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Mirrors > Home > MPE Home > Th. List > chordthmlem2 | Structured version Visualization version GIF version |
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 25404, where P = B, and using angrtmuld 25380 to observe that QMP is right iff QMB is. (Contributed by David Moews, 28-Feb-2017.) |
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) |
Ref | Expression |
---|---|
chordthmlem2 | ⊢ (𝜑 → ((𝑄 − 𝑀)𝐹(𝑃 − 𝑀)) ∈ {(π / 2), -(π / 2)}) |
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 11705 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ | |
8 | 7 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℝ) |
9 | 2ne0 11735 | . . . . . . . . . 10 ⊢ 2 ≠ 0 | |
10 | 9 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → 2 ≠ 0) |
11 | 8, 10 | rereccld 11461 | . . . . . . . 8 ⊢ (𝜑 → (1 / 2) ∈ ℝ) |
12 | chordthmlem2.X | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℝ) | |
13 | 11, 12 | resubcld 11062 | . . . . . . 7 ⊢ (𝜑 → ((1 / 2) − 𝑋) ∈ ℝ) |
14 | 13 | recnd 10663 | . . . . . 6 ⊢ (𝜑 → ((1 / 2) − 𝑋) ∈ ℂ) |
15 | 3, 2 | subcld 10991 | . . . . . 6 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℂ) |
16 | 11 | recnd 10663 | . . . . . . . . 9 ⊢ (𝜑 → (1 / 2) ∈ ℂ) |
17 | 12 | recnd 10663 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
18 | 16, 17, 15 | subdird 11091 | . . . . . . . 8 ⊢ (𝜑 → (((1 / 2) − 𝑋) · (𝐵 − 𝐴)) = (((1 / 2) · (𝐵 − 𝐴)) − (𝑋 · (𝐵 − 𝐴)))) |
19 | 2cnd 11709 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ∈ ℂ) | |
20 | 3, 19, 10 | divcan4d 11416 | . . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐵 · 2) / 2) = 𝐵) |
21 | 3 | times2d 11875 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 · 2) = (𝐵 + 𝐵)) |
22 | 21 | oveq1d 7165 | . . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐵 · 2) / 2) = ((𝐵 + 𝐵) / 2)) |
23 | 20, 22 | eqtr3d 2858 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 = ((𝐵 + 𝐵) / 2)) |
24 | 23, 5 | oveq12d 7168 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 − 𝑀) = (((𝐵 + 𝐵) / 2) − ((𝐴 + 𝐵) / 2))) |
25 | 3, 3 | addcld 10654 | . . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 + 𝐵) ∈ ℂ) |
26 | 2, 3 | addcld 10654 | . . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ) |
27 | 25, 26, 19, 10 | divsubdird 11449 | . . . . . . . . . . 11 ⊢ (𝜑 → (((𝐵 + 𝐵) − (𝐴 + 𝐵)) / 2) = (((𝐵 + 𝐵) / 2) − ((𝐴 + 𝐵) / 2))) |
28 | 3, 2, 3 | pnpcan2d 11029 | . . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐵 + 𝐵) − (𝐴 + 𝐵)) = (𝐵 − 𝐴)) |
29 | 28 | oveq1d 7165 | . . . . . . . . . . 11 ⊢ (𝜑 → (((𝐵 + 𝐵) − (𝐴 + 𝐵)) / 2) = ((𝐵 − 𝐴) / 2)) |
30 | 24, 27, 29 | 3eqtr2d 2862 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐵 − 𝑀) = ((𝐵 − 𝐴) / 2)) |
31 | 15, 19, 10 | divrec2d 11414 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝐵 − 𝐴) / 2) = ((1 / 2) · (𝐵 − 𝐴))) |
32 | 30, 31 | eqtrd 2856 | . . . . . . . . 9 ⊢ (𝜑 → (𝐵 − 𝑀) = ((1 / 2) · (𝐵 − 𝐴))) |
33 | chordthmlem2.P | . . . . . . . . . 10 ⊢ (𝜑 → 𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵))) | |
34 | 17, 2 | mulcld 10655 | . . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 · 𝐴) ∈ ℂ) |
35 | 1cnd 10630 | . . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℂ) | |
36 | 35, 17 | subcld 10991 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → (1 − 𝑋) ∈ ℂ) |
37 | 36, 3 | mulcld 10655 | . . . . . . . . . . . . 13 ⊢ (𝜑 → ((1 − 𝑋) · 𝐵) ∈ ℂ) |
38 | 34, 37 | addcld 10654 | . . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ∈ ℂ) |
39 | 33, 38 | eqeltrd 2913 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
40 | 2, 39, 3, 17 | affineequiv 25395 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ↔ (𝐵 − 𝑃) = (𝑋 · (𝐵 − 𝐴)))) |
41 | 33, 40 | mpbid 234 | . . . . . . . . 9 ⊢ (𝜑 → (𝐵 − 𝑃) = (𝑋 · (𝐵 − 𝐴))) |
42 | 32, 41 | oveq12d 7168 | . . . . . . . 8 ⊢ (𝜑 → ((𝐵 − 𝑀) − (𝐵 − 𝑃)) = (((1 / 2) · (𝐵 − 𝐴)) − (𝑋 · (𝐵 − 𝐴)))) |
43 | 26 | halfcld 11876 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℂ) |
44 | 5, 43 | eqeltrd 2913 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
45 | 3, 44, 39 | nnncan1d 11025 | . . . . . . . 8 ⊢ (𝜑 → ((𝐵 − 𝑀) − (𝐵 − 𝑃)) = (𝑃 − 𝑀)) |
46 | 18, 42, 45 | 3eqtr2rd 2863 | . . . . . . 7 ⊢ (𝜑 → (𝑃 − 𝑀) = (((1 / 2) − 𝑋) · (𝐵 − 𝐴))) |
47 | chordthmlem2.PneM | . . . . . . . 8 ⊢ (𝜑 → 𝑃 ≠ 𝑀) | |
48 | 39, 44, 47 | subne0d 11000 | . . . . . . 7 ⊢ (𝜑 → (𝑃 − 𝑀) ≠ 0) |
49 | 46, 48 | eqnetrrd 3084 | . . . . . 6 ⊢ (𝜑 → (((1 / 2) − 𝑋) · (𝐵 − 𝐴)) ≠ 0) |
50 | 14, 15, 49 | mulne0bbd 11290 | . . . . 5 ⊢ (𝜑 → (𝐵 − 𝐴) ≠ 0) |
51 | 3, 2, 50 | subne0ad 11002 | . . . 4 ⊢ (𝜑 → 𝐵 ≠ 𝐴) |
52 | 51 | necomd 3071 | . . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
53 | chordthmlem2.QneM | . . 3 ⊢ (𝜑 → 𝑄 ≠ 𝑀) | |
54 | 1, 2, 3, 4, 5, 6, 52, 53 | chordthmlem 25404 | . 2 ⊢ (𝜑 → ((𝑄 − 𝑀)𝐹(𝐵 − 𝑀)) ∈ {(π / 2), -(π / 2)}) |
55 | 4, 44 | subcld 10991 | . . 3 ⊢ (𝜑 → (𝑄 − 𝑀) ∈ ℂ) |
56 | 39, 44 | subcld 10991 | . . 3 ⊢ (𝜑 → (𝑃 − 𝑀) ∈ ℂ) |
57 | 3, 44 | subcld 10991 | . . 3 ⊢ (𝜑 → (𝐵 − 𝑀) ∈ ℂ) |
58 | 4, 44, 53 | subne0d 11000 | . . 3 ⊢ (𝜑 → (𝑄 − 𝑀) ≠ 0) |
59 | 19, 10 | recne0d 11404 | . . . . 5 ⊢ (𝜑 → (1 / 2) ≠ 0) |
60 | 16, 15, 59, 50 | mulne0d 11286 | . . . 4 ⊢ (𝜑 → ((1 / 2) · (𝐵 − 𝐴)) ≠ 0) |
61 | 32, 60 | eqnetrd 3083 | . . 3 ⊢ (𝜑 → (𝐵 − 𝑀) ≠ 0) |
62 | 32, 46 | oveq12d 7168 | . . . . 5 ⊢ (𝜑 → ((𝐵 − 𝑀) / (𝑃 − 𝑀)) = (((1 / 2) · (𝐵 − 𝐴)) / (((1 / 2) − 𝑋) · (𝐵 − 𝐴)))) |
63 | 14, 15, 49 | mulne0bad 11289 | . . . . . 6 ⊢ (𝜑 → ((1 / 2) − 𝑋) ≠ 0) |
64 | 16, 14, 15, 63, 50 | divcan5rd 11437 | . . . . 5 ⊢ (𝜑 → (((1 / 2) · (𝐵 − 𝐴)) / (((1 / 2) − 𝑋) · (𝐵 − 𝐴))) = ((1 / 2) / ((1 / 2) − 𝑋))) |
65 | 62, 64 | eqtrd 2856 | . . . 4 ⊢ (𝜑 → ((𝐵 − 𝑀) / (𝑃 − 𝑀)) = ((1 / 2) / ((1 / 2) − 𝑋))) |
66 | 11, 13, 63 | redivcld 11462 | . . . 4 ⊢ (𝜑 → ((1 / 2) / ((1 / 2) − 𝑋)) ∈ ℝ) |
67 | 65, 66 | eqeltrd 2913 | . . 3 ⊢ (𝜑 → ((𝐵 − 𝑀) / (𝑃 − 𝑀)) ∈ ℝ) |
68 | 1, 55, 56, 57, 58, 48, 61, 67 | angrtmuld 25380 | . 2 ⊢ (𝜑 → (((𝑄 − 𝑀)𝐹(𝑃 − 𝑀)) ∈ {(π / 2), -(π / 2)} ↔ ((𝑄 − 𝑀)𝐹(𝐵 − 𝑀)) ∈ {(π / 2), -(π / 2)})) |
69 | 54, 68 | mpbird 259 | 1 ⊢ (𝜑 → ((𝑄 − 𝑀)𝐹(𝑃 − 𝑀)) ∈ {(π / 2), -(π / 2)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∖ cdif 3932 {csn 4560 {cpr 4562 ‘cfv 6349 (class class class)co 7150 ∈ cmpo 7152 ℂcc 10529 ℝcr 10530 0cc0 10531 1c1 10532 + caddc 10534 · cmul 10536 − cmin 10864 -cneg 10865 / cdiv 11291 2c2 11686 ℑcim 14451 abscabs 14587 πcpi 15414 logclog 25132 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ioc 12737 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-mod 13232 df-seq 13364 df-exp 13424 df-fac 13628 df-bc 13657 df-hash 13685 df-shft 14420 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-limsup 14822 df-clim 14839 df-rlim 14840 df-sum 15037 df-ef 15415 df-sin 15417 df-cos 15418 df-pi 15420 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-rest 16690 df-topn 16691 df-0g 16709 df-gsum 16710 df-topgen 16711 df-pt 16712 df-prds 16715 df-xrs 16769 df-qtop 16774 df-imas 16775 df-xps 16777 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-mulg 18219 df-cntz 18441 df-cmn 18902 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-fbas 20536 df-fg 20537 df-cnfld 20540 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-cld 21621 df-ntr 21622 df-cls 21623 df-nei 21700 df-lp 21738 df-perf 21739 df-cn 21829 df-cnp 21830 df-haus 21917 df-tx 22164 df-hmeo 22357 df-fil 22448 df-fm 22540 df-flim 22541 df-flf 22542 df-xms 22924 df-ms 22925 df-tms 22926 df-cncf 23480 df-limc 24458 df-dv 24459 df-log 25134 |
This theorem is referenced by: chordthmlem3 25406 |
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