| 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 26875, where P = B, and using angrtmuld 26851 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 12340 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ | |
| 8 | 7 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℝ) |
| 9 | 2ne0 12370 | . . . . . . . . . 10 ⊢ 2 ≠ 0 | |
| 10 | 9 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → 2 ≠ 0) |
| 11 | 8, 10 | rereccld 12094 | . . . . . . . 8 ⊢ (𝜑 → (1 / 2) ∈ ℝ) |
| 12 | chordthmlem2.X | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℝ) | |
| 13 | 11, 12 | resubcld 11691 | . . . . . . 7 ⊢ (𝜑 → ((1 / 2) − 𝑋) ∈ ℝ) |
| 14 | 13 | recnd 11289 | . . . . . 6 ⊢ (𝜑 → ((1 / 2) − 𝑋) ∈ ℂ) |
| 15 | 3, 2 | subcld 11620 | . . . . . 6 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℂ) |
| 16 | 11 | recnd 11289 | . . . . . . . . 9 ⊢ (𝜑 → (1 / 2) ∈ ℂ) |
| 17 | 12 | recnd 11289 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
| 18 | 16, 17, 15 | subdird 11720 | . . . . . . . 8 ⊢ (𝜑 → (((1 / 2) − 𝑋) · (𝐵 − 𝐴)) = (((1 / 2) · (𝐵 − 𝐴)) − (𝑋 · (𝐵 − 𝐴)))) |
| 19 | 2cnd 12344 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ∈ ℂ) | |
| 20 | 3, 19, 10 | divcan4d 12049 | . . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐵 · 2) / 2) = 𝐵) |
| 21 | 3 | times2d 12510 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 · 2) = (𝐵 + 𝐵)) |
| 22 | 21 | oveq1d 7446 | . . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐵 · 2) / 2) = ((𝐵 + 𝐵) / 2)) |
| 23 | 20, 22 | eqtr3d 2779 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 = ((𝐵 + 𝐵) / 2)) |
| 24 | 23, 5 | oveq12d 7449 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 − 𝑀) = (((𝐵 + 𝐵) / 2) − ((𝐴 + 𝐵) / 2))) |
| 25 | 3, 3 | addcld 11280 | . . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 + 𝐵) ∈ ℂ) |
| 26 | 2, 3 | addcld 11280 | . . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ) |
| 27 | 25, 26, 19, 10 | divsubdird 12082 | . . . . . . . . . . 11 ⊢ (𝜑 → (((𝐵 + 𝐵) − (𝐴 + 𝐵)) / 2) = (((𝐵 + 𝐵) / 2) − ((𝐴 + 𝐵) / 2))) |
| 28 | 3, 2, 3 | pnpcan2d 11658 | . . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐵 + 𝐵) − (𝐴 + 𝐵)) = (𝐵 − 𝐴)) |
| 29 | 28 | oveq1d 7446 | . . . . . . . . . . 11 ⊢ (𝜑 → (((𝐵 + 𝐵) − (𝐴 + 𝐵)) / 2) = ((𝐵 − 𝐴) / 2)) |
| 30 | 24, 27, 29 | 3eqtr2d 2783 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐵 − 𝑀) = ((𝐵 − 𝐴) / 2)) |
| 31 | 15, 19, 10 | divrec2d 12047 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝐵 − 𝐴) / 2) = ((1 / 2) · (𝐵 − 𝐴))) |
| 32 | 30, 31 | eqtrd 2777 | . . . . . . . . 9 ⊢ (𝜑 → (𝐵 − 𝑀) = ((1 / 2) · (𝐵 − 𝐴))) |
| 33 | chordthmlem2.P | . . . . . . . . . 10 ⊢ (𝜑 → 𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵))) | |
| 34 | 17, 2 | mulcld 11281 | . . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 · 𝐴) ∈ ℂ) |
| 35 | 1cnd 11256 | . . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℂ) | |
| 36 | 35, 17 | subcld 11620 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → (1 − 𝑋) ∈ ℂ) |
| 37 | 36, 3 | mulcld 11281 | . . . . . . . . . . . . 13 ⊢ (𝜑 → ((1 − 𝑋) · 𝐵) ∈ ℂ) |
| 38 | 34, 37 | addcld 11280 | . . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ∈ ℂ) |
| 39 | 33, 38 | eqeltrd 2841 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
| 40 | 2, 39, 3, 17 | affineequiv 26866 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ↔ (𝐵 − 𝑃) = (𝑋 · (𝐵 − 𝐴)))) |
| 41 | 33, 40 | mpbid 232 | . . . . . . . . 9 ⊢ (𝜑 → (𝐵 − 𝑃) = (𝑋 · (𝐵 − 𝐴))) |
| 42 | 32, 41 | oveq12d 7449 | . . . . . . . 8 ⊢ (𝜑 → ((𝐵 − 𝑀) − (𝐵 − 𝑃)) = (((1 / 2) · (𝐵 − 𝐴)) − (𝑋 · (𝐵 − 𝐴)))) |
| 43 | 26 | halfcld 12511 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℂ) |
| 44 | 5, 43 | eqeltrd 2841 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
| 45 | 3, 44, 39 | nnncan1d 11654 | . . . . . . . 8 ⊢ (𝜑 → ((𝐵 − 𝑀) − (𝐵 − 𝑃)) = (𝑃 − 𝑀)) |
| 46 | 18, 42, 45 | 3eqtr2rd 2784 | . . . . . . 7 ⊢ (𝜑 → (𝑃 − 𝑀) = (((1 / 2) − 𝑋) · (𝐵 − 𝐴))) |
| 47 | chordthmlem2.PneM | . . . . . . . 8 ⊢ (𝜑 → 𝑃 ≠ 𝑀) | |
| 48 | 39, 44, 47 | subne0d 11629 | . . . . . . 7 ⊢ (𝜑 → (𝑃 − 𝑀) ≠ 0) |
| 49 | 46, 48 | eqnetrrd 3009 | . . . . . 6 ⊢ (𝜑 → (((1 / 2) − 𝑋) · (𝐵 − 𝐴)) ≠ 0) |
| 50 | 14, 15, 49 | mulne0bbd 11919 | . . . . 5 ⊢ (𝜑 → (𝐵 − 𝐴) ≠ 0) |
| 51 | 3, 2, 50 | subne0ad 11631 | . . . 4 ⊢ (𝜑 → 𝐵 ≠ 𝐴) |
| 52 | 51 | necomd 2996 | . . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
| 53 | chordthmlem2.QneM | . . 3 ⊢ (𝜑 → 𝑄 ≠ 𝑀) | |
| 54 | 1, 2, 3, 4, 5, 6, 52, 53 | chordthmlem 26875 | . 2 ⊢ (𝜑 → ((𝑄 − 𝑀)𝐹(𝐵 − 𝑀)) ∈ {(π / 2), -(π / 2)}) |
| 55 | 4, 44 | subcld 11620 | . . 3 ⊢ (𝜑 → (𝑄 − 𝑀) ∈ ℂ) |
| 56 | 39, 44 | subcld 11620 | . . 3 ⊢ (𝜑 → (𝑃 − 𝑀) ∈ ℂ) |
| 57 | 3, 44 | subcld 11620 | . . 3 ⊢ (𝜑 → (𝐵 − 𝑀) ∈ ℂ) |
| 58 | 4, 44, 53 | subne0d 11629 | . . 3 ⊢ (𝜑 → (𝑄 − 𝑀) ≠ 0) |
| 59 | 19, 10 | recne0d 12037 | . . . . 5 ⊢ (𝜑 → (1 / 2) ≠ 0) |
| 60 | 16, 15, 59, 50 | mulne0d 11915 | . . . 4 ⊢ (𝜑 → ((1 / 2) · (𝐵 − 𝐴)) ≠ 0) |
| 61 | 32, 60 | eqnetrd 3008 | . . 3 ⊢ (𝜑 → (𝐵 − 𝑀) ≠ 0) |
| 62 | 32, 46 | oveq12d 7449 | . . . . 5 ⊢ (𝜑 → ((𝐵 − 𝑀) / (𝑃 − 𝑀)) = (((1 / 2) · (𝐵 − 𝐴)) / (((1 / 2) − 𝑋) · (𝐵 − 𝐴)))) |
| 63 | 14, 15, 49 | mulne0bad 11918 | . . . . . 6 ⊢ (𝜑 → ((1 / 2) − 𝑋) ≠ 0) |
| 64 | 16, 14, 15, 63, 50 | divcan5rd 12070 | . . . . 5 ⊢ (𝜑 → (((1 / 2) · (𝐵 − 𝐴)) / (((1 / 2) − 𝑋) · (𝐵 − 𝐴))) = ((1 / 2) / ((1 / 2) − 𝑋))) |
| 65 | 62, 64 | eqtrd 2777 | . . . 4 ⊢ (𝜑 → ((𝐵 − 𝑀) / (𝑃 − 𝑀)) = ((1 / 2) / ((1 / 2) − 𝑋))) |
| 66 | 11, 13, 63 | redivcld 12095 | . . . 4 ⊢ (𝜑 → ((1 / 2) / ((1 / 2) − 𝑋)) ∈ ℝ) |
| 67 | 65, 66 | eqeltrd 2841 | . . 3 ⊢ (𝜑 → ((𝐵 − 𝑀) / (𝑃 − 𝑀)) ∈ ℝ) |
| 68 | 1, 55, 56, 57, 58, 48, 61, 67 | angrtmuld 26851 | . 2 ⊢ (𝜑 → (((𝑄 − 𝑀)𝐹(𝑃 − 𝑀)) ∈ {(π / 2), -(π / 2)} ↔ ((𝑄 − 𝑀)𝐹(𝐵 − 𝑀)) ∈ {(π / 2), -(π / 2)})) |
| 69 | 54, 68 | mpbird 257 | 1 ⊢ (𝜑 → ((𝑄 − 𝑀)𝐹(𝑃 − 𝑀)) ∈ {(π / 2), -(π / 2)}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∖ cdif 3948 {csn 4626 {cpr 4628 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 − cmin 11492 -cneg 11493 / cdiv 11920 2c2 12321 ℑcim 15137 abscabs 15273 πcpi 16102 logclog 26596 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 df-sin 16105 df-cos 16106 df-pi 16108 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-limc 25901 df-dv 25902 df-log 26598 |
| This theorem is referenced by: chordthmlem3 26877 |
| Copyright terms: Public domain | W3C validator |