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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 24359, where P = B, and using angrtmuld 24338 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 10967 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ | |
8 | 7 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℝ) |
9 | 2ne0 10990 | . . . . . . . . . 10 ⊢ 2 ≠ 0 | |
10 | 9 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → 2 ≠ 0) |
11 | 8, 10 | rereccld 10731 | . . . . . . . 8 ⊢ (𝜑 → (1 / 2) ∈ ℝ) |
12 | chordthmlem2.X | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℝ) | |
13 | 11, 12 | resubcld 10337 | . . . . . . 7 ⊢ (𝜑 → ((1 / 2) − 𝑋) ∈ ℝ) |
14 | 13 | recnd 9947 | . . . . . 6 ⊢ (𝜑 → ((1 / 2) − 𝑋) ∈ ℂ) |
15 | 3, 2 | subcld 10271 | . . . . . 6 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℂ) |
16 | 11 | recnd 9947 | . . . . . . . . 9 ⊢ (𝜑 → (1 / 2) ∈ ℂ) |
17 | 12 | recnd 9947 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
18 | 16, 17, 15 | subdird 10366 | . . . . . . . 8 ⊢ (𝜑 → (((1 / 2) − 𝑋) · (𝐵 − 𝐴)) = (((1 / 2) · (𝐵 − 𝐴)) − (𝑋 · (𝐵 − 𝐴)))) |
19 | 2cnd 10970 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ∈ ℂ) | |
20 | 3, 19, 10 | divcan4d 10686 | . . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐵 · 2) / 2) = 𝐵) |
21 | 3 | times2d 11153 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 · 2) = (𝐵 + 𝐵)) |
22 | 21 | oveq1d 6564 | . . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐵 · 2) / 2) = ((𝐵 + 𝐵) / 2)) |
23 | 20, 22 | eqtr3d 2646 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 = ((𝐵 + 𝐵) / 2)) |
24 | 23, 5 | oveq12d 6567 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 − 𝑀) = (((𝐵 + 𝐵) / 2) − ((𝐴 + 𝐵) / 2))) |
25 | 3, 3 | addcld 9938 | . . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 + 𝐵) ∈ ℂ) |
26 | 2, 3 | addcld 9938 | . . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ) |
27 | 25, 26, 19, 10 | divsubdird 10719 | . . . . . . . . . . 11 ⊢ (𝜑 → (((𝐵 + 𝐵) − (𝐴 + 𝐵)) / 2) = (((𝐵 + 𝐵) / 2) − ((𝐴 + 𝐵) / 2))) |
28 | 3, 2, 3 | pnpcan2d 10309 | . . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐵 + 𝐵) − (𝐴 + 𝐵)) = (𝐵 − 𝐴)) |
29 | 28 | oveq1d 6564 | . . . . . . . . . . 11 ⊢ (𝜑 → (((𝐵 + 𝐵) − (𝐴 + 𝐵)) / 2) = ((𝐵 − 𝐴) / 2)) |
30 | 24, 27, 29 | 3eqtr2d 2650 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐵 − 𝑀) = ((𝐵 − 𝐴) / 2)) |
31 | 15, 19, 10 | divrec2d 10684 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝐵 − 𝐴) / 2) = ((1 / 2) · (𝐵 − 𝐴))) |
32 | 30, 31 | eqtrd 2644 | . . . . . . . . 9 ⊢ (𝜑 → (𝐵 − 𝑀) = ((1 / 2) · (𝐵 − 𝐴))) |
33 | chordthmlem2.P | . . . . . . . . . 10 ⊢ (𝜑 → 𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵))) | |
34 | 17, 2 | mulcld 9939 | . . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 · 𝐴) ∈ ℂ) |
35 | 1cnd 9935 | . . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℂ) | |
36 | 35, 17 | subcld 10271 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → (1 − 𝑋) ∈ ℂ) |
37 | 36, 3 | mulcld 9939 | . . . . . . . . . . . . 13 ⊢ (𝜑 → ((1 − 𝑋) · 𝐵) ∈ ℂ) |
38 | 34, 37 | addcld 9938 | . . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ∈ ℂ) |
39 | 33, 38 | eqeltrd 2688 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
40 | 2, 39, 3, 17 | affineequiv 24353 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ↔ (𝐵 − 𝑃) = (𝑋 · (𝐵 − 𝐴)))) |
41 | 33, 40 | mpbid 221 | . . . . . . . . 9 ⊢ (𝜑 → (𝐵 − 𝑃) = (𝑋 · (𝐵 − 𝐴))) |
42 | 32, 41 | oveq12d 6567 | . . . . . . . 8 ⊢ (𝜑 → ((𝐵 − 𝑀) − (𝐵 − 𝑃)) = (((1 / 2) · (𝐵 − 𝐴)) − (𝑋 · (𝐵 − 𝐴)))) |
43 | 26 | halfcld 11154 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℂ) |
44 | 5, 43 | eqeltrd 2688 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
45 | 3, 44, 39 | nnncan1d 10305 | . . . . . . . 8 ⊢ (𝜑 → ((𝐵 − 𝑀) − (𝐵 − 𝑃)) = (𝑃 − 𝑀)) |
46 | 18, 42, 45 | 3eqtr2rd 2651 | . . . . . . 7 ⊢ (𝜑 → (𝑃 − 𝑀) = (((1 / 2) − 𝑋) · (𝐵 − 𝐴))) |
47 | chordthmlem2.PneM | . . . . . . . 8 ⊢ (𝜑 → 𝑃 ≠ 𝑀) | |
48 | 39, 44, 47 | subne0d 10280 | . . . . . . 7 ⊢ (𝜑 → (𝑃 − 𝑀) ≠ 0) |
49 | 46, 48 | eqnetrrd 2850 | . . . . . 6 ⊢ (𝜑 → (((1 / 2) − 𝑋) · (𝐵 − 𝐴)) ≠ 0) |
50 | 14, 15, 49 | mulne0bbd 10562 | . . . . 5 ⊢ (𝜑 → (𝐵 − 𝐴) ≠ 0) |
51 | 3, 2, 50 | subne0ad 10282 | . . . 4 ⊢ (𝜑 → 𝐵 ≠ 𝐴) |
52 | 51 | necomd 2837 | . . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
53 | chordthmlem2.QneM | . . 3 ⊢ (𝜑 → 𝑄 ≠ 𝑀) | |
54 | 1, 2, 3, 4, 5, 6, 52, 53 | chordthmlem 24359 | . 2 ⊢ (𝜑 → ((𝑄 − 𝑀)𝐹(𝐵 − 𝑀)) ∈ {(π / 2), -(π / 2)}) |
55 | 4, 44 | subcld 10271 | . . 3 ⊢ (𝜑 → (𝑄 − 𝑀) ∈ ℂ) |
56 | 39, 44 | subcld 10271 | . . 3 ⊢ (𝜑 → (𝑃 − 𝑀) ∈ ℂ) |
57 | 3, 44 | subcld 10271 | . . 3 ⊢ (𝜑 → (𝐵 − 𝑀) ∈ ℂ) |
58 | 4, 44, 53 | subne0d 10280 | . . 3 ⊢ (𝜑 → (𝑄 − 𝑀) ≠ 0) |
59 | 19, 10 | recne0d 10674 | . . . . 5 ⊢ (𝜑 → (1 / 2) ≠ 0) |
60 | 16, 15, 59, 50 | mulne0d 10558 | . . . 4 ⊢ (𝜑 → ((1 / 2) · (𝐵 − 𝐴)) ≠ 0) |
61 | 32, 60 | eqnetrd 2849 | . . 3 ⊢ (𝜑 → (𝐵 − 𝑀) ≠ 0) |
62 | 32, 46 | oveq12d 6567 | . . . . 5 ⊢ (𝜑 → ((𝐵 − 𝑀) / (𝑃 − 𝑀)) = (((1 / 2) · (𝐵 − 𝐴)) / (((1 / 2) − 𝑋) · (𝐵 − 𝐴)))) |
63 | 14, 15, 49 | mulne0bad 10561 | . . . . . 6 ⊢ (𝜑 → ((1 / 2) − 𝑋) ≠ 0) |
64 | 16, 14, 15, 63, 50 | divcan5rd 10707 | . . . . 5 ⊢ (𝜑 → (((1 / 2) · (𝐵 − 𝐴)) / (((1 / 2) − 𝑋) · (𝐵 − 𝐴))) = ((1 / 2) / ((1 / 2) − 𝑋))) |
65 | 62, 64 | eqtrd 2644 | . . . 4 ⊢ (𝜑 → ((𝐵 − 𝑀) / (𝑃 − 𝑀)) = ((1 / 2) / ((1 / 2) − 𝑋))) |
66 | 11, 13, 63 | redivcld 10732 | . . . 4 ⊢ (𝜑 → ((1 / 2) / ((1 / 2) − 𝑋)) ∈ ℝ) |
67 | 65, 66 | eqeltrd 2688 | . . 3 ⊢ (𝜑 → ((𝐵 − 𝑀) / (𝑃 − 𝑀)) ∈ ℝ) |
68 | 1, 55, 56, 57, 58, 48, 61, 67 | angrtmuld 24338 | . 2 ⊢ (𝜑 → (((𝑄 − 𝑀)𝐹(𝑃 − 𝑀)) ∈ {(π / 2), -(π / 2)} ↔ ((𝑄 − 𝑀)𝐹(𝐵 − 𝑀)) ∈ {(π / 2), -(π / 2)})) |
69 | 54, 68 | mpbird 246 | 1 ⊢ (𝜑 → ((𝑄 − 𝑀)𝐹(𝑃 − 𝑀)) ∈ {(π / 2), -(π / 2)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∖ cdif 3537 {csn 4125 {cpr 4127 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 − cmin 10145 -cneg 10146 / cdiv 10563 2c2 10947 ℑcim 13686 abscabs 13822 πcpi 14636 logclog 24105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-sin 14639 df-cos 14640 df-pi 14642 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-limc 23436 df-dv 23437 df-log 24107 |
This theorem is referenced by: chordthmlem3 24361 |
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