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 27048

Description: If 𝑀 is the midpoint of AB and AQ = BQ, then QMB is a right angle. The proof uses ssscongptld 27038 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 26756	. . . . . 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 11243	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ)
8	7	halfcld 12504	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℂ)
9	4, 8	eqeltrd 2865	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℂ)
10	3, 9	subcld 11584	. . . . . . 7 ⊢ (𝜑 → (𝑄 − 𝑀) ∈ ℂ)
11		chordthmlem.QneM	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ≠ 𝑀)
12	3, 9, 11	subne0d 11593	. . . . . . 7 ⊢ (𝜑 → (𝑄 − 𝑀) ≠ 0)
13	6, 9	subcld 11584	. . . . . . 7 ⊢ (𝜑 → (𝐵 − 𝑀) ∈ ℂ)
14	4	oveq1d 7434	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀 · 2) = (((𝐴 + 𝐵) / 2) · 2))
15	9	times2d 12503	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀 · 2) = (𝑀 + 𝑀))
16		2cnd 12334	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ∈ ℂ)
17		2ne0 12362	. . . . . . . . . . . . . . . 16 ⊢ 2 ≠ 0
18	17	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ≠ 0)
19	7, 16, 18	divcan1d 12007	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((𝐴 + 𝐵) / 2) · 2) = (𝐴 + 𝐵))
20	14, 15, 19	3eqtr3d 2808	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑀 + 𝑀) = (𝐴 + 𝐵))
21		chordthmlem.AneB	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
22	5, 6, 6, 21	addneintr2d 11433	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 + 𝐵) ≠ (𝐵 + 𝐵))
23	20, 22	eqnetrd 3027	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑀 + 𝑀) ≠ (𝐵 + 𝐵))
24	23	neneqd 2965	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (𝑀 + 𝑀) = (𝐵 + 𝐵))
25		oveq12 7428	. . . . . . . . . . . 12 ⊢ ((𝑀 = 𝐵 ∧ 𝑀 = 𝐵) → (𝑀 + 𝑀) = (𝐵 + 𝐵))
26	25	anidms 577	. . . . . . . . . . 11 ⊢ (𝑀 = 𝐵 → (𝑀 + 𝑀) = (𝐵 + 𝐵))
27	24, 26	nsyl 141	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑀 = 𝐵)
28	27	neqned 2967	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ≠ 𝐵)
29	28	necomd 3015	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ≠ 𝑀)
30	6, 9, 29	subne0d 11593	. . . . . . 7 ⊢ (𝜑 → (𝐵 − 𝑀) ≠ 0)
31	2, 10, 12, 13, 30	angcld 27021	. . . . . 6 ⊢ (𝜑 → ((𝑄 − 𝑀)𝐹(𝐵 − 𝑀)) ∈ (-π(,]π))
32	1, 31	sselid 3936	. . . . 5 ⊢ (𝜑 → ((𝑄 − 𝑀)𝐹(𝐵 − 𝑀)) ∈ ℝ)
33	32	recnd 11252	. . . 4 ⊢ (𝜑 → ((𝑄 − 𝑀)𝐹(𝐵 − 𝑀)) ∈ ℂ)
34	33	coscld 16209	. . 3 ⊢ (𝜑 → (cos‘((𝑄 − 𝑀)𝐹(𝐵 − 𝑀))) ∈ ℂ)
35	6, 9	negsubdi2d 11600	. . . . . . 7 ⊢ (𝜑 → -(𝐵 − 𝑀) = (𝑀 − 𝐵))
36	9, 9, 5, 6	addsubeq4d 11635	. . . . . . . 8 ⊢ (𝜑 → ((𝑀 + 𝑀) = (𝐴 + 𝐵) ↔ (𝐴 − 𝑀) = (𝑀 − 𝐵)))
37	20, 36	mpbid 235	. . . . . . 7 ⊢ (𝜑 → (𝐴 − 𝑀) = (𝑀 − 𝐵))
38	35, 37	eqtr4d 2803	. . . . . 6 ⊢ (𝜑 → -(𝐵 − 𝑀) = (𝐴 − 𝑀))
39	38	oveq2d 7435	. . . . 5 ⊢ (𝜑 → ((𝑄 − 𝑀)𝐹-(𝐵 − 𝑀)) = ((𝑄 − 𝑀)𝐹(𝐴 − 𝑀)))
40	39	fveq2d 6889	. . . 4 ⊢ (𝜑 → (cos‘((𝑄 − 𝑀)𝐹-(𝐵 − 𝑀))) = (cos‘((𝑄 − 𝑀)𝐹(𝐴 − 𝑀))))
41	2, 10, 12, 13, 30	cosangneg2d 27023	. . . 4 ⊢ (𝜑 → (cos‘((𝑄 − 𝑀)𝐹-(𝐵 − 𝑀))) = -(cos‘((𝑄 − 𝑀)𝐹(𝐵 − 𝑀))))
42	5, 5, 6, 21	addneintrd 11432	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 + 𝐴) ≠ (𝐴 + 𝐵))
43	42, 20	neeqtrrd 3034	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 + 𝐴) ≠ (𝑀 + 𝑀))
44	43	necomd 3015	. . . . . . . 8 ⊢ (𝜑 → (𝑀 + 𝑀) ≠ (𝐴 + 𝐴))
45	44	neneqd 2965	. . . . . . 7 ⊢ (𝜑 → ¬ (𝑀 + 𝑀) = (𝐴 + 𝐴))
46		oveq12 7428	. . . . . . . 8 ⊢ ((𝑀 = 𝐴 ∧ 𝑀 = 𝐴) → (𝑀 + 𝑀) = (𝐴 + 𝐴))
47	46	anidms 577	. . . . . . 7 ⊢ (𝑀 = 𝐴 → (𝑀 + 𝑀) = (𝐴 + 𝐴))
48	45, 47	nsyl 141	. . . . . 6 ⊢ (𝜑 → ¬ 𝑀 = 𝐴)
49	48	neqned 2967	. . . . 5 ⊢ (𝜑 → 𝑀 ≠ 𝐴)
50		eqidd 2766	. . . . 5 ⊢ (𝜑 → (abs‘(𝑄 − 𝑀)) = (abs‘(𝑄 − 𝑀)))
51	5, 9	subcld 11584	. . . . . . 7 ⊢ (𝜑 → (𝐴 − 𝑀) ∈ ℂ)
52	51	absnegd 15527	. . . . . 6 ⊢ (𝜑 → (abs‘-(𝐴 − 𝑀)) = (abs‘(𝐴 − 𝑀)))
53	5, 9	negsubdi2d 11600	. . . . . . 7 ⊢ (𝜑 → -(𝐴 − 𝑀) = (𝑀 − 𝐴))
54	53	fveq2d 6889	. . . . . 6 ⊢ (𝜑 → (abs‘-(𝐴 − 𝑀)) = (abs‘(𝑀 − 𝐴)))
55	37	fveq2d 6889	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐴 − 𝑀)) = (abs‘(𝑀 − 𝐵)))
56	52, 54, 55	3eqtr3d 2808	. . . . 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 27038	. . . 4 ⊢ (𝜑 → (cos‘((𝑄 − 𝑀)𝐹(𝐴 − 𝑀))) = (cos‘((𝑄 − 𝑀)𝐹(𝐵 − 𝑀))))
59	40, 41, 58	3eqtr3rd 2809	. . 3 ⊢ (𝜑 → (cos‘((𝑄 − 𝑀)𝐹(𝐵 − 𝑀))) = -(cos‘((𝑄 − 𝑀)𝐹(𝐵 − 𝑀))))
60	34, 59	eqnegad 11952	. 2 ⊢ (𝜑 → (cos‘((𝑄 − 𝑀)𝐹(𝐵 − 𝑀))) = 0)
61		coseq0negpitopi 26719	. . 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 18	. 2 ⊢ (𝜑 → ((cos‘((𝑄 − 𝑀)𝐹(𝐵 − 𝑀))) = 0 ↔ ((𝑄 − 𝑀)𝐹(𝐵 − 𝑀)) ∈ {(π / 2), -(π / 2)}))
63	60, 62	mpbid 235	1 ⊢ (𝜑 → ((𝑄 − 𝑀)𝐹(𝐵 − 𝑀)) ∈ {(π / 2), -(π / 2)})

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 = 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 + caddc 11118 · cmul 11120 − cmin 11456 -cneg 11457 / cdiv 11886 2c2 12310 (,]cioc 13389 ℑcim 15173 abscabs 15309 cosccos 16140 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: chordthmlem2 27049

W3C validator