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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 466 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝐺 ∈ TarskiG) |
6 | tglngval.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
7 | 6 | adantr 466 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ 𝑃) |
8 | tgcolg.z | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑃) | |
9 | 8 | adantr 466 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑍 ∈ 𝑃) |
10 | tgidinside.1 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐼𝑌)) | |
11 | 10 | adantr 466 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑍 ∈ (𝑋𝐼𝑌)) |
12 | simpr 471 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌) | |
13 | 12 | oveq2d 6807 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑋𝐼𝑋) = (𝑋𝐼𝑌)) |
14 | 11, 13 | eleqtrrd 2853 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑍 ∈ (𝑋𝐼𝑋)) |
15 | 1, 2, 3, 5, 7, 9, 14 | axtgbtwnid 25579 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 = 𝑍) |
16 | lnxfr.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
17 | 16 | adantr 466 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝐴 ∈ 𝑃) |
18 | tgidinside.2 | . . . . 5 ⊢ (𝜑 → (𝑋 − 𝑍) = (𝑋 − 𝐴)) | |
19 | 18 | adantr 466 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑋 − 𝑍) = (𝑋 − 𝐴)) |
20 | 1, 2, 3, 5, 7, 9, 7, 17, 19, 15 | tgcgreq 25591 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 = 𝐴) |
21 | 15, 20 | eqtr3d 2807 | . 2 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑍 = 𝐴) |
22 | tglngval.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
23 | 4 | adantr 466 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝐺 ∈ TarskiG) |
24 | 6 | adantr 466 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ∈ 𝑃) |
25 | tglngval.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
26 | 25 | adantr 466 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑌 ∈ 𝑃) |
27 | 8 | adantr 466 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑍 ∈ 𝑃) |
28 | lnxfr.r | . . 3 ⊢ ∼ = (cgrG‘𝐺) | |
29 | 16 | adantr 466 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝐴 ∈ 𝑃) |
30 | lnxfr.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
31 | 30 | adantr 466 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝐵 ∈ 𝑃) |
32 | simpr 471 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ≠ 𝑌) | |
33 | 1, 22, 3, 4, 6, 8, 25, 10 | btwncolg3 25666 | . . . 4 ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍)) |
34 | 33 | adantr 466 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍)) |
35 | 18 | adantr 466 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑋 − 𝑍) = (𝑋 − 𝐴)) |
36 | tgidinside.3 | . . . 4 ⊢ (𝜑 → (𝑌 − 𝑍) = (𝑌 − 𝐴)) | |
37 | 36 | adantr 466 | . . 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 25679 | . 2 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑍 = 𝐴) |
39 | 21, 38 | pm2.61dane 3030 | 1 ⊢ (𝜑 → 𝑍 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∨ wo 836 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 ‘cfv 6029 (class class class)co 6791 Basecbs 16057 distcds 16151 TarskiGcstrkg 25543 Itvcitv 25549 LineGclng 25550 cgrGccgrg 25619 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7094 ax-cnex 10192 ax-resscn 10193 ax-1cn 10194 ax-icn 10195 ax-addcl 10196 ax-addrcl 10197 ax-mulcl 10198 ax-mulrcl 10199 ax-mulcom 10200 ax-addass 10201 ax-mulass 10202 ax-distr 10203 ax-i2m1 10204 ax-1ne0 10205 ax-1rid 10206 ax-rnegex 10207 ax-rrecex 10208 ax-cnre 10209 ax-pre-lttri 10210 ax-pre-lttrn 10211 ax-pre-ltadd 10212 ax-pre-mulgt0 10213 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5821 df-ord 5867 df-on 5868 df-lim 5869 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-riota 6752 df-ov 6794 df-oprab 6795 df-mpt2 6796 df-om 7211 df-1st 7313 df-2nd 7314 df-wrecs 7557 df-recs 7619 df-rdg 7657 df-1o 7711 df-oadd 7715 df-er 7894 df-pm 8010 df-en 8108 df-dom 8109 df-sdom 8110 df-fin 8111 df-card 8963 df-cda 9190 df-pnf 10276 df-mnf 10277 df-xr 10278 df-ltxr 10279 df-le 10280 df-sub 10468 df-neg 10469 df-nn 11221 df-2 11279 df-3 11280 df-n0 11493 df-xnn0 11564 df-z 11578 df-uz 11887 df-fz 12527 df-fzo 12667 df-hash 13315 df-word 13488 df-concat 13490 df-s1 13491 df-s2 13795 df-s3 13796 df-trkgc 25561 df-trkgb 25562 df-trkgcb 25563 df-trkg 25566 df-cgrg 25620 |
This theorem is referenced by: miduniq 25794 ragflat2 25812 |
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