	addcld 11149	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ∈ ℂ)
39	33, 38	eqeltrd 2834	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ ℂ)
40	2, 39, 3, 17	affineequiv 26787	. . . . . . . . . 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 7374	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 − 𝑀) − (𝐵 − 𝑃)) = (((1 / 2) · (𝐵 − 𝐴)) − (𝑋 · (𝐵 − 𝐴))))
43	26	halfcld 12384	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℂ)
44	5, 43	eqeltrd 2834	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ)
45	3, 44, 39	nnncan1d 11524	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 − 𝑀) − (𝐵 − 𝑃)) = (𝑃 − 𝑀))
46	18, 42, 45	3eqtr2rd 2776	. . . . . . 7 ⊢ (𝜑 → (𝑃 − 𝑀) = (((1 / 2) − 𝑋) · (𝐵 − 𝐴)))
47		chordthmlem2.PneM	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ≠ 𝑀)
48	39, 44, 47	subne0d 11499	. . . . . . 7 ⊢ (𝜑 → (𝑃 − 𝑀) ≠ 0)
49	46, 48	eqnetrrd 2998	. . . . . 6 ⊢ (𝜑 → (((1 / 2) − 𝑋) · (𝐵 − 𝐴)) ≠ 0)
50	14, 15, 49	mulne0bbd 11791	. . . . 5 ⊢ (𝜑 → (𝐵 − 𝐴) ≠ 0)
51	3, 2, 50	subne0ad 11501	. . . 4 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
52	51	necomd 2985	. . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
53		chordthmlem2.QneM	. . 3 ⊢ (𝜑 → 𝑄 ≠ 𝑀)
54	1, 2, 3, 4, 5, 6, 52, 53	chordthmlem 26796	. 2 ⊢ (𝜑 → ((𝑄 − 𝑀)𝐹(𝐵 − 𝑀)) ∈ {(π / 2), -(π / 2)})
55	4, 44	subcld 11490	. . 3 ⊢ (𝜑 → (𝑄 − 𝑀) ∈ ℂ)
56	39, 44	subcld 11490	. . 3 ⊢ (𝜑 → (𝑃 − 𝑀) ∈ ℂ)
57	3, 44	subcld 11490	. . 3 ⊢ (𝜑 → (𝐵 − 𝑀) ∈ ℂ)
58	4, 44, 53	subne0d 11499	. . 3 ⊢ (𝜑 → (𝑄 − 𝑀) ≠ 0)
59	19, 10	recne0d 11909	. . . . 5 ⊢ (𝜑 → (1 / 2) ≠ 0)
60	16, 15, 59, 50	mulne0d 11787	. . . 4 ⊢ (𝜑 → ((1 / 2) · (𝐵 − 𝐴)) ≠ 0)
61	32, 60	eqnetrd 2997	. . 3 ⊢ (𝜑 → (𝐵 − 𝑀) ≠ 0)
62	32, 46	oveq12d 7374	. . . . 5 ⊢ (𝜑 → ((𝐵 − 𝑀) / (𝑃 − 𝑀)) = (((1 / 2) · (𝐵 − 𝐴)) / (((1 / 2) − 𝑋) · (𝐵 − 𝐴))))
63	14, 15, 49	mulne0bad 11790	. . . . . 6 ⊢ (𝜑 → ((1 / 2) − 𝑋) ≠ 0)
64	16, 14, 15, 63, 50	divcan5rd 11942	. . . . 5 ⊢ (𝜑 → (((1 / 2) · (𝐵 − 𝐴)) / (((1 / 2) − 𝑋) · (𝐵 − 𝐴))) = ((1 / 2) / ((1 / 2) − 𝑋)))
65	62, 64	eqtrd 2769	. . . 4 ⊢ (𝜑 → ((𝐵 − 𝑀) / (𝑃 − 𝑀)) = ((1 / 2) / ((1 / 2) − 𝑋)))
66	11, 13, 63	redivcld 11967	. . . 4 ⊢ (𝜑 → ((1 / 2) / ((1 / 2) − 𝑋)) ∈ ℝ)
67	65, 66	eqeltrd 2834	. . 3 ⊢ (𝜑 → ((𝐵 − 𝑀) / (𝑃 − 𝑀)) ∈ ℝ)
68	1, 55, 56, 57, 58, 48, 61, 67	angrtmuld 26772	. 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 1541 ∈ wcel 2113 ≠ wne 2930 ∖ cdif 3896 {csn 4578 {cpr 4580 ‘cfv 6490 (class class class)co 7356 ∈ cmpo 7358 ℂcc 11022 ℝcr 11023 0cc0 11024 1c1 11025 + caddc 11027 · cmul 11029 − cmin 11362 -cneg 11363 / cdiv 11792 2c2 12198 ℑcim 15019 abscabs 15155 πcpi 15987 logclog 26517

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-inf2 9548 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102 ax-addf 11103

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 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-iin 4947 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8763 df-pm 8764 df-ixp 8834 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-fsupp 9263 df-fi 9312 df-sup 9343 df-inf 9344 df-oi 9413 df-card 9849 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-z 12487 df-dec 12606 df-uz 12750 df-q 12860 df-rp 12904 df-xneg 13024 df-xadd 13025 df-xmul 13026 df-ioo 13263 df-ioc 13264 df-ico 13265 df-icc 13266 df-fz 13422 df-fzo 13569 df-fl 13710 df-mod 13788 df-seq 13923 df-exp 13983 df-fac 14195 df-bc 14224 df-hash 14252 df-shft 14988 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-limsup 15392 df-clim 15409 df-rlim 15410 df-sum 15608 df-ef 15988 df-sin 15990 df-cos 15991 df-pi 15993 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-starv 17190 df-sca 17191 df-vsca 17192 df-ip 17193 df-tset 17194 df-ple 17195 df-ds 17197 df-unif 17198 df-hom 17199 df-cco 17200 df-rest 17340 df-topn 17341 df-0g 17359 df-gsum 17360 df-topgen 17361 df-pt 17362 df-prds 17365 df-xrs 17421 df-qtop 17426 df-imas 17427 df-xps 17429 df-mre 17503 df-mrc 17504 df-acs 17506 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18707 df-mulg 18996 df-cntz 19244 df-cmn 19709 df-psmet 21299 df-xmet 21300 df-met 21301 df-bl 21302 df-mopn 21303 df-fbas 21304 df-fg 21305 df-cnfld 21308 df-top 22836 df-topon 22853 df-topsp 22875 df-bases 22888 df-cld 22961 df-ntr 22962 df-cls 22963 df-nei 23040 df-lp 23078 df-perf 23079 df-cn 23169 df-cnp 23170 df-haus 23257 df-tx 23504 df-hmeo 23697 df-fil 23788 df-fm 23880 df-flim 23881 df-flf 23882 df-xms 24262 df-ms 24263 df-tms 24264 df-cncf 24825 df-limc 25821 df-dv 25822 df-log 26519

This theorem is referenced by: chordthmlem3 26798

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