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Mirrors > Home > MPE Home > Th. List > tgbtwntriv2 | Structured version Visualization version GIF version |
Description: Betweenness always holds for the second endpoint. Theorem 3.1 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 15-Mar-2019.) |
Ref | Expression |
---|---|
tkgeom.p | ⊢ 𝑃 = (Base‘𝐺) |
tkgeom.d | ⊢ − = (dist‘𝐺) |
tkgeom.i | ⊢ 𝐼 = (Itv‘𝐺) |
tkgeom.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
tgbtwntriv2.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
tgbtwntriv2.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
Ref | Expression |
---|---|
tgbtwntriv2 | ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprl 769 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵))) → 𝐵 ∈ (𝐴𝐼𝑥)) | |
2 | tkgeom.p | . . . . . 6 ⊢ 𝑃 = (Base‘𝐺) | |
3 | tkgeom.d | . . . . . 6 ⊢ − = (dist‘𝐺) | |
4 | tkgeom.i | . . . . . 6 ⊢ 𝐼 = (Itv‘𝐺) | |
5 | tkgeom.g | . . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
6 | 5 | ad2antrr 724 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵)) → 𝐺 ∈ TarskiG) |
7 | tgbtwntriv2.2 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
8 | 7 | ad2antrr 724 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵)) → 𝐵 ∈ 𝑃) |
9 | simplr 767 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵)) → 𝑥 ∈ 𝑃) | |
10 | simpr 487 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵)) → (𝐵 − 𝑥) = (𝐵 − 𝐵)) | |
11 | 2, 3, 4, 6, 8, 9, 8, 10 | axtgcgrid 26243 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵)) → 𝐵 = 𝑥) |
12 | 11 | adantrl 714 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵))) → 𝐵 = 𝑥) |
13 | 12 | oveq2d 7166 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵))) → (𝐴𝐼𝐵) = (𝐴𝐼𝑥)) |
14 | 1, 13 | eleqtrrd 2916 | . 2 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵))) → 𝐵 ∈ (𝐴𝐼𝐵)) |
15 | tgbtwntriv2.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
16 | 2, 3, 4, 5, 15, 7, 7, 7 | axtgsegcon 26244 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵))) |
17 | 14, 16 | r19.29a 3289 | 1 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 distcds 16568 TarskiGcstrkg 26210 Itvcitv 26216 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-nul 5202 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-iota 6308 df-fv 6357 df-ov 7153 df-trkgc 26228 df-trkgcb 26230 df-trkg 26233 |
This theorem is referenced by: tgbtwncom 26268 tgbtwntriv1 26271 tgcolg 26334 legid 26367 hlid 26389 lnhl 26395 tglinerflx2 26414 mirreu3 26434 mirconn 26458 symquadlem 26469 outpasch 26535 hlpasch 26536 |
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