chordthmlem2 - Metamath Proof Explorer

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Theorem chordthmlem2 24781

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 24780, where P = B, and using angrtmuld 24759 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 11292	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
8	7	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℝ)
9		2ne0 11315	. . . . . . . . . 10 ⊢ 2 ≠ 0
10	9	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ≠ 0)
11	8, 10	rereccld 11054	. . . . . . . 8 ⊢ (𝜑 → (1 / 2) ∈ ℝ)
12		chordthmlem2.X	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℝ)
13	11, 12	resubcld 10660	. . . . . . 7 ⊢ (𝜑 → ((1 / 2) − 𝑋) ∈ ℝ)
14	13	recnd 10270	. . . . . 6 ⊢ (𝜑 → ((1 / 2) − 𝑋) ∈ ℂ)
15	3, 2	subcld 10594	. . . . . 6 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℂ)
16	11	recnd 10270	. . . . . . . . 9 ⊢ (𝜑 → (1 / 2) ∈ ℂ)
17	12	recnd 10270	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℂ)
18	16, 17, 15	subdird 10689	. . . . . . . 8 ⊢ (𝜑 → (((1 / 2) − 𝑋) · (𝐵 − 𝐴)) = (((1 / 2) · (𝐵 − 𝐴)) − (𝑋 · (𝐵 − 𝐴))))
19		2cnd 11295	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ∈ ℂ)
20	3, 19, 10	divcan4d 11009	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐵 · 2) / 2) = 𝐵)
21	3	times2d 11478	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 · 2) = (𝐵 + 𝐵))
22	21	oveq1d 6808	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐵 · 2) / 2) = ((𝐵 + 𝐵) / 2))
23	20, 22	eqtr3d 2807	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 = ((𝐵 + 𝐵) / 2))
24	23, 5	oveq12d 6811	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 − 𝑀) = (((𝐵 + 𝐵) / 2) − ((𝐴 + 𝐵) / 2)))
25	3, 3	addcld 10261	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 + 𝐵) ∈ ℂ)
26	2, 3	addcld 10261	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ)
27	25, 26, 19, 10	divsubdird 11042	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐵 + 𝐵) − (𝐴 + 𝐵)) / 2) = (((𝐵 + 𝐵) / 2) − ((𝐴 + 𝐵) / 2)))
28	3, 2, 3	pnpcan2d 10632	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐵 + 𝐵) − (𝐴 + 𝐵)) = (𝐵 − 𝐴))
29	28	oveq1d 6808	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐵 + 𝐵) − (𝐴 + 𝐵)) / 2) = ((𝐵 − 𝐴) / 2))
30	24, 27, 29	3eqtr2d 2811	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 − 𝑀) = ((𝐵 − 𝐴) / 2))
31	15, 19, 10	divrec2d 11007	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐵 − 𝐴) / 2) = ((1 / 2) · (𝐵 − 𝐴)))
32	30, 31	eqtrd 2805	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 − 𝑀) = ((1 / 2) · (𝐵 − 𝐴)))
33		chordthmlem2.P	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)))
34	17, 2	mulcld 10262	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 · 𝐴) ∈ ℂ)
35		1cnd 10258	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℂ)
36	35, 17	subcld 10594	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1 − 𝑋) ∈ ℂ)
37	36, 3	mulcld 10262	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((1 − 𝑋) · 𝐵) ∈ ℂ)
38	34, 37	addcld 10261	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ∈ ℂ)
39	33, 38	eqeltrd 2850	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ ℂ)
40	2, 39, 3, 17	affineequiv 24774	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ↔ (𝐵 − 𝑃) = (𝑋 · (𝐵 − 𝐴))))
41	33, 40	mpbid 222	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 − 𝑃) = (𝑋 · (𝐵 − 𝐴)))
42	32, 41	oveq12d 6811	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 − 𝑀) − (𝐵 − 𝑃)) = (((1 / 2) · (𝐵 − 𝐴)) − (𝑋 · (𝐵 − 𝐴))))
43	26	halfcld 11479	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℂ)
44	5, 43	eqeltrd 2850	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ)
45	3, 44, 39	nnncan1d 10628	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 − 𝑀) − (𝐵 − 𝑃)) = (𝑃 − 𝑀))
46	18, 42, 45	3eqtr2rd 2812	. . . . . . 7 ⊢ (𝜑 → (𝑃 − 𝑀) = (((1 / 2) − 𝑋) · (𝐵 − 𝐴)))
47		chordthmlem2.PneM	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ≠ 𝑀)
48	39, 44, 47	subne0d 10603	. . . . . . 7 ⊢ (𝜑 → (𝑃 − 𝑀) ≠ 0)
49	46, 48	eqnetrrd 3011	. . . . . 6 ⊢ (𝜑 → (((1 / 2) − 𝑋) · (𝐵 − 𝐴)) ≠ 0)
50	14, 15, 49	mulne0bbd 10885	. . . . 5 ⊢ (𝜑 → (𝐵 − 𝐴) ≠ 0)
51	3, 2, 50	subne0ad 10605	. . . 4 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
52	51	necomd 2998	. . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
53		chordthmlem2.QneM	. . 3 ⊢ (𝜑 → 𝑄 ≠ 𝑀)
54	1, 2, 3, 4, 5, 6, 52, 53	chordthmlem 24780	. 2 ⊢ (𝜑 → ((𝑄 − 𝑀)𝐹(𝐵 − 𝑀)) ∈ {(π / 2), -(π / 2)})
55	4, 44	subcld 10594	. . 3 ⊢ (𝜑 → (𝑄 − 𝑀) ∈ ℂ)
56	39, 44	subcld 10594	. . 3 ⊢ (𝜑 → (𝑃 − 𝑀) ∈ ℂ)
57	3, 44	subcld 10594	. . 3 ⊢ (𝜑 → (𝐵 − 𝑀) ∈ ℂ)
58	4, 44, 53	subne0d 10603	. . 3 ⊢ (𝜑 → (𝑄 − 𝑀) ≠ 0)
59	19, 10	recne0d 10997	. . . . 5 ⊢ (𝜑 → (1 / 2) ≠ 0)
60	16, 15, 59, 50	mulne0d 10881	. . . 4 ⊢ (𝜑 → ((1 / 2) · (𝐵 − 𝐴)) ≠ 0)
61	32, 60	eqnetrd 3010	. . 3 ⊢ (𝜑 → (𝐵 − 𝑀) ≠ 0)
62	32, 46	oveq12d 6811	. . . . 5 ⊢ (𝜑 → ((𝐵 − 𝑀) / (𝑃 − 𝑀)) = (((1 / 2) · (𝐵 − 𝐴)) / (((1 / 2) − 𝑋) · (𝐵 − 𝐴))))
63	14, 15, 49	mulne0bad 10884	. . . . . 6 ⊢ (𝜑 → ((1 / 2) − 𝑋) ≠ 0)
64	16, 14, 15, 63, 50	divcan5rd 11030	. . . . 5 ⊢ (𝜑 → (((1 / 2) · (𝐵 − 𝐴)) / (((1 / 2) − 𝑋) · (𝐵 − 𝐴))) = ((1 / 2) / ((1 / 2) − 𝑋)))
65	62, 64	eqtrd 2805	. . . 4 ⊢ (𝜑 → ((𝐵 − 𝑀) / (𝑃 − 𝑀)) = ((1 / 2) / ((1 / 2) − 𝑋)))
66	11, 13, 63	redivcld 11055	. . . 4 ⊢ (𝜑 → ((1 / 2) / ((1 / 2) − 𝑋)) ∈ ℝ)
67	65, 66	eqeltrd 2850	. . 3 ⊢ (𝜑 → ((𝐵 − 𝑀) / (𝑃 − 𝑀)) ∈ ℝ)
68	1, 55, 56, 57, 58, 48, 61, 67	angrtmuld 24759	. 2 ⊢ (𝜑 → (((𝑄 − 𝑀)𝐹(𝑃 − 𝑀)) ∈ {(π / 2), -(π / 2)} ↔ ((𝑄 − 𝑀)𝐹(𝐵 − 𝑀)) ∈ {(π / 2), -(π / 2)}))
69	54, 68	mpbird 247	1 ⊢ (𝜑 → ((𝑄 − 𝑀)𝐹(𝑃 − 𝑀)) ∈ {(π / 2), -(π / 2)})

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 ∖ cdif 3720 {csn 4316 {cpr 4318 ‘cfv 6031 (class class class)co 6793 ↦ cmpt2 6795 ℂcc 10136 ℝcr 10137 0cc0 10138 1c1 10139 + caddc 10141 · cmul 10143 − cmin 10468 -cneg 10469 / cdiv 10886 2c2 11272 ℑcim 14046 abscabs 14182 πcpi 15003 logclog 24522

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-inf2 8702 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-pre-sup 10216 ax-addf 10217 ax-mulf 10218

This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-iin 4657 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-isom 6040 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-of 7044 df-om 7213 df-1st 7315 df-2nd 7316 df-supp 7447 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-2o 7714 df-oadd 7717 df-er 7896 df-map 8011 df-pm 8012 df-ixp 8063 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-fsupp 8432 df-fi 8473 df-sup 8504 df-inf 8505 df-oi 8571 df-card 8965 df-cda 9192 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-nn 11223 df-2 11281 df-3 11282 df-4 11283 df-5 11284 df-6 11285 df-7 11286 df-8 11287 df-9 11288 df-n0 11495 df-z 11580 df-dec 11696 df-uz 11889 df-q 11992 df-rp 12036 df-xneg 12151 df-xadd 12152 df-xmul 12153 df-ioo 12384 df-ioc 12385 df-ico 12386 df-icc 12387 df-fz 12534 df-fzo 12674 df-fl 12801 df-mod 12877 df-seq 13009 df-exp 13068 df-fac 13265 df-bc 13294 df-hash 13322 df-shft 14015 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-abs 14184 df-limsup 14410 df-clim 14427 df-rlim 14428 df-sum 14625 df-ef 15004 df-sin 15006 df-cos 15007 df-pi 15009 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-ress 16072 df-plusg 16162 df-mulr 16163 df-starv 16164 df-sca 16165 df-vsca 16166 df-ip 16167 df-tset 16168 df-ple 16169 df-ds 16172 df-unif 16173 df-hom 16174 df-cco 16175 df-rest 16291 df-topn 16292 df-0g 16310 df-gsum 16311 df-topgen 16312 df-pt 16313 df-prds 16316 df-xrs 16370 df-qtop 16375 df-imas 16376 df-xps 16378 df-mre 16454 df-mrc 16455 df-acs 16457 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-submnd 17544 df-mulg 17749 df-cntz 17957 df-cmn 18402 df-psmet 19953 df-xmet 19954 df-met 19955 df-bl 19956 df-mopn 19957 df-fbas 19958 df-fg 19959 df-cnfld 19962 df-top 20919 df-topon 20936 df-topsp 20958 df-bases 20971 df-cld 21044 df-ntr 21045 df-cls 21046 df-nei 21123 df-lp 21161 df-perf 21162 df-cn 21252 df-cnp 21253 df-haus 21340 df-tx 21586 df-hmeo 21779 df-fil 21870 df-fm 21962 df-flim 21963 df-flf 21964 df-xms 22345 df-ms 22346 df-tms 22347 df-cncf 22901 df-limc 23850 df-dv 23851 df-log 24524

This theorem is referenced by: chordthmlem3 24782

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