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| 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 726 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐺 ∈ TarskiG) | 
| 6 | tgbtwnintr.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
| 7 | 6 | ad2antrr 726 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐵 ∈ 𝑃) | 
| 8 | simplr 769 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥 ∈ 𝑃) | |
| 9 | simprr 773 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥 ∈ (𝐵𝐼𝐵)) | |
| 10 | 1, 2, 3, 5, 7, 8, 9 | axtgbtwnid 28474 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐵 = 𝑥) | 
| 11 | simprl 771 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥 ∈ (𝐴𝐼𝐶)) | |
| 12 | 10, 11 | eqeltrd 2841 | . 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 28475 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) | 
| 19 | 12, 18 | r19.29a 3162 | 1 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 distcds 17306 TarskiGcstrkg 28435 Itvcitv 28441 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-trkgb 28457 df-trkg 28461 | 
| This theorem is referenced by: tgbtwnexch3 28502 tgbtwnexch2 28504 tgbtwnconn1lem3 28582 tgbtwnconn3 28585 tgbtwnconn22 28587 tglineeltr 28639 mirconn 28686 | 
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