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Mirrors > Home > MPE Home > Th. List > chordthmlem5 | Structured version Visualization version GIF version |
Description: If P is on the segment AB and AQ = BQ, then PA · PB = BQ 2 − PQ 2 . This follows from two uses of chordthmlem3 25689 to show that PQ 2 = QM 2 + PM 2 and BQ 2 = QM 2 + BM 2 , so BQ 2 − PQ 2 = (QM 2 + BM 2 ) − (QM 2 + PM 2 ) = BM 2 − PM 2 , which equals PA · PB by chordthmlem4 25690. (Contributed by David Moews, 28-Feb-2017.) |
Ref | Expression |
---|---|
chordthmlem5.A | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
chordthmlem5.B | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
chordthmlem5.Q | ⊢ (𝜑 → 𝑄 ∈ ℂ) |
chordthmlem5.X | ⊢ (𝜑 → 𝑋 ∈ (0[,]1)) |
chordthmlem5.P | ⊢ (𝜑 → 𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵))) |
chordthmlem5.ABequidistQ | ⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) = (abs‘(𝐵 − 𝑄))) |
Ref | Expression |
---|---|
chordthmlem5 | ⊢ (𝜑 → ((abs‘(𝑃 − 𝐴)) · (abs‘(𝑃 − 𝐵))) = (((abs‘(𝐵 − 𝑄))↑2) − ((abs‘(𝑃 − 𝑄))↑2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chordthmlem5.Q | . . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℂ) | |
2 | chordthmlem5.A | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
3 | chordthmlem5.B | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
4 | 2, 3 | addcld 10835 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ) |
5 | 4 | halfcld 12058 | . . . . . . 7 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℂ) |
6 | 1, 5 | subcld 11172 | . . . . . 6 ⊢ (𝜑 → (𝑄 − ((𝐴 + 𝐵) / 2)) ∈ ℂ) |
7 | 6 | abscld 14983 | . . . . 5 ⊢ (𝜑 → (abs‘(𝑄 − ((𝐴 + 𝐵) / 2))) ∈ ℝ) |
8 | 7 | recnd 10844 | . . . 4 ⊢ (𝜑 → (abs‘(𝑄 − ((𝐴 + 𝐵) / 2))) ∈ ℂ) |
9 | 8 | sqcld 13697 | . . 3 ⊢ (𝜑 → ((abs‘(𝑄 − ((𝐴 + 𝐵) / 2)))↑2) ∈ ℂ) |
10 | 3, 5 | subcld 11172 | . . . . . 6 ⊢ (𝜑 → (𝐵 − ((𝐴 + 𝐵) / 2)) ∈ ℂ) |
11 | 10 | abscld 14983 | . . . . 5 ⊢ (𝜑 → (abs‘(𝐵 − ((𝐴 + 𝐵) / 2))) ∈ ℝ) |
12 | 11 | recnd 10844 | . . . 4 ⊢ (𝜑 → (abs‘(𝐵 − ((𝐴 + 𝐵) / 2))) ∈ ℂ) |
13 | 12 | sqcld 13697 | . . 3 ⊢ (𝜑 → ((abs‘(𝐵 − ((𝐴 + 𝐵) / 2)))↑2) ∈ ℂ) |
14 | chordthmlem5.P | . . . . . . . 8 ⊢ (𝜑 → 𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵))) | |
15 | unitssre 13070 | . . . . . . . . . . . 12 ⊢ (0[,]1) ⊆ ℝ | |
16 | chordthmlem5.X | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ (0[,]1)) | |
17 | 15, 16 | sseldi 3889 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ ℝ) |
18 | 17 | recnd 10844 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
19 | 18, 2 | mulcld 10836 | . . . . . . . . 9 ⊢ (𝜑 → (𝑋 · 𝐴) ∈ ℂ) |
20 | 1cnd 10811 | . . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℂ) | |
21 | 20, 18 | subcld 11172 | . . . . . . . . . 10 ⊢ (𝜑 → (1 − 𝑋) ∈ ℂ) |
22 | 21, 3 | mulcld 10836 | . . . . . . . . 9 ⊢ (𝜑 → ((1 − 𝑋) · 𝐵) ∈ ℂ) |
23 | 19, 22 | addcld 10835 | . . . . . . . 8 ⊢ (𝜑 → ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ∈ ℂ) |
24 | 14, 23 | eqeltrd 2834 | . . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
25 | 24, 5 | subcld 11172 | . . . . . 6 ⊢ (𝜑 → (𝑃 − ((𝐴 + 𝐵) / 2)) ∈ ℂ) |
26 | 25 | abscld 14983 | . . . . 5 ⊢ (𝜑 → (abs‘(𝑃 − ((𝐴 + 𝐵) / 2))) ∈ ℝ) |
27 | 26 | recnd 10844 | . . . 4 ⊢ (𝜑 → (abs‘(𝑃 − ((𝐴 + 𝐵) / 2))) ∈ ℂ) |
28 | 27 | sqcld 13697 | . . 3 ⊢ (𝜑 → ((abs‘(𝑃 − ((𝐴 + 𝐵) / 2)))↑2) ∈ ℂ) |
29 | 9, 13, 28 | pnpcand 11209 | . 2 ⊢ (𝜑 → ((((abs‘(𝑄 − ((𝐴 + 𝐵) / 2)))↑2) + ((abs‘(𝐵 − ((𝐴 + 𝐵) / 2)))↑2)) − (((abs‘(𝑄 − ((𝐴 + 𝐵) / 2)))↑2) + ((abs‘(𝑃 − ((𝐴 + 𝐵) / 2)))↑2))) = (((abs‘(𝐵 − ((𝐴 + 𝐵) / 2)))↑2) − ((abs‘(𝑃 − ((𝐴 + 𝐵) / 2)))↑2))) |
30 | 0red 10819 | . . . 4 ⊢ (𝜑 → 0 ∈ ℝ) | |
31 | eqidd 2735 | . . . 4 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) = ((𝐴 + 𝐵) / 2)) | |
32 | 2 | mul02d 11013 | . . . . . 6 ⊢ (𝜑 → (0 · 𝐴) = 0) |
33 | 20 | subid1d 11161 | . . . . . . . 8 ⊢ (𝜑 → (1 − 0) = 1) |
34 | 33 | oveq1d 7217 | . . . . . . 7 ⊢ (𝜑 → ((1 − 0) · 𝐵) = (1 · 𝐵)) |
35 | 3 | mulid2d 10834 | . . . . . . 7 ⊢ (𝜑 → (1 · 𝐵) = 𝐵) |
36 | 34, 35 | eqtrd 2774 | . . . . . 6 ⊢ (𝜑 → ((1 − 0) · 𝐵) = 𝐵) |
37 | 32, 36 | oveq12d 7220 | . . . . 5 ⊢ (𝜑 → ((0 · 𝐴) + ((1 − 0) · 𝐵)) = (0 + 𝐵)) |
38 | 3 | addid2d 11016 | . . . . 5 ⊢ (𝜑 → (0 + 𝐵) = 𝐵) |
39 | 37, 38 | eqtr2d 2775 | . . . 4 ⊢ (𝜑 → 𝐵 = ((0 · 𝐴) + ((1 − 0) · 𝐵))) |
40 | chordthmlem5.ABequidistQ | . . . 4 ⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) = (abs‘(𝐵 − 𝑄))) | |
41 | 2, 3, 1, 30, 31, 39, 40 | chordthmlem3 25689 | . . 3 ⊢ (𝜑 → ((abs‘(𝐵 − 𝑄))↑2) = (((abs‘(𝑄 − ((𝐴 + 𝐵) / 2)))↑2) + ((abs‘(𝐵 − ((𝐴 + 𝐵) / 2)))↑2))) |
42 | 2, 3, 1, 17, 31, 14, 40 | chordthmlem3 25689 | . . 3 ⊢ (𝜑 → ((abs‘(𝑃 − 𝑄))↑2) = (((abs‘(𝑄 − ((𝐴 + 𝐵) / 2)))↑2) + ((abs‘(𝑃 − ((𝐴 + 𝐵) / 2)))↑2))) |
43 | 41, 42 | oveq12d 7220 | . 2 ⊢ (𝜑 → (((abs‘(𝐵 − 𝑄))↑2) − ((abs‘(𝑃 − 𝑄))↑2)) = ((((abs‘(𝑄 − ((𝐴 + 𝐵) / 2)))↑2) + ((abs‘(𝐵 − ((𝐴 + 𝐵) / 2)))↑2)) − (((abs‘(𝑄 − ((𝐴 + 𝐵) / 2)))↑2) + ((abs‘(𝑃 − ((𝐴 + 𝐵) / 2)))↑2)))) |
44 | 2, 3, 16, 31, 14 | chordthmlem4 25690 | . 2 ⊢ (𝜑 → ((abs‘(𝑃 − 𝐴)) · (abs‘(𝑃 − 𝐵))) = (((abs‘(𝐵 − ((𝐴 + 𝐵) / 2)))↑2) − ((abs‘(𝑃 − ((𝐴 + 𝐵) / 2)))↑2))) |
45 | 29, 43, 44 | 3eqtr4rd 2785 | 1 ⊢ (𝜑 → ((abs‘(𝑃 − 𝐴)) · (abs‘(𝑃 − 𝐵))) = (((abs‘(𝐵 − 𝑄))↑2) − ((abs‘(𝑃 − 𝑄))↑2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ‘cfv 6369 (class class class)co 7202 ℂcc 10710 ℝcr 10711 0cc0 10712 1c1 10713 + caddc 10715 · cmul 10717 − cmin 11045 / cdiv 11472 2c2 11868 [,]cicc 12921 ↑cexp 13618 abscabs 14780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-inf2 9245 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 ax-addf 10791 ax-mulf 10792 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-iin 4897 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-of 7458 df-om 7634 df-1st 7750 df-2nd 7751 df-supp 7893 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-2o 8192 df-er 8380 df-map 8499 df-pm 8500 df-ixp 8568 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-fsupp 8975 df-fi 9016 df-sup 9047 df-inf 9048 df-oi 9115 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-dec 12277 df-uz 12422 df-q 12528 df-rp 12570 df-xneg 12687 df-xadd 12688 df-xmul 12689 df-ioo 12922 df-ioc 12923 df-ico 12924 df-icc 12925 df-fz 13079 df-fzo 13222 df-fl 13350 df-mod 13426 df-seq 13558 df-exp 13619 df-fac 13823 df-bc 13852 df-hash 13880 df-shft 14613 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-limsup 15015 df-clim 15032 df-rlim 15033 df-sum 15233 df-ef 15610 df-sin 15612 df-cos 15613 df-pi 15615 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-starv 16782 df-sca 16783 df-vsca 16784 df-ip 16785 df-tset 16786 df-ple 16787 df-ds 16789 df-unif 16790 df-hom 16791 df-cco 16792 df-rest 16899 df-topn 16900 df-0g 16918 df-gsum 16919 df-topgen 16920 df-pt 16921 df-prds 16924 df-xrs 16979 df-qtop 16984 df-imas 16985 df-xps 16987 df-mre 17061 df-mrc 17062 df-acs 17064 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-submnd 18191 df-mulg 18461 df-cntz 18683 df-cmn 19144 df-psmet 20327 df-xmet 20328 df-met 20329 df-bl 20330 df-mopn 20331 df-fbas 20332 df-fg 20333 df-cnfld 20336 df-top 21763 df-topon 21780 df-topsp 21802 df-bases 21815 df-cld 21888 df-ntr 21889 df-cls 21890 df-nei 21967 df-lp 22005 df-perf 22006 df-cn 22096 df-cnp 22097 df-haus 22184 df-tx 22431 df-hmeo 22624 df-fil 22715 df-fm 22807 df-flim 22808 df-flf 22809 df-xms 23190 df-ms 23191 df-tms 23192 df-cncf 23747 df-limc 24735 df-dv 24736 df-log 25417 |
This theorem is referenced by: chordthm 25692 chordthmALT 42178 |
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