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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 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝐺 ∈ TarskiG) |
6 | tglngval.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
7 | 6 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ 𝑃) |
8 | tgcolg.z | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑃) | |
9 | 8 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑍 ∈ 𝑃) |
10 | tgidinside.1 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐼𝑌)) | |
11 | 10 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑍 ∈ (𝑋𝐼𝑌)) |
12 | simpr 476 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌) | |
13 | 12 | oveq2d 6565 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑋𝐼𝑋) = (𝑋𝐼𝑌)) |
14 | 11, 13 | eleqtrrd 2691 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑍 ∈ (𝑋𝐼𝑋)) |
15 | 1, 2, 3, 5, 7, 9, 14 | axtgbtwnid 25165 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 = 𝑍) |
16 | lnxfr.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
17 | 16 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝐴 ∈ 𝑃) |
18 | tgidinside.2 | . . . . 5 ⊢ (𝜑 → (𝑋 − 𝑍) = (𝑋 − 𝐴)) | |
19 | 18 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑋 − 𝑍) = (𝑋 − 𝐴)) |
20 | 1, 2, 3, 5, 7, 9, 7, 17, 19, 15 | tgcgreq 25177 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 = 𝐴) |
21 | 15, 20 | eqtr3d 2646 | . 2 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑍 = 𝐴) |
22 | tglngval.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
23 | 4 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝐺 ∈ TarskiG) |
24 | 6 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ∈ 𝑃) |
25 | tglngval.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
26 | 25 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑌 ∈ 𝑃) |
27 | 8 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑍 ∈ 𝑃) |
28 | lnxfr.r | . . 3 ⊢ ∼ = (cgrG‘𝐺) | |
29 | 16 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝐴 ∈ 𝑃) |
30 | lnxfr.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
31 | 30 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝐵 ∈ 𝑃) |
32 | simpr 476 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ≠ 𝑌) | |
33 | 1, 22, 3, 4, 6, 8, 25, 10 | btwncolg3 25252 | . . . 4 ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍)) |
34 | 33 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍)) |
35 | 18 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑋 − 𝑍) = (𝑋 − 𝐴)) |
36 | tgidinside.3 | . . . 4 ⊢ (𝜑 → (𝑌 − 𝑍) = (𝑌 − 𝐴)) | |
37 | 36 | adantr 480 | . . 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 25265 | . 2 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑍 = 𝐴) |
39 | 21, 38 | pm2.61dane 2869 | 1 ⊢ (𝜑 → 𝑍 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 distcds 15777 TarskiGcstrkg 25129 Itvcitv 25135 LineGclng 25136 cgrGccgrg 25205 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-concat 13156 df-s1 13157 df-s2 13444 df-s3 13445 df-trkgc 25147 df-trkgb 25148 df-trkgcb 25149 df-trkg 25152 df-cgrg 25206 |
This theorem is referenced by: miduniq 25380 ragflat2 25398 |
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