![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > tgidinside | Structured version Visualization version GIF version |
Description: Law for finding a point inside a segment. Theorem 4.19 of [Schwabhauser] p. 38. (Contributed by Thierry Arnoux, 28-Apr-2019.) |
Ref | Expression |
---|---|
tglngval.p | ⊢ 𝑃 = (Base‘𝐺) |
tglngval.l | ⊢ 𝐿 = (LineG‘𝐺) |
tglngval.i | ⊢ 𝐼 = (Itv‘𝐺) |
tglngval.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
tglngval.x | ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
tglngval.y | ⊢ (𝜑 → 𝑌 ∈ 𝑃) |
tgcolg.z | ⊢ (𝜑 → 𝑍 ∈ 𝑃) |
lnxfr.r | ⊢ ∼ = (cgrG‘𝐺) |
lnxfr.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
lnxfr.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
lnxfr.d | ⊢ − = (dist‘𝐺) |
tgidinside.1 | ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐼𝑌)) |
tgidinside.2 | ⊢ (𝜑 → (𝑋 − 𝑍) = (𝑋 − 𝐴)) |
tgidinside.3 | ⊢ (𝜑 → (𝑌 − 𝑍) = (𝑌 − 𝐴)) |
Ref | Expression |
---|---|
tgidinside | ⊢ (𝜑 → 𝑍 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tglngval.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
2 | lnxfr.d | . . . 4 ⊢ − = (dist‘𝐺) | |
3 | tglngval.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | tglngval.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | 4 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝐺 ∈ TarskiG) |
6 | tglngval.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
7 | 6 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ 𝑃) |
8 | tgcolg.z | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑃) | |
9 | 8 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑍 ∈ 𝑃) |
10 | tgidinside.1 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐼𝑌)) | |
11 | 10 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑍 ∈ (𝑋𝐼𝑌)) |
12 | simpr 479 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌) | |
13 | 12 | oveq2d 6926 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑋𝐼𝑋) = (𝑋𝐼𝑌)) |
14 | 11, 13 | eleqtrrd 2909 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑍 ∈ (𝑋𝐼𝑋)) |
15 | 1, 2, 3, 5, 7, 9, 14 | axtgbtwnid 25785 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 = 𝑍) |
16 | lnxfr.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
17 | 16 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝐴 ∈ 𝑃) |
18 | tgidinside.2 | . . . . 5 ⊢ (𝜑 → (𝑋 − 𝑍) = (𝑋 − 𝐴)) | |
19 | 18 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑋 − 𝑍) = (𝑋 − 𝐴)) |
20 | 1, 2, 3, 5, 7, 9, 7, 17, 19, 15 | tgcgreq 25801 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 = 𝐴) |
21 | 15, 20 | eqtr3d 2863 | . 2 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑍 = 𝐴) |
22 | tglngval.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
23 | 4 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝐺 ∈ TarskiG) |
24 | 6 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ∈ 𝑃) |
25 | tglngval.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
26 | 25 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑌 ∈ 𝑃) |
27 | 8 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑍 ∈ 𝑃) |
28 | lnxfr.r | . . 3 ⊢ ∼ = (cgrG‘𝐺) | |
29 | 16 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝐴 ∈ 𝑃) |
30 | lnxfr.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
31 | 30 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝐵 ∈ 𝑃) |
32 | simpr 479 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ≠ 𝑌) | |
33 | 1, 22, 3, 4, 6, 8, 25, 10 | btwncolg3 25876 | . . . 4 ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍)) |
34 | 33 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍)) |
35 | 18 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑋 − 𝑍) = (𝑋 − 𝐴)) |
36 | tgidinside.3 | . . . 4 ⊢ (𝜑 → (𝑌 − 𝑍) = (𝑌 − 𝐴)) | |
37 | 36 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑌 − 𝑍) = (𝑌 − 𝐴)) |
38 | 1, 22, 3, 23, 24, 26, 27, 28, 29, 31, 2, 32, 34, 35, 37 | lnid 25889 | . 2 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑍 = 𝐴) |
39 | 21, 38 | pm2.61dane 3086 | 1 ⊢ (𝜑 → 𝑍 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∨ wo 878 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 ‘cfv 6127 (class class class)co 6910 Basecbs 16229 distcds 16321 TarskiGcstrkg 25749 Itvcitv 25755 LineGclng 25756 cgrGccgrg 25829 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-pm 8130 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-card 9085 df-cda 9312 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-n0 11626 df-xnn0 11698 df-z 11712 df-uz 11976 df-fz 12627 df-fzo 12768 df-hash 13418 df-word 13582 df-concat 13638 df-s1 13663 df-s2 13976 df-s3 13977 df-trkgc 25767 df-trkgb 25768 df-trkgcb 25769 df-trkg 25772 df-cgrg 25830 |
This theorem is referenced by: miduniq 26004 ragflat2 26022 |
Copyright terms: Public domain | W3C validator |