chordthmlem2 - Metamath Proof Explorer

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Theorem chordthmlem2 27049

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 27048, where P = B, and using angrtmuld 27024 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 12330	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
8	7	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℝ)
9		2ne0 12362	. . . . . . . . . 10 ⊢ 2 ≠ 0
10	9	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ≠ 0)
11	8, 10	rereccld 12057	. . . . . . . 8 ⊢ (𝜑 → (1 / 2) ∈ ℝ)
12		chordthmlem2.X	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℝ)
13	11, 12	resubcld 11657	. . . . . . 7 ⊢ (𝜑 → ((1 / 2) − 𝑋) ∈ ℝ)
14	13	recnd 11252	. . . . . 6 ⊢ (𝜑 → ((1 / 2) − 𝑋) ∈ ℂ)
15	3, 2	subcld 11584	. . . . . 6 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℂ)
16	11	recnd 11252	. . . . . . . . 9 ⊢ (𝜑 → (1 / 2) ∈ ℂ)
17	12	recnd 11252	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℂ)
18	16, 17, 15	subdird 11686	. . . . . . . 8 ⊢ (𝜑 → (((1 / 2) − 𝑋) · (𝐵 − 𝐴)) = (((1 / 2) · (𝐵 − 𝐴)) − (𝑋 · (𝐵 − 𝐴))))
19		2cnd 12334	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ∈ ℂ)
20	3, 19, 10	divcan4d 12012	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐵 · 2) / 2) = 𝐵)
21	3	times2d 12503	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 · 2) = (𝐵 + 𝐵))
22	21	oveq1d 7434	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐵 · 2) / 2) = ((𝐵 + 𝐵) / 2))
23	20, 22	eqtr3d 2802	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 = ((𝐵 + 𝐵) / 2))
24	23, 5	oveq12d 7437	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 − 𝑀) = (((𝐵 + 𝐵) / 2) − ((𝐴 + 𝐵) / 2)))
25	3, 3	addcld 11243	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 + 𝐵) ∈ ℂ)
26	2, 3	addcld 11243	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ)
27	25, 26, 19, 10	divsubdird 12045	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐵 + 𝐵) − (𝐴 + 𝐵)) / 2) = (((𝐵 + 𝐵) / 2) − ((𝐴 + 𝐵) / 2)))
28	3, 2, 3	pnpcan2d 11622	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐵 + 𝐵) − (𝐴 + 𝐵)) = (𝐵 − 𝐴))
29	28	oveq1d 7434	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐵 + 𝐵) − (𝐴 + 𝐵)) / 2) = ((𝐵 − 𝐴) / 2))
30	24, 27, 29	3eqtr2d 2806	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 − 𝑀) = ((𝐵 − 𝐴) / 2))
31	15, 19, 10	divrec2d 12010	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐵 − 𝐴) / 2) = ((1 / 2) · (𝐵 − 𝐴)))
32	30, 31	eqtrd 2800	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 − 𝑀) = ((1 / 2) · (𝐵 − 𝐴)))
33		chordthmlem2.P	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)))
34	17, 2	mulcld 11244	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 · 𝐴) ∈ ℂ)
35		1cnd 11217	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℂ)
36	35, 17	subcld 11584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1 − 𝑋) ∈ ℂ)
37	36, 3	mulcld 11244	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((1 − 𝑋) · 𝐵) ∈ ℂ)
38	34, 37	addcld 11243	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ∈ ℂ)
39	33, 38	eqeltrd 2865	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ ℂ)
40	2, 39, 3, 17	affineequiv 27039	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ↔ (𝐵 − 𝑃) = (𝑋 · (𝐵 − 𝐴))))
41	33, 40	mpbid 235	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 − 𝑃) = (𝑋 · (𝐵 − 𝐴)))
42	32, 41	oveq12d 7437	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 − 𝑀) − (𝐵 − 𝑃)) = (((1 / 2) · (𝐵 − 𝐴)) − (𝑋 · (𝐵 − 𝐴))))
43	26	halfcld 12504	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℂ)
44	5, 43	eqeltrd 2865	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ)
45	3, 44, 39	nnncan1d 11618	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 − 𝑀) − (𝐵 − 𝑃)) = (𝑃 − 𝑀))
46	18, 42, 45	3eqtr2rd 2807	. . . . . . 7 ⊢ (𝜑 → (𝑃 − 𝑀) = (((1 / 2) − 𝑋) · (𝐵 − 𝐴)))
47		chordthmlem2.PneM	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ≠ 𝑀)
48	39, 44, 47	subne0d 11593	. . . . . . 7 ⊢ (𝜑 → (𝑃 − 𝑀) ≠ 0)
49	46, 48	eqnetrrd 3028	. . . . . 6 ⊢ (𝜑 → (((1 / 2) − 𝑋) · (𝐵 − 𝐴)) ≠ 0)
50	14, 15, 49	mulne0bbd 11885	. . . . 5 ⊢ (𝜑 → (𝐵 − 𝐴) ≠ 0)
51	3, 2, 50	subne0ad 11595	. . . 4 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
52	51	necomd 3015	. . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
53		chordthmlem2.QneM	. . 3 ⊢ (𝜑 → 𝑄 ≠ 𝑀)
54	1, 2, 3, 4, 5, 6, 52, 53	chordthmlem 27048	. 2 ⊢ (𝜑 → ((𝑄 − 𝑀)𝐹(𝐵 − 𝑀)) ∈ {(π / 2), -(π / 2)})
55	4, 44	subcld 11584	. . 3 ⊢ (𝜑 → (𝑄 − 𝑀) ∈ ℂ)
56	39, 44	subcld 11584	. . 3 ⊢ (𝜑 → (𝑃 − 𝑀) ∈ ℂ)
57	3, 44	subcld 11584	. . 3 ⊢ (𝜑 → (𝐵 − 𝑀) ∈ ℂ)
58	4, 44, 53	subne0d 11593	. . 3 ⊢ (𝜑 → (𝑄 − 𝑀) ≠ 0)
59	19, 10	recne0d 12000	. . . . 5 ⊢ (𝜑 → (1 / 2) ≠ 0)
60	16, 15, 59, 50	mulne0d 11881	. . . 4 ⊢ (𝜑 → ((1 / 2) · (𝐵 − 𝐴)) ≠ 0)
61	32, 60	eqnetrd 3027	. . 3 ⊢ (𝜑 → (𝐵 − 𝑀) ≠ 0)
62	32, 46	oveq12d 7437	. . . . 5 ⊢ (𝜑 → ((𝐵 − 𝑀) / (𝑃 − 𝑀)) = (((1 / 2) · (𝐵 − 𝐴)) / (((1 / 2) − 𝑋) · (𝐵 − 𝐴))))
63	14, 15, 49	mulne0bad 11884	. . . . . 6 ⊢ (𝜑 → ((1 / 2) − 𝑋) ≠ 0)
64	16, 14, 15, 63, 50	divcan5rd 12033	. . . . 5 ⊢ (𝜑 → (((1 / 2) · (𝐵 − 𝐴)) / (((1 / 2) − 𝑋) · (𝐵 − 𝐴))) = ((1 / 2) / ((1 / 2) − 𝑋)))
65	62, 64	eqtrd 2800	. . . 4 ⊢ (𝜑 → ((𝐵 − 𝑀) / (𝑃 − 𝑀)) = ((1 / 2) / ((1 / 2) − 𝑋)))
66	11, 13, 63	redivcld 12058	. . . 4 ⊢ (𝜑 → ((1 / 2) / ((1 / 2) − 𝑋)) ∈ ℝ)
67	65, 66	eqeltrd 2865	. . 3 ⊢ (𝜑 → ((𝐵 − 𝑀) / (𝑃 − 𝑀)) ∈ ℝ)
68	1, 55, 56, 57, 58, 48, 61, 67	angrtmuld 27024	. 2 ⊢ (𝜑 → (((𝑄 − 𝑀)𝐹(𝑃 − 𝑀)) ∈ {(π / 2), -(π / 2)} ↔ ((𝑄 − 𝑀)𝐹(𝐵 − 𝑀)) ∈ {(π / 2), -(π / 2)}))
69	54, 68	mpbird 260	1 ⊢ (𝜑 → ((𝑄 − 𝑀)𝐹(𝑃 − 𝑀)) ∈ {(π / 2), -(π / 2)})

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 ∖ cdif 3903 {csn 4591 {cpr 4593 ‘cfv 6540 (class class class)co 7419 ∈ cmpo 7421 ℂcc 11113 ℝcr 11114 0cc0 11115 1c1 11116 + caddc 11118 · cmul 11120 − cmin 11456 -cneg 11457 / cdiv 11886 2c2 12310 ℑcim 15173 abscabs 15309 πcpi 16142 logclog 26770

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-inf2 9617 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192 ax-pre-sup 11193 ax-addf 11194

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8163 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-er 8700 df-map 8832 df-pm 8833 df-ixp 8902 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-fsupp 9329 df-fi 9378 df-sup 9409 df-inf 9410 df-oi 9479 df-card 9941 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-div 11887 df-nn 12249 df-2 12318 df-3 12319 df-4 12320 df-5 12321 df-6 12322 df-7 12323 df-8 12324 df-9 12325 df-n0 12520 df-z 12607 df-dec 12728 df-uz 12879 df-q 12989 df-rp 13033 df-xneg 13153 df-xadd 13154 df-xmul 13155 df-ioo 13392 df-ioc 13393 df-ico 13394 df-icc 13395 df-fz 13552 df-fzo 13700 df-fl 13843 df-mod 13921 df-seq 14056 df-exp 14116 df-fac 14328 df-bc 14357 df-hash 14385 df-shft 15128 df-cj 15174 df-re 15175 df-im 15176 df-sqrt 15310 df-abs 15311 df-limsup 15546 df-clim 15563 df-rlim 15564 df-sum 15762 df-ef 16143 df-sin 16145 df-cos 16146 df-pi 16148 df-struct 17229 df-sets 17246 df-slot 17264 df-ndx 17276 df-base 17292 df-ress 17313 df-plusg 17345 df-mulr 17346 df-starv 17347 df-sca 17348 df-vsca 17349 df-ip 17350 df-tset 17351 df-ple 17352 df-ds 17354 df-unif 17355 df-hom 17356 df-cco 17357 df-rest 17497 df-topn 17498 df-0g 17516 df-gsum 17517 df-topgen 17518 df-pt 17519 df-prds 17522 df-xrs 17578 df-qtop 17583 df-imas 17584 df-xps 17586 df-mre 17660 df-mrc 17661 df-acs 17663 df-mgm 18720 df-sgrp 18809 df-mnd 18825 df-submnd 18879 df-mulg 19178 df-cntz 19431 df-cmn 19896 df-psmet 21564 df-xmet 21565 df-met 21566 df-bl 21567 df-mopn 21568 df-fbas 21569 df-fg 21570 df-cnfld 21573 df-top 23101 df-topon 23118 df-topsp 23140 df-bases 23153 df-cld 23226 df-ntr 23227 df-cls 23228 df-nei 23305 df-lp 23343 df-perf 23344 df-cn 23434 df-cnp 23435 df-haus 23522 df-tx 23770 df-hmeo 23963 df-fil 24054 df-fm 24146 df-flim 24147 df-flf 24148 df-xms 24528 df-ms 24529 df-tms 24530 df-cncf 25088 df-limc 26076 df-dv 26077 df-log 26772

This theorem is used by: chordthmlem3 27050

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