chordthmlem2 - Metamath Proof Explorer

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Theorem chordthmlem2 26815

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 26814, where P = B, and using angrtmuld 26790 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 12246	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
8	7	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℝ)
9		2ne0 12276	. . . . . . . . . 10 ⊢ 2 ≠ 0
10	9	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ≠ 0)
11	8, 10	rereccld 11973	. . . . . . . 8 ⊢ (𝜑 → (1 / 2) ∈ ℝ)
12		chordthmlem2.X	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℝ)
13	11, 12	resubcld 11569	. . . . . . 7 ⊢ (𝜑 → ((1 / 2) − 𝑋) ∈ ℝ)
14	13	recnd 11164	. . . . . 6 ⊢ (𝜑 → ((1 / 2) − 𝑋) ∈ ℂ)
15	3, 2	subcld 11496	. . . . . 6 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℂ)
16	11	recnd 11164	. . . . . . . . 9 ⊢ (𝜑 → (1 / 2) ∈ ℂ)
17	12	recnd 11164	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℂ)
18	16, 17, 15	subdird 11598	. . . . . . . 8 ⊢ (𝜑 → (((1 / 2) − 𝑋) · (𝐵 − 𝐴)) = (((1 / 2) · (𝐵 − 𝐴)) − (𝑋 · (𝐵 − 𝐴))))
19		2cnd 12250	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ∈ ℂ)
20	3, 19, 10	divcan4d 11928	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐵 · 2) / 2) = 𝐵)
21	3	times2d 12412	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 · 2) = (𝐵 + 𝐵))
22	21	oveq1d 7371	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐵 · 2) / 2) = ((𝐵 + 𝐵) / 2))
23	20, 22	eqtr3d 2776	. . . . . . . . . . . 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 11155	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 + 𝐵) ∈ ℂ)
26	2, 3	addcld 11155	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ)
27	25, 26, 19, 10	divsubdird 11961	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐵 + 𝐵) − (𝐴 + 𝐵)) / 2) = (((𝐵 + 𝐵) / 2) − ((𝐴 + 𝐵) / 2)))
28	3, 2, 3	pnpcan2d 11534	. . . . . . . . . . . 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 2780	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 − 𝑀) = ((𝐵 − 𝐴) / 2))
31	15, 19, 10	divrec2d 11926	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐵 − 𝐴) / 2) = ((1 / 2) · (𝐵 − 𝐴)))
32	30, 31	eqtrd 2774	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 − 𝑀) = ((1 / 2) · (𝐵 − 𝐴)))
33		chordthmlem2.P	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)))
34	17, 2	mulcld 11156	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 · 𝐴) ∈ ℂ)
35		1cnd 11130	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℂ)
36	35, 17	subcld 11496	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1 − 𝑋) ∈ ℂ)
37	36, 3	mulcld 11156	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((1 − 𝑋) · 𝐵) ∈ ℂ)
38	34, 37	addcld 11155	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ∈ ℂ)
39	33, 38	eqeltrd 2839	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ ℂ)
40	2, 39, 3, 17	affineequiv 26805	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ↔ (𝐵 − 𝑃) = (𝑋 · (𝐵 − 𝐴))))
41	33, 40	mpbid 233	. . . . . . . . 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 12413	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℂ)
44	5, 43	eqeltrd 2839	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ)
45	3, 44, 39	nnncan1d 11530	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 − 𝑀) − (𝐵 − 𝑃)) = (𝑃 − 𝑀))
46	18, 42, 45	3eqtr2rd 2781	. . . . . . 7 ⊢ (𝜑 → (𝑃 − 𝑀) = (((1 / 2) − 𝑋) · (𝐵 − 𝐴)))
47		chordthmlem2.PneM	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ≠ 𝑀)
48	39, 44, 47	subne0d 11505	. . . . . . 7 ⊢ (𝜑 → (𝑃 − 𝑀) ≠ 0)
49	46, 48	eqnetrrd 3002	. . . . . 6 ⊢ (𝜑 → (((1 / 2) − 𝑋) · (𝐵 − 𝐴)) ≠ 0)
50	14, 15, 49	mulne0bbd 11797	. . . . 5 ⊢ (𝜑 → (𝐵 − 𝐴) ≠ 0)
51	3, 2, 50	subne0ad 11507	. . . 4 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
52	51	necomd 2989	. . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
53		chordthmlem2.QneM	. . 3 ⊢ (𝜑 → 𝑄 ≠ 𝑀)
54	1, 2, 3, 4, 5, 6, 52, 53	chordthmlem 26814	. 2 ⊢ (𝜑 → ((𝑄 − 𝑀)𝐹(𝐵 − 𝑀)) ∈ {(π / 2), -(π / 2)})
55	4, 44	subcld 11496	. . 3 ⊢ (𝜑 → (𝑄 − 𝑀) ∈ ℂ)
56	39, 44	subcld 11496	. . 3 ⊢ (𝜑 → (𝑃 − 𝑀) ∈ ℂ)
57	3, 44	subcld 11496	. . 3 ⊢ (𝜑 → (𝐵 − 𝑀) ∈ ℂ)
58	4, 44, 53	subne0d 11505	. . 3 ⊢ (𝜑 → (𝑄 − 𝑀) ≠ 0)
59	19, 10	recne0d 11916	. . . . 5 ⊢ (𝜑 → (1 / 2) ≠ 0)
60	16, 15, 59, 50	mulne0d 11793	. . . 4 ⊢ (𝜑 → ((1 / 2) · (𝐵 − 𝐴)) ≠ 0)
61	32, 60	eqnetrd 3001	. . 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 11796	. . . . . 6 ⊢ (𝜑 → ((1 / 2) − 𝑋) ≠ 0)
64	16, 14, 15, 63, 50	divcan5rd 11949	. . . . 5 ⊢ (𝜑 → (((1 / 2) · (𝐵 − 𝐴)) / (((1 / 2) − 𝑋) · (𝐵 − 𝐴))) = ((1 / 2) / ((1 / 2) − 𝑋)))
65	62, 64	eqtrd 2774	. . . 4 ⊢ (𝜑 → ((𝐵 − 𝑀) / (𝑃 − 𝑀)) = ((1 / 2) / ((1 / 2) − 𝑋)))
66	11, 13, 63	redivcld 11974	. . . 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 26790	. 2 ⊢ (𝜑 → (((𝑄 − 𝑀)𝐹(𝑃 − 𝑀)) ∈ {(π / 2), -(π / 2)} ↔ ((𝑄 − 𝑀)𝐹(𝐵 − 𝑀)) ∈ {(π / 2), -(π / 2)}))
69	54, 68	mpbird 258	1 ⊢ (𝜑 → ((𝑄 − 𝑀)𝐹(𝑃 − 𝑀)) ∈ {(π / 2), -(π / 2)})

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 ∖ cdif 3880 {csn 4555 {cpr 4557 ‘cfv 6485 (class class class)co 7356 ∈ cmpo 7358 ℂcc 11027 ℝcr 11028 0cc0 11029 1c1 11030 + caddc 11032 · cmul 11034 − cmin 11368 -cneg 11369 / cdiv 11798 2c2 12227 ℑcim 15051 abscabs 15187 πcpi 16022 logclog 26536

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-inf2 9553 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 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 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ioc 13294 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-mod 13820 df-seq 13955 df-exp 14015 df-fac 14227 df-bc 14256 df-hash 14284 df-shft 15020 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15424 df-clim 15441 df-rlim 15442 df-sum 15640 df-ef 16023 df-sin 16025 df-cos 16026 df-pi 16028 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-mulg 19035 df-cntz 19283 df-cmn 19748 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-fbas 21344 df-fg 21345 df-cnfld 21348 df-top 22877 df-topon 22894 df-topsp 22916 df-bases 22929 df-cld 23002 df-ntr 23003 df-cls 23004 df-nei 23081 df-lp 23119 df-perf 23120 df-cn 23210 df-cnp 23211 df-haus 23298 df-tx 23545 df-hmeo 23738 df-fil 23829 df-fm 23921 df-flim 23922 df-flf 23923 df-xms 24303 df-ms 24304 df-tms 24305 df-cncf 24863 df-limc 25851 df-dv 25852 df-log 26538

This theorem is referenced by: chordthmlem3 26816

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