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Mirrors > Home > MPE Home > Th. List > tgbtwnintr | Structured version Visualization version GIF version |
Description: Inner transitivity law for betweenness. Left-hand side of Theorem 3.5 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 18-Mar-2019.) |
Ref | Expression |
---|---|
tkgeom.p | ⊢ 𝑃 = (Base‘𝐺) |
tkgeom.d | ⊢ − = (dist‘𝐺) |
tkgeom.i | ⊢ 𝐼 = (Itv‘𝐺) |
tkgeom.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
tgbtwnintr.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
tgbtwnintr.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
tgbtwnintr.3 | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
tgbtwnintr.4 | ⊢ (𝜑 → 𝐷 ∈ 𝑃) |
tgbtwnintr.5 | ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐼𝐷)) |
tgbtwnintr.6 | ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐷)) |
Ref | Expression |
---|---|
tgbtwnintr | ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tkgeom.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
2 | tkgeom.d | . . . 4 ⊢ − = (dist‘𝐺) | |
3 | tkgeom.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | tkgeom.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | 4 | ad2antrr 723 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐺 ∈ TarskiG) |
6 | tgbtwnintr.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
7 | 6 | ad2antrr 723 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐵 ∈ 𝑃) |
8 | simplr 766 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥 ∈ 𝑃) | |
9 | simprr 770 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥 ∈ (𝐵𝐼𝐵)) | |
10 | 1, 2, 3, 5, 7, 8, 9 | axtgbtwnid 28189 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐵 = 𝑥) |
11 | simprl 768 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥 ∈ (𝐴𝐼𝐶)) | |
12 | 10, 11 | eqeltrd 2825 | . 2 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐵 ∈ (𝐴𝐼𝐶)) |
13 | tgbtwnintr.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
14 | tgbtwnintr.4 | . . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
15 | tgbtwnintr.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
16 | tgbtwnintr.5 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐼𝐷)) | |
17 | tgbtwnintr.6 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐷)) | |
18 | 1, 2, 3, 4, 6, 13, 14, 15, 6, 16, 17 | axtgpasch 28190 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) |
19 | 12, 18 | r19.29a 3154 | 1 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ‘cfv 6534 (class class class)co 7402 Basecbs 17145 distcds 17207 TarskiGcstrkg 28150 Itvcitv 28156 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-nul 5297 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3771 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-iota 6486 df-fv 6542 df-ov 7405 df-trkgb 28172 df-trkg 28176 |
This theorem is referenced by: tgbtwnexch3 28217 tgbtwnexch2 28219 tgbtwnconn1lem3 28297 tgbtwnconn3 28300 tgbtwnconn22 28302 tglineeltr 28354 mirconn 28401 |
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