chordthmlem - Metamath Proof Explorer

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Theorem chordthmlem 26814

Description: If 𝑀 is the midpoint of AB and AQ = BQ, then QMB is a right angle. The proof uses ssscongptld 26804 to observe that, since AMQ and BMQ have equal sides, the angles QMB and QMA must be equal. Since they are supplementary, both must be right angles. (Contributed by David Moews, 28-Feb-2017.)

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

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

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

**Proof of Theorem chordthmlem**
Step	Hyp	Ref	Expression
1		negpitopissre 26522	. . . . . 6 ⊢ (-π(,]π) ⊆ ℝ
2		chordthmlem.angdef	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))
3		chordthmlem.Q	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ ℂ)
4		chordthmlem.M	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 = ((𝐴 + 𝐵) / 2))
5		chordthmlem.A	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℂ)
6		chordthmlem.B	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℂ)
7	5, 6	addcld 11155	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ)
8	7	halfcld 12413	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℂ)
9	4, 8	eqeltrd 2839	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℂ)
10	3, 9	subcld 11496	. . . . . . 7 ⊢ (𝜑 → (𝑄 − 𝑀) ∈ ℂ)
11		chordthmlem.QneM	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ≠ 𝑀)
12	3, 9, 11	subne0d 11505	. . . . . . 7 ⊢ (𝜑 → (𝑄 − 𝑀) ≠ 0)
13	6, 9	subcld 11496	. . . . . . 7 ⊢ (𝜑 → (𝐵 − 𝑀) ∈ ℂ)
14	4	oveq1d 7371	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀 · 2) = (((𝐴 + 𝐵) / 2) · 2))
15	9	times2d 12412	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀 · 2) = (𝑀 + 𝑀))
16		2cnd 12250	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ∈ ℂ)
17		2ne0 12276	. . . . . . . . . . . . . . . 16 ⊢ 2 ≠ 0
18	17	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ≠ 0)
19	7, 16, 18	divcan1d 11923	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((𝐴 + 𝐵) / 2) · 2) = (𝐴 + 𝐵))
20	14, 15, 19	3eqtr3d 2782	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑀 + 𝑀) = (𝐴 + 𝐵))
21		chordthmlem.AneB	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
22	5, 6, 6, 21	addneintr2d 11345	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 + 𝐵) ≠ (𝐵 + 𝐵))
23	20, 22	eqnetrd 3001	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑀 + 𝑀) ≠ (𝐵 + 𝐵))
24	23	neneqd 2939	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (𝑀 + 𝑀) = (𝐵 + 𝐵))
25		oveq12 7365	. . . . . . . . . . . 12 ⊢ ((𝑀 = 𝐵 ∧ 𝑀 = 𝐵) → (𝑀 + 𝑀) = (𝐵 + 𝐵))
26	25	anidms 571	. . . . . . . . . . 11 ⊢ (𝑀 = 𝐵 → (𝑀 + 𝑀) = (𝐵 + 𝐵))
27	24, 26	nsyl 140	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑀 = 𝐵)
28	27	neqned 2941	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ≠ 𝐵)
29	28	necomd 2989	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ≠ 𝑀)
30	6, 9, 29	subne0d 11505	. . . . . . 7 ⊢ (𝜑 → (𝐵 − 𝑀) ≠ 0)
31	2, 10, 12, 13, 30	angcld 26787	. . . . . 6 ⊢ (𝜑 → ((𝑄 − 𝑀)𝐹(𝐵 − 𝑀)) ∈ (-π(,]π))
32	1, 31	sselid 3913	. . . . 5 ⊢ (𝜑 → ((𝑄 − 𝑀)𝐹(𝐵 − 𝑀)) ∈ ℝ)
33	32	recnd 11164	. . . 4 ⊢ (𝜑 → ((𝑄 − 𝑀)𝐹(𝐵 − 𝑀)) ∈ ℂ)
34	33	coscld 16089	. . 3 ⊢ (𝜑 → (cos‘((𝑄 − 𝑀)𝐹(𝐵 − 𝑀))) ∈ ℂ)
35	6, 9	negsubdi2d 11512	. . . . . . 7 ⊢ (𝜑 → -(𝐵 − 𝑀) = (𝑀 − 𝐵))
36	9, 9, 5, 6	addsubeq4d 11547	. . . . . . . 8 ⊢ (𝜑 → ((𝑀 + 𝑀) = (𝐴 + 𝐵) ↔ (𝐴 − 𝑀) = (𝑀 − 𝐵)))
37	20, 36	mpbid 233	. . . . . . 7 ⊢ (𝜑 → (𝐴 − 𝑀) = (𝑀 − 𝐵))
38	35, 37	eqtr4d 2777	. . . . . 6 ⊢ (𝜑 → -(𝐵 − 𝑀) = (𝐴 − 𝑀))
39	38	oveq2d 7372	. . . . 5 ⊢ (𝜑 → ((𝑄 − 𝑀)𝐹-(𝐵 − 𝑀)) = ((𝑄 − 𝑀)𝐹(𝐴 − 𝑀)))
40	39	fveq2d 6831	. . . 4 ⊢ (𝜑 → (cos‘((𝑄 − 𝑀)𝐹-(𝐵 − 𝑀))) = (cos‘((𝑄 − 𝑀)𝐹(𝐴 − 𝑀))))
41	2, 10, 12, 13, 30	cosangneg2d 26789	. . . 4 ⊢ (𝜑 → (cos‘((𝑄 − 𝑀)𝐹-(𝐵 − 𝑀))) = -(cos‘((𝑄 − 𝑀)𝐹(𝐵 − 𝑀))))
42	5, 5, 6, 21	addneintrd 11344	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 + 𝐴) ≠ (𝐴 + 𝐵))
43	42, 20	neeqtrrd 3008	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 + 𝐴) ≠ (𝑀 + 𝑀))
44	43	necomd 2989	. . . . . . . 8 ⊢ (𝜑 → (𝑀 + 𝑀) ≠ (𝐴 + 𝐴))
45	44	neneqd 2939	. . . . . . 7 ⊢ (𝜑 → ¬ (𝑀 + 𝑀) = (𝐴 + 𝐴))
46		oveq12 7365	. . . . . . . 8 ⊢ ((𝑀 = 𝐴 ∧ 𝑀 = 𝐴) → (𝑀 + 𝑀) = (𝐴 + 𝐴))
47	46	anidms 571	. . . . . . 7 ⊢ (𝑀 = 𝐴 → (𝑀 + 𝑀) = (𝐴 + 𝐴))
48	45, 47	nsyl 140	. . . . . 6 ⊢ (𝜑 → ¬ 𝑀 = 𝐴)
49	48	neqned 2941	. . . . 5 ⊢ (𝜑 → 𝑀 ≠ 𝐴)
50		eqidd 2740	. . . . 5 ⊢ (𝜑 → (abs‘(𝑄 − 𝑀)) = (abs‘(𝑄 − 𝑀)))
51	5, 9	subcld 11496	. . . . . . 7 ⊢ (𝜑 → (𝐴 − 𝑀) ∈ ℂ)
52	51	absnegd 15405	. . . . . 6 ⊢ (𝜑 → (abs‘-(𝐴 − 𝑀)) = (abs‘(𝐴 − 𝑀)))
53	5, 9	negsubdi2d 11512	. . . . . . 7 ⊢ (𝜑 → -(𝐴 − 𝑀) = (𝑀 − 𝐴))
54	53	fveq2d 6831	. . . . . 6 ⊢ (𝜑 → (abs‘-(𝐴 − 𝑀)) = (abs‘(𝑀 − 𝐴)))
55	37	fveq2d 6831	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐴 − 𝑀)) = (abs‘(𝑀 − 𝐵)))
56	52, 54, 55	3eqtr3d 2782	. . . . 5 ⊢ (𝜑 → (abs‘(𝑀 − 𝐴)) = (abs‘(𝑀 − 𝐵)))
57		chordthmlem.ABequidistQ	. . . . 5 ⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) = (abs‘(𝐵 − 𝑄)))
58	2, 3, 9, 5, 3, 9, 6, 11, 49, 11, 28, 50, 56, 57	ssscongptld 26804	. . . 4 ⊢ (𝜑 → (cos‘((𝑄 − 𝑀)𝐹(𝐴 − 𝑀))) = (cos‘((𝑄 − 𝑀)𝐹(𝐵 − 𝑀))))
59	40, 41, 58	3eqtr3rd 2783	. . 3 ⊢ (𝜑 → (cos‘((𝑄 − 𝑀)𝐹(𝐵 − 𝑀))) = -(cos‘((𝑄 − 𝑀)𝐹(𝐵 − 𝑀))))
60	34, 59	eqnegad 11868	. 2 ⊢ (𝜑 → (cos‘((𝑄 − 𝑀)𝐹(𝐵 − 𝑀))) = 0)
61		coseq0negpitopi 26485	. . 3 ⊢ (((𝑄 − 𝑀)𝐹(𝐵 − 𝑀)) ∈ (-π(,]π) → ((cos‘((𝑄 − 𝑀)𝐹(𝐵 − 𝑀))) = 0 ↔ ((𝑄 − 𝑀)𝐹(𝐵 − 𝑀)) ∈ {(π / 2), -(π / 2)}))
62	31, 61	syl 17	. 2 ⊢ (𝜑 → ((cos‘((𝑄 − 𝑀)𝐹(𝐵 − 𝑀))) = 0 ↔ ((𝑄 − 𝑀)𝐹(𝐵 − 𝑀)) ∈ {(π / 2), -(π / 2)}))
63	60, 62	mpbid 233	1 ⊢ (𝜑 → ((𝑄 − 𝑀)𝐹(𝐵 − 𝑀)) ∈ {(π / 2), -(π / 2)})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 = 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 + caddc 11032 · cmul 11034 − cmin 11368 -cneg 11369 / cdiv 11798 2c2 12227 (,]cioc 13290 ℑcim 15051 abscabs 15187 cosccos 16020 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: chordthmlem2 26815

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