chordthmlem - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > chordthmlem		Structured version Visualization version GIF version

Theorem chordthmlem 25980

Description: If 𝑀 is the midpoint of AB and AQ = BQ, then QMB is a right angle. The proof uses ssscongptld 25970 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 25694	. . . . . 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 10992	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ)
8	7	halfcld 12216	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℂ)
9	4, 8	eqeltrd 2839	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℂ)
10	3, 9	subcld 11330	. . . . . . 7 ⊢ (𝜑 → (𝑄 − 𝑀) ∈ ℂ)
11		chordthmlem.QneM	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ≠ 𝑀)
12	3, 9, 11	subne0d 11339	. . . . . . 7 ⊢ (𝜑 → (𝑄 − 𝑀) ≠ 0)
13	6, 9	subcld 11330	. . . . . . 7 ⊢ (𝜑 → (𝐵 − 𝑀) ∈ ℂ)
14	4	oveq1d 7292	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀 · 2) = (((𝐴 + 𝐵) / 2) · 2))
15	9	times2d 12215	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀 · 2) = (𝑀 + 𝑀))
16		2cnd 12049	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ∈ ℂ)
17		2ne0 12075	. . . . . . . . . . . . . . . 16 ⊢ 2 ≠ 0
18	17	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ≠ 0)
19	7, 16, 18	divcan1d 11750	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((𝐴 + 𝐵) / 2) · 2) = (𝐴 + 𝐵))
20	14, 15, 19	3eqtr3d 2786	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑀 + 𝑀) = (𝐴 + 𝐵))
21		chordthmlem.AneB	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
22	5, 6, 6, 21	addneintr2d 11181	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 + 𝐵) ≠ (𝐵 + 𝐵))
23	20, 22	eqnetrd 3011	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑀 + 𝑀) ≠ (𝐵 + 𝐵))
24	23	neneqd 2948	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (𝑀 + 𝑀) = (𝐵 + 𝐵))
25		oveq12 7286	. . . . . . . . . . . 12 ⊢ ((𝑀 = 𝐵 ∧ 𝑀 = 𝐵) → (𝑀 + 𝑀) = (𝐵 + 𝐵))
26	25	anidms 567	. . . . . . . . . . 11 ⊢ (𝑀 = 𝐵 → (𝑀 + 𝑀) = (𝐵 + 𝐵))
27	24, 26	nsyl 140	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑀 = 𝐵)
28	27	neqned 2950	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ≠ 𝐵)
29	28	necomd 2999	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ≠ 𝑀)
30	6, 9, 29	subne0d 11339	. . . . . . 7 ⊢ (𝜑 → (𝐵 − 𝑀) ≠ 0)
31	2, 10, 12, 13, 30	angcld 25953	. . . . . 6 ⊢ (𝜑 → ((𝑄 − 𝑀)𝐹(𝐵 − 𝑀)) ∈ (-π(,]π))
32	1, 31	sselid 3920	. . . . 5 ⊢ (𝜑 → ((𝑄 − 𝑀)𝐹(𝐵 − 𝑀)) ∈ ℝ)
33	32	recnd 11001	. . . 4 ⊢ (𝜑 → ((𝑄 − 𝑀)𝐹(𝐵 − 𝑀)) ∈ ℂ)
34	33	coscld 15838	. . 3 ⊢ (𝜑 → (cos‘((𝑄 − 𝑀)𝐹(𝐵 − 𝑀))) ∈ ℂ)
35	6, 9	negsubdi2d 11346	. . . . . . 7 ⊢ (𝜑 → -(𝐵 − 𝑀) = (𝑀 − 𝐵))
36	9, 9, 5, 6	addsubeq4d 11381	. . . . . . . 8 ⊢ (𝜑 → ((𝑀 + 𝑀) = (𝐴 + 𝐵) ↔ (𝐴 − 𝑀) = (𝑀 − 𝐵)))
37	20, 36	mpbid 231	. . . . . . 7 ⊢ (𝜑 → (𝐴 − 𝑀) = (𝑀 − 𝐵))
38	35, 37	eqtr4d 2781	. . . . . 6 ⊢ (𝜑 → -(𝐵 − 𝑀) = (𝐴 − 𝑀))
39	38	oveq2d 7293	. . . . 5 ⊢ (𝜑 → ((𝑄 − 𝑀)𝐹-(𝐵 − 𝑀)) = ((𝑄 − 𝑀)𝐹(𝐴 − 𝑀)))
40	39	fveq2d 6780	. . . 4 ⊢ (𝜑 → (cos‘((𝑄 − 𝑀)𝐹-(𝐵 − 𝑀))) = (cos‘((𝑄 − 𝑀)𝐹(𝐴 − 𝑀))))
41	2, 10, 12, 13, 30	cosangneg2d 25955	. . . 4 ⊢ (𝜑 → (cos‘((𝑄 − 𝑀)𝐹-(𝐵 − 𝑀))) = -(cos‘((𝑄 − 𝑀)𝐹(𝐵 − 𝑀))))
42	5, 5, 6, 21	addneintrd 11180	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 + 𝐴) ≠ (𝐴 + 𝐵))
43	42, 20	neeqtrrd 3018	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 + 𝐴) ≠ (𝑀 + 𝑀))
44	43	necomd 2999	. . . . . . . 8 ⊢ (𝜑 → (𝑀 + 𝑀) ≠ (𝐴 + 𝐴))
45	44	neneqd 2948	. . . . . . 7 ⊢ (𝜑 → ¬ (𝑀 + 𝑀) = (𝐴 + 𝐴))
46		oveq12 7286	. . . . . . . 8 ⊢ ((𝑀 = 𝐴 ∧ 𝑀 = 𝐴) → (𝑀 + 𝑀) = (𝐴 + 𝐴))
47	46	anidms 567	. . . . . . 7 ⊢ (𝑀 = 𝐴 → (𝑀 + 𝑀) = (𝐴 + 𝐴))
48	45, 47	nsyl 140	. . . . . 6 ⊢ (𝜑 → ¬ 𝑀 = 𝐴)
49	48	neqned 2950	. . . . 5 ⊢ (𝜑 → 𝑀 ≠ 𝐴)
50		eqidd 2739	. . . . 5 ⊢ (𝜑 → (abs‘(𝑄 − 𝑀)) = (abs‘(𝑄 − 𝑀)))
51	5, 9	subcld 11330	. . . . . . 7 ⊢ (𝜑 → (𝐴 − 𝑀) ∈ ℂ)
52	51	absnegd 15159	. . . . . 6 ⊢ (𝜑 → (abs‘-(𝐴 − 𝑀)) = (abs‘(𝐴 − 𝑀)))
53	5, 9	negsubdi2d 11346	. . . . . . 7 ⊢ (𝜑 → -(𝐴 − 𝑀) = (𝑀 − 𝐴))
54	53	fveq2d 6780	. . . . . 6 ⊢ (𝜑 → (abs‘-(𝐴 − 𝑀)) = (abs‘(𝑀 − 𝐴)))
55	37	fveq2d 6780	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐴 − 𝑀)) = (abs‘(𝑀 − 𝐵)))
56	52, 54, 55	3eqtr3d 2786	. . . . 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 25970	. . . 4 ⊢ (𝜑 → (cos‘((𝑄 − 𝑀)𝐹(𝐴 − 𝑀))) = (cos‘((𝑄 − 𝑀)𝐹(𝐵 − 𝑀))))
59	40, 41, 58	3eqtr3rd 2787	. . 3 ⊢ (𝜑 → (cos‘((𝑄 − 𝑀)𝐹(𝐵 − 𝑀))) = -(cos‘((𝑄 − 𝑀)𝐹(𝐵 − 𝑀))))
60	34, 59	eqnegad 11695	. 2 ⊢ (𝜑 → (cos‘((𝑄 − 𝑀)𝐹(𝐵 − 𝑀))) = 0)
61		coseq0negpitopi 25658	. . 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 231	1 ⊢ (𝜑 → ((𝑄 − 𝑀)𝐹(𝐵 − 𝑀)) ∈ {(π / 2), -(π / 2)})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∖ cdif 3885 {csn 4563 {cpr 4565 ‘cfv 6435 (class class class)co 7277 ∈ cmpo 7279 ℂcc 10867 ℝcr 10868 0cc0 10869 + caddc 10872 · cmul 10874 − cmin 11203 -cneg 11204 / cdiv 11630 2c2 12026 (,]cioc 13078 ℑcim 14807 abscabs 14943 cosccos 15772 πcpi 15774 logclog 25708

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5211 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-inf2 9397 ax-cnex 10925 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-mulcom 10933 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 ax-pre-mulgt0 10946 ax-pre-sup 10947 ax-addf 10948 ax-mulf 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-tp 4568 df-op 4570 df-uni 4842 df-int 4882 df-iun 4928 df-iin 4929 df-br 5077 df-opab 5139 df-mpt 5160 df-tr 5194 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-se 5547 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6204 df-ord 6271 df-on 6272 df-lim 6273 df-suc 6274 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-isom 6444 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7976 df-frecs 8095 df-wrecs 8126 df-recs 8200 df-rdg 8239 df-1o 8295 df-2o 8296 df-er 8496 df-map 8615 df-pm 8616 df-ixp 8684 df-en 8732 df-dom 8733 df-sdom 8734 df-fin 8735 df-fsupp 9127 df-fi 9168 df-sup 9199 df-inf 9200 df-oi 9267 df-card 9695 df-pnf 11009 df-mnf 11010 df-xr 11011 df-ltxr 11012 df-le 11013 df-sub 11205 df-neg 11206 df-div 11631 df-nn 11972 df-2 12034 df-3 12035 df-4 12036 df-5 12037 df-6 12038 df-7 12039 df-8 12040 df-9 12041 df-n0 12232 df-z 12318 df-dec 12436 df-uz 12581 df-q 12687 df-rp 12729 df-xneg 12846 df-xadd 12847 df-xmul 12848 df-ioo 13081 df-ioc 13082 df-ico 13083 df-icc 13084 df-fz 13238 df-fzo 13381 df-fl 13510 df-mod 13588 df-seq 13720 df-exp 13781 df-fac 13986 df-bc 14015 df-hash 14043 df-shft 14776 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-limsup 15178 df-clim 15195 df-rlim 15196 df-sum 15396 df-ef 15775 df-sin 15777 df-cos 15778 df-pi 15780 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-starv 16975 df-sca 16976 df-vsca 16977 df-ip 16978 df-tset 16979 df-ple 16980 df-ds 16982 df-unif 16983 df-hom 16984 df-cco 16985 df-rest 17131 df-topn 17132 df-0g 17150 df-gsum 17151 df-topgen 17152 df-pt 17153 df-prds 17156 df-xrs 17211 df-qtop 17216 df-imas 17217 df-xps 17219 df-mre 17293 df-mrc 17294 df-acs 17296 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-submnd 18429 df-mulg 18699 df-cntz 18921 df-cmn 19386 df-psmet 20587 df-xmet 20588 df-met 20589 df-bl 20590 df-mopn 20591 df-fbas 20592 df-fg 20593 df-cnfld 20596 df-top 22041 df-topon 22058 df-topsp 22080 df-bases 22094 df-cld 22168 df-ntr 22169 df-cls 22170 df-nei 22247 df-lp 22285 df-perf 22286 df-cn 22376 df-cnp 22377 df-haus 22464 df-tx 22711 df-hmeo 22904 df-fil 22995 df-fm 23087 df-flim 23088 df-flf 23089 df-xms 23471 df-ms 23472 df-tms 23473 df-cncf 24039 df-limc 25028 df-dv 25029 df-log 25710

This theorem is referenced by: chordthmlem2 25981

W3C validator