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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 719 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐺 ∈ TarskiG) |
6 | tgbtwnintr.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
7 | 6 | ad2antrr 719 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐵 ∈ 𝑃) |
8 | simplr 787 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥 ∈ 𝑃) | |
9 | simprr 791 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥 ∈ (𝐵𝐼𝐵)) | |
10 | 1, 2, 3, 5, 7, 8, 9 | axtgbtwnid 25778 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐵 = 𝑥) |
11 | simprl 789 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥 ∈ (𝐴𝐼𝐶)) | |
12 | 10, 11 | eqeltrd 2906 | . 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 25779 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) |
19 | 12, 18 | r19.29a 3288 | 1 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ‘cfv 6123 (class class class)co 6905 Basecbs 16222 distcds 16314 TarskiGcstrkg 25742 Itvcitv 25748 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-nul 5013 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-iota 6086 df-fv 6131 df-ov 6908 df-trkgb 25761 df-trkg 25765 |
This theorem is referenced by: tgbtwnexch3 25806 tgbtwnexch2 25808 tgbtwnconn1lem3 25886 tgbtwnconn3 25889 tgbtwnconn22 25891 tglineeltr 25943 mirconn 25990 |
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