|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > tgbtwnexch2 | Structured version Visualization version GIF version | ||
| Description: Exchange the outer point of two betweenness statements. Right-hand side of Theorem 3.5 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 23-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) | 
| tgbtwnexch2.1 | ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷)) | 
| tgbtwnexch2.2 | ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷)) | 
| Ref | Expression | 
|---|---|
| tgbtwnexch2 | ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐵 = 𝐶) | |
| 2 | tgbtwnexch2.1 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷)) | |
| 3 | 2 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐵 ∈ (𝐴𝐼𝐷)) | 
| 4 | 1, 3 | eqeltrrd 2841 | . 2 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐶 ∈ (𝐴𝐼𝐷)) | 
| 5 | tkgeom.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
| 6 | tkgeom.d | . . 3 ⊢ − = (dist‘𝐺) | |
| 7 | tkgeom.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
| 8 | tkgeom.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
| 9 | 8 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐺 ∈ TarskiG) | 
| 10 | tgbtwnintr.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
| 11 | 10 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐴 ∈ 𝑃) | 
| 12 | tgbtwnintr.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
| 13 | 12 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ 𝑃) | 
| 14 | tgbtwnintr.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
| 15 | 14 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ 𝑃) | 
| 16 | tgbtwnintr.4 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
| 17 | 16 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐷 ∈ 𝑃) | 
| 18 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ≠ 𝐶) | |
| 19 | tgbtwnexch2.2 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷)) | |
| 20 | 19 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ (𝐵𝐼𝐷)) | 
| 21 | 2 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (𝐴𝐼𝐷)) | 
| 22 | 5, 6, 7, 9, 15, 13, 11, 17, 20, 21 | tgbtwnintr 28502 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (𝐶𝐼𝐴)) | 
| 23 | 5, 6, 7, 9, 15, 13, 11, 22 | tgbtwncom 28497 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (𝐴𝐼𝐶)) | 
| 24 | 5, 6, 7, 9, 11, 13, 15, 17, 18, 23, 20 | tgbtwnouttr2 28504 | . 2 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ (𝐴𝐼𝐷)) | 
| 25 | 4, 24 | pm2.61dane 3028 | 1 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 distcds 17307 TarskiGcstrkg 28436 Itvcitv 28442 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-nul 5305 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-ov 7435 df-trkgc 28457 df-trkgb 28458 df-trkgcb 28459 df-trkg 28462 | 
| This theorem is referenced by: tgbtwnexch 28507 tgtrisegint 28508 tgbtwnconn1lem3 28583 legtri3 28599 miriso 28679 | 
| Copyright terms: Public domain | W3C validator |