Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 24559, where P = B, and using angrtmuld 24538 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 11090 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ | |
8 | 7 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℝ) |
9 | 2ne0 11113 | . . . . . . . . . 10 ⊢ 2 ≠ 0 | |
10 | 9 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → 2 ≠ 0) |
11 | 8, 10 | rereccld 10852 | . . . . . . . 8 ⊢ (𝜑 → (1 / 2) ∈ ℝ) |
12 | chordthmlem2.X | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℝ) | |
13 | 11, 12 | resubcld 10458 | . . . . . . 7 ⊢ (𝜑 → ((1 / 2) − 𝑋) ∈ ℝ) |
14 | 13 | recnd 10068 | . . . . . 6 ⊢ (𝜑 → ((1 / 2) − 𝑋) ∈ ℂ) |
15 | 3, 2 | subcld 10392 | . . . . . 6 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℂ) |
16 | 11 | recnd 10068 | . . . . . . . . 9 ⊢ (𝜑 → (1 / 2) ∈ ℂ) |
17 | 12 | recnd 10068 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
18 | 16, 17, 15 | subdird 10487 | . . . . . . . 8 ⊢ (𝜑 → (((1 / 2) − 𝑋) · (𝐵 − 𝐴)) = (((1 / 2) · (𝐵 − 𝐴)) − (𝑋 · (𝐵 − 𝐴)))) |
19 | 2cnd 11093 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ∈ ℂ) | |
20 | 3, 19, 10 | divcan4d 10807 | . . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐵 · 2) / 2) = 𝐵) |
21 | 3 | times2d 11276 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 · 2) = (𝐵 + 𝐵)) |
22 | 21 | oveq1d 6665 | . . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐵 · 2) / 2) = ((𝐵 + 𝐵) / 2)) |
23 | 20, 22 | eqtr3d 2658 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 = ((𝐵 + 𝐵) / 2)) |
24 | 23, 5 | oveq12d 6668 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 − 𝑀) = (((𝐵 + 𝐵) / 2) − ((𝐴 + 𝐵) / 2))) |
25 | 3, 3 | addcld 10059 | . . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 + 𝐵) ∈ ℂ) |
26 | 2, 3 | addcld 10059 | . . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ) |
27 | 25, 26, 19, 10 | divsubdird 10840 | . . . . . . . . . . 11 ⊢ (𝜑 → (((𝐵 + 𝐵) − (𝐴 + 𝐵)) / 2) = (((𝐵 + 𝐵) / 2) − ((𝐴 + 𝐵) / 2))) |
28 | 3, 2, 3 | pnpcan2d 10430 | . . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐵 + 𝐵) − (𝐴 + 𝐵)) = (𝐵 − 𝐴)) |
29 | 28 | oveq1d 6665 | . . . . . . . . . . 11 ⊢ (𝜑 → (((𝐵 + 𝐵) − (𝐴 + 𝐵)) / 2) = ((𝐵 − 𝐴) / 2)) |
30 | 24, 27, 29 | 3eqtr2d 2662 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐵 − 𝑀) = ((𝐵 − 𝐴) / 2)) |
31 | 15, 19, 10 | divrec2d 10805 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝐵 − 𝐴) / 2) = ((1 / 2) · (𝐵 − 𝐴))) |
32 | 30, 31 | eqtrd 2656 | . . . . . . . . 9 ⊢ (𝜑 → (𝐵 − 𝑀) = ((1 / 2) · (𝐵 − 𝐴))) |
33 | chordthmlem2.P | . . . . . . . . . 10 ⊢ (𝜑 → 𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵))) | |
34 | 17, 2 | mulcld 10060 | . . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 · 𝐴) ∈ ℂ) |
35 | 1cnd 10056 | . . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℂ) | |
36 | 35, 17 | subcld 10392 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → (1 − 𝑋) ∈ ℂ) |
37 | 36, 3 | mulcld 10060 | . . . . . . . . . . . . 13 ⊢ (𝜑 → ((1 − 𝑋) · 𝐵) ∈ ℂ) |
38 | 34, 37 | addcld 10059 | . . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ∈ ℂ) |
39 | 33, 38 | eqeltrd 2701 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
40 | 2, 39, 3, 17 | affineequiv 24553 | . . . . . . . . . 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 6668 | . . . . . . . 8 ⊢ (𝜑 → ((𝐵 − 𝑀) − (𝐵 − 𝑃)) = (((1 / 2) · (𝐵 − 𝐴)) − (𝑋 · (𝐵 − 𝐴)))) |
43 | 26 | halfcld 11277 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℂ) |
44 | 5, 43 | eqeltrd 2701 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
45 | 3, 44, 39 | nnncan1d 10426 | . . . . . . . 8 ⊢ (𝜑 → ((𝐵 − 𝑀) − (𝐵 − 𝑃)) = (𝑃 − 𝑀)) |
46 | 18, 42, 45 | 3eqtr2rd 2663 | . . . . . . 7 ⊢ (𝜑 → (𝑃 − 𝑀) = (((1 / 2) − 𝑋) · (𝐵 − 𝐴))) |
47 | chordthmlem2.PneM | . . . . . . . 8 ⊢ (𝜑 → 𝑃 ≠ 𝑀) | |
48 | 39, 44, 47 | subne0d 10401 | . . . . . . 7 ⊢ (𝜑 → (𝑃 − 𝑀) ≠ 0) |
49 | 46, 48 | eqnetrrd 2862 | . . . . . 6 ⊢ (𝜑 → (((1 / 2) − 𝑋) · (𝐵 − 𝐴)) ≠ 0) |
50 | 14, 15, 49 | mulne0bbd 10683 | . . . . 5 ⊢ (𝜑 → (𝐵 − 𝐴) ≠ 0) |
51 | 3, 2, 50 | subne0ad 10403 | . . . 4 ⊢ (𝜑 → 𝐵 ≠ 𝐴) |
52 | 51 | necomd 2849 | . . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
53 | chordthmlem2.QneM | . . 3 ⊢ (𝜑 → 𝑄 ≠ 𝑀) | |
54 | 1, 2, 3, 4, 5, 6, 52, 53 | chordthmlem 24559 | . 2 ⊢ (𝜑 → ((𝑄 − 𝑀)𝐹(𝐵 − 𝑀)) ∈ {(π / 2), -(π / 2)}) |
55 | 4, 44 | subcld 10392 | . . 3 ⊢ (𝜑 → (𝑄 − 𝑀) ∈ ℂ) |
56 | 39, 44 | subcld 10392 | . . 3 ⊢ (𝜑 → (𝑃 − 𝑀) ∈ ℂ) |
57 | 3, 44 | subcld 10392 | . . 3 ⊢ (𝜑 → (𝐵 − 𝑀) ∈ ℂ) |
58 | 4, 44, 53 | subne0d 10401 | . . 3 ⊢ (𝜑 → (𝑄 − 𝑀) ≠ 0) |
59 | 19, 10 | recne0d 10795 | . . . . 5 ⊢ (𝜑 → (1 / 2) ≠ 0) |
60 | 16, 15, 59, 50 | mulne0d 10679 | . . . 4 ⊢ (𝜑 → ((1 / 2) · (𝐵 − 𝐴)) ≠ 0) |
61 | 32, 60 | eqnetrd 2861 | . . 3 ⊢ (𝜑 → (𝐵 − 𝑀) ≠ 0) |
62 | 32, 46 | oveq12d 6668 | . . . . 5 ⊢ (𝜑 → ((𝐵 − 𝑀) / (𝑃 − 𝑀)) = (((1 / 2) · (𝐵 − 𝐴)) / (((1 / 2) − 𝑋) · (𝐵 − 𝐴)))) |
63 | 14, 15, 49 | mulne0bad 10682 | . . . . . 6 ⊢ (𝜑 → ((1 / 2) − 𝑋) ≠ 0) |
64 | 16, 14, 15, 63, 50 | divcan5rd 10828 | . . . . 5 ⊢ (𝜑 → (((1 / 2) · (𝐵 − 𝐴)) / (((1 / 2) − 𝑋) · (𝐵 − 𝐴))) = ((1 / 2) / ((1 / 2) − 𝑋))) |
65 | 62, 64 | eqtrd 2656 | . . . 4 ⊢ (𝜑 → ((𝐵 − 𝑀) / (𝑃 − 𝑀)) = ((1 / 2) / ((1 / 2) − 𝑋))) |
66 | 11, 13, 63 | redivcld 10853 | . . . 4 ⊢ (𝜑 → ((1 / 2) / ((1 / 2) − 𝑋)) ∈ ℝ) |
67 | 65, 66 | eqeltrd 2701 | . . 3 ⊢ (𝜑 → ((𝐵 − 𝑀) / (𝑃 − 𝑀)) ∈ ℝ) |
68 | 1, 55, 56, 57, 58, 48, 61, 67 | angrtmuld 24538 | . 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 1483 ∈ wcel 1990 ≠ wne 2794 ∖ cdif 3571 {csn 4177 {cpr 4179 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 − cmin 10266 -cneg 10267 / cdiv 10684 2c2 11070 ℑcim 13838 abscabs 13974 πcpi 14797 logclog 24301 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-sin 14800 df-cos 14801 df-pi 14803 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631 df-log 24303 |
This theorem is referenced by: chordthmlem3 24561 |
Copyright terms: Public domain | W3C validator |