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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 25414 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 25415. (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 10662 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ) |
5 | 4 | halfcld 11885 | . . . . . . 7 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℂ) |
6 | 1, 5 | subcld 10999 | . . . . . 6 ⊢ (𝜑 → (𝑄 − ((𝐴 + 𝐵) / 2)) ∈ ℂ) |
7 | 6 | abscld 14798 | . . . . 5 ⊢ (𝜑 → (abs‘(𝑄 − ((𝐴 + 𝐵) / 2))) ∈ ℝ) |
8 | 7 | recnd 10671 | . . . 4 ⊢ (𝜑 → (abs‘(𝑄 − ((𝐴 + 𝐵) / 2))) ∈ ℂ) |
9 | 8 | sqcld 13511 | . . 3 ⊢ (𝜑 → ((abs‘(𝑄 − ((𝐴 + 𝐵) / 2)))↑2) ∈ ℂ) |
10 | 3, 5 | subcld 10999 | . . . . . 6 ⊢ (𝜑 → (𝐵 − ((𝐴 + 𝐵) / 2)) ∈ ℂ) |
11 | 10 | abscld 14798 | . . . . 5 ⊢ (𝜑 → (abs‘(𝐵 − ((𝐴 + 𝐵) / 2))) ∈ ℝ) |
12 | 11 | recnd 10671 | . . . 4 ⊢ (𝜑 → (abs‘(𝐵 − ((𝐴 + 𝐵) / 2))) ∈ ℂ) |
13 | 12 | sqcld 13511 | . . 3 ⊢ (𝜑 → ((abs‘(𝐵 − ((𝐴 + 𝐵) / 2)))↑2) ∈ ℂ) |
14 | chordthmlem5.P | . . . . . . . 8 ⊢ (𝜑 → 𝑃 = ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵))) | |
15 | unitssre 12888 | . . . . . . . . . . . 12 ⊢ (0[,]1) ⊆ ℝ | |
16 | chordthmlem5.X | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ (0[,]1)) | |
17 | 15, 16 | sseldi 3967 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ ℝ) |
18 | 17 | recnd 10671 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
19 | 18, 2 | mulcld 10663 | . . . . . . . . 9 ⊢ (𝜑 → (𝑋 · 𝐴) ∈ ℂ) |
20 | 1cnd 10638 | . . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℂ) | |
21 | 20, 18 | subcld 10999 | . . . . . . . . . 10 ⊢ (𝜑 → (1 − 𝑋) ∈ ℂ) |
22 | 21, 3 | mulcld 10663 | . . . . . . . . 9 ⊢ (𝜑 → ((1 − 𝑋) · 𝐵) ∈ ℂ) |
23 | 19, 22 | addcld 10662 | . . . . . . . 8 ⊢ (𝜑 → ((𝑋 · 𝐴) + ((1 − 𝑋) · 𝐵)) ∈ ℂ) |
24 | 14, 23 | eqeltrd 2915 | . . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
25 | 24, 5 | subcld 10999 | . . . . . 6 ⊢ (𝜑 → (𝑃 − ((𝐴 + 𝐵) / 2)) ∈ ℂ) |
26 | 25 | abscld 14798 | . . . . 5 ⊢ (𝜑 → (abs‘(𝑃 − ((𝐴 + 𝐵) / 2))) ∈ ℝ) |
27 | 26 | recnd 10671 | . . . 4 ⊢ (𝜑 → (abs‘(𝑃 − ((𝐴 + 𝐵) / 2))) ∈ ℂ) |
28 | 27 | sqcld 13511 | . . 3 ⊢ (𝜑 → ((abs‘(𝑃 − ((𝐴 + 𝐵) / 2)))↑2) ∈ ℂ) |
29 | 9, 13, 28 | pnpcand 11036 | . 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 10646 | . . . 4 ⊢ (𝜑 → 0 ∈ ℝ) | |
31 | eqidd 2824 | . . . 4 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) = ((𝐴 + 𝐵) / 2)) | |
32 | 2 | mul02d 10840 | . . . . . 6 ⊢ (𝜑 → (0 · 𝐴) = 0) |
33 | 20 | subid1d 10988 | . . . . . . . 8 ⊢ (𝜑 → (1 − 0) = 1) |
34 | 33 | oveq1d 7173 | . . . . . . 7 ⊢ (𝜑 → ((1 − 0) · 𝐵) = (1 · 𝐵)) |
35 | 3 | mulid2d 10661 | . . . . . . 7 ⊢ (𝜑 → (1 · 𝐵) = 𝐵) |
36 | 34, 35 | eqtrd 2858 | . . . . . 6 ⊢ (𝜑 → ((1 − 0) · 𝐵) = 𝐵) |
37 | 32, 36 | oveq12d 7176 | . . . . 5 ⊢ (𝜑 → ((0 · 𝐴) + ((1 − 0) · 𝐵)) = (0 + 𝐵)) |
38 | 3 | addid2d 10843 | . . . . 5 ⊢ (𝜑 → (0 + 𝐵) = 𝐵) |
39 | 37, 38 | eqtr2d 2859 | . . . 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 25414 | . . 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 25414 | . . 3 ⊢ (𝜑 → ((abs‘(𝑃 − 𝑄))↑2) = (((abs‘(𝑄 − ((𝐴 + 𝐵) / 2)))↑2) + ((abs‘(𝑃 − ((𝐴 + 𝐵) / 2)))↑2))) |
43 | 41, 42 | oveq12d 7176 | . 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 25415 | . 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 2869 | 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 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 ℝcr 10538 0cc0 10539 1c1 10540 + caddc 10542 · cmul 10544 − cmin 10872 / cdiv 11299 2c2 11695 [,]cicc 12744 ↑cexp 13432 abscabs 14595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-shft 14428 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-limsup 14830 df-clim 14847 df-rlim 14848 df-sum 15045 df-ef 15423 df-sin 15425 df-cos 15426 df-pi 15428 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-xrs 16777 df-qtop 16782 df-imas 16783 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-mulg 18227 df-cntz 18449 df-cmn 18910 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-lp 21746 df-perf 21747 df-cn 21837 df-cnp 21838 df-haus 21925 df-tx 22172 df-hmeo 22365 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-xms 22932 df-ms 22933 df-tms 22934 df-cncf 23488 df-limc 24466 df-dv 24467 df-log 25142 |
This theorem is referenced by: chordthm 25417 chordthmALT 41274 |
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