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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 26182, where P = B, and using angrtmuld 26158 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 12227 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ | |
8 | 7 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℝ) |
9 | 2ne0 12257 | . . . . . . . . . 10 ⊢ 2 ≠ 0 | |
10 | 9 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → 2 ≠ 0) |
11 | 8, 10 | rereccld 11982 | . . . . . . . 8 ⊢ (𝜑 → (1 / 2) ∈ ℝ) |
12 | chordthmlem2.X | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℝ) | |
13 | 11, 12 | resubcld 11583 | . . . . . . 7 ⊢ (𝜑 → ((1 / 2) − 𝑋) ∈ ℝ) |
14 | 13 | recnd 11183 | . . . . . 6 ⊢ (𝜑 → ((1 / 2) − 𝑋) ∈ ℂ) |
15 | 3, 2 | subcld 11512 | . . . . . 6 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℂ) |
16 | 11 | recnd 11183 | . . . . . . . . 9 ⊢ (𝜑 → (1 / 2) ∈ ℂ) |
17 | 12 | recnd 11183 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
18 | 16, 17, 15 | subdird 11612 | . . . . . . . 8 ⊢ (𝜑 → (((1 / 2) − 𝑋) · (𝐵 − 𝐴)) = (((1 / 2) · (𝐵 − 𝐴)) − (𝑋 · (𝐵 − 𝐴)))) |
19 | 2cnd 12231 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ∈ ℂ) | |
20 | 3, 19, 10 | divcan4d 11937 | . . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐵 · 2) / 2) = 𝐵) |
21 | 3 | times2d 12397 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 · 2) = (𝐵 + 𝐵)) |
22 | 21 | oveq1d 7372 | . . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐵 · 2) / 2) = ((𝐵 + 𝐵) / 2)) |
23 | 20, 22 | eqtr3d 2778 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 = ((𝐵 + 𝐵) / 2)) |
24 | 23, 5 | oveq12d 7375 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 − 𝑀) = (((𝐵 + 𝐵) / 2) − ((𝐴 + 𝐵) / 2))) |
25 | 3, 3 | addcld 11174 | . . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 + 𝐵) ∈ ℂ) |
26 | 2, 3 | addcld 11174 | . . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ) |
27 | 25, 26, 19, 10 | divsubdird 11970 | . . . . . . . . . . 11 ⊢ (𝜑 → (((𝐵 + 𝐵) − (𝐴 + 𝐵)) / 2) = (((𝐵 + 𝐵) / 2) − ((𝐴 + 𝐵) / 2))) |
28 | 3, 2, 3 | pnpcan2d 11550 | . . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐵 + 𝐵) − (𝐴 + 𝐵)) = (𝐵 − 𝐴)) |
29 | 28 | oveq1d 7372 | . . . . . . . . . . 11 ⊢ (𝜑 → (((𝐵 + 𝐵) − (𝐴 + 𝐵)) / 2) = ((𝐵 − 𝐴) / 2)) |
30 | 24, 27, 29 | 3eqtr2d 2782 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐵 − 𝑀) = ((𝐵 − 𝐴) / 2)) |
31 | 15, 19, 10 | divrec2d 11935 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝐵 − 𝐴) / 2) = ((1 / 2) · (𝐵 − 𝐴))) |
32 | 30, 31 | eqtrd 2776 | . . . . . . . . 9 ⊢ (𝜑 → (𝐵 − 𝑀) = ((1 / 2) · (𝐵 − 𝐴))) |
33 | chordthmlem2.P | . . . . . . . . . 10 ⊢ (𝜑 → 𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵))) | |
34 | 17, 2 | mulcld 11175 | . . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 · 𝐴) ∈ ℂ) |
35 | 1cnd 11150 | . . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℂ) | |
36 | 35, 17 | subcld 11512 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → (1 − 𝑋) ∈ ℂ) |
37 | 36, 3 | mulcld 11175 | . . . . . . . . . . . . 13 ⊢ (𝜑 → ((1 − 𝑋) · 𝐵) ∈ ℂ) |
38 | 34, 37 | addcld 11174 | . . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ∈ ℂ) |
39 | 33, 38 | eqeltrd 2838 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
40 | 2, 39, 3, 17 | affineequiv 26173 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ↔ (𝐵 − 𝑃) = (𝑋 · (𝐵 − 𝐴)))) |
41 | 33, 40 | mpbid 231 | . . . . . . . . 9 ⊢ (𝜑 → (𝐵 − 𝑃) = (𝑋 · (𝐵 − 𝐴))) |
42 | 32, 41 | oveq12d 7375 | . . . . . . . 8 ⊢ (𝜑 → ((𝐵 − 𝑀) − (𝐵 − 𝑃)) = (((1 / 2) · (𝐵 − 𝐴)) − (𝑋 · (𝐵 − 𝐴)))) |
43 | 26 | halfcld 12398 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℂ) |
44 | 5, 43 | eqeltrd 2838 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
45 | 3, 44, 39 | nnncan1d 11546 | . . . . . . . 8 ⊢ (𝜑 → ((𝐵 − 𝑀) − (𝐵 − 𝑃)) = (𝑃 − 𝑀)) |
46 | 18, 42, 45 | 3eqtr2rd 2783 | . . . . . . 7 ⊢ (𝜑 → (𝑃 − 𝑀) = (((1 / 2) − 𝑋) · (𝐵 − 𝐴))) |
47 | chordthmlem2.PneM | . . . . . . . 8 ⊢ (𝜑 → 𝑃 ≠ 𝑀) | |
48 | 39, 44, 47 | subne0d 11521 | . . . . . . 7 ⊢ (𝜑 → (𝑃 − 𝑀) ≠ 0) |
49 | 46, 48 | eqnetrrd 3012 | . . . . . 6 ⊢ (𝜑 → (((1 / 2) − 𝑋) · (𝐵 − 𝐴)) ≠ 0) |
50 | 14, 15, 49 | mulne0bbd 11811 | . . . . 5 ⊢ (𝜑 → (𝐵 − 𝐴) ≠ 0) |
51 | 3, 2, 50 | subne0ad 11523 | . . . 4 ⊢ (𝜑 → 𝐵 ≠ 𝐴) |
52 | 51 | necomd 2999 | . . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
53 | chordthmlem2.QneM | . . 3 ⊢ (𝜑 → 𝑄 ≠ 𝑀) | |
54 | 1, 2, 3, 4, 5, 6, 52, 53 | chordthmlem 26182 | . 2 ⊢ (𝜑 → ((𝑄 − 𝑀)𝐹(𝐵 − 𝑀)) ∈ {(π / 2), -(π / 2)}) |
55 | 4, 44 | subcld 11512 | . . 3 ⊢ (𝜑 → (𝑄 − 𝑀) ∈ ℂ) |
56 | 39, 44 | subcld 11512 | . . 3 ⊢ (𝜑 → (𝑃 − 𝑀) ∈ ℂ) |
57 | 3, 44 | subcld 11512 | . . 3 ⊢ (𝜑 → (𝐵 − 𝑀) ∈ ℂ) |
58 | 4, 44, 53 | subne0d 11521 | . . 3 ⊢ (𝜑 → (𝑄 − 𝑀) ≠ 0) |
59 | 19, 10 | recne0d 11925 | . . . . 5 ⊢ (𝜑 → (1 / 2) ≠ 0) |
60 | 16, 15, 59, 50 | mulne0d 11807 | . . . 4 ⊢ (𝜑 → ((1 / 2) · (𝐵 − 𝐴)) ≠ 0) |
61 | 32, 60 | eqnetrd 3011 | . . 3 ⊢ (𝜑 → (𝐵 − 𝑀) ≠ 0) |
62 | 32, 46 | oveq12d 7375 | . . . . 5 ⊢ (𝜑 → ((𝐵 − 𝑀) / (𝑃 − 𝑀)) = (((1 / 2) · (𝐵 − 𝐴)) / (((1 / 2) − 𝑋) · (𝐵 − 𝐴)))) |
63 | 14, 15, 49 | mulne0bad 11810 | . . . . . 6 ⊢ (𝜑 → ((1 / 2) − 𝑋) ≠ 0) |
64 | 16, 14, 15, 63, 50 | divcan5rd 11958 | . . . . 5 ⊢ (𝜑 → (((1 / 2) · (𝐵 − 𝐴)) / (((1 / 2) − 𝑋) · (𝐵 − 𝐴))) = ((1 / 2) / ((1 / 2) − 𝑋))) |
65 | 62, 64 | eqtrd 2776 | . . . 4 ⊢ (𝜑 → ((𝐵 − 𝑀) / (𝑃 − 𝑀)) = ((1 / 2) / ((1 / 2) − 𝑋))) |
66 | 11, 13, 63 | redivcld 11983 | . . . 4 ⊢ (𝜑 → ((1 / 2) / ((1 / 2) − 𝑋)) ∈ ℝ) |
67 | 65, 66 | eqeltrd 2838 | . . 3 ⊢ (𝜑 → ((𝐵 − 𝑀) / (𝑃 − 𝑀)) ∈ ℝ) |
68 | 1, 55, 56, 57, 58, 48, 61, 67 | angrtmuld 26158 | . 2 ⊢ (𝜑 → (((𝑄 − 𝑀)𝐹(𝑃 − 𝑀)) ∈ {(π / 2), -(π / 2)} ↔ ((𝑄 − 𝑀)𝐹(𝐵 − 𝑀)) ∈ {(π / 2), -(π / 2)})) |
69 | 54, 68 | mpbird 256 | 1 ⊢ (𝜑 → ((𝑄 − 𝑀)𝐹(𝑃 − 𝑀)) ∈ {(π / 2), -(π / 2)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∖ cdif 3907 {csn 4586 {cpr 4588 ‘cfv 6496 (class class class)co 7357 ∈ cmpo 7359 ℂcc 11049 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 · cmul 11056 − cmin 11385 -cneg 11386 / cdiv 11812 2c2 12208 ℑcim 14983 abscabs 15119 πcpi 15949 logclog 25910 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ioc 13269 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-fac 14174 df-bc 14203 df-hash 14231 df-shft 14952 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-limsup 15353 df-clim 15370 df-rlim 15371 df-sum 15571 df-ef 15950 df-sin 15952 df-cos 15953 df-pi 15955 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-limc 25230 df-dv 25231 df-log 25912 |
This theorem is referenced by: chordthmlem3 26184 |
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