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| Mirrors > Home > MPE Home > Th. List > tgbtwnswapid | Structured version Visualization version GIF version | ||
| Description: If you can swap the first two arguments of a betweenness statement, then those arguments are identical. Theorem 3.4 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 16-Mar-2019.) | 
| Ref | Expression | 
|---|---|
| tkgeom.p | ⊢ 𝑃 = (Base‘𝐺) | 
| tkgeom.d | ⊢ − = (dist‘𝐺) | 
| tkgeom.i | ⊢ 𝐼 = (Itv‘𝐺) | 
| tkgeom.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) | 
| tgbtwnswapid.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑃) | 
| tgbtwnswapid.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑃) | 
| tgbtwnswapid.3 | ⊢ (𝜑 → 𝐶 ∈ 𝑃) | 
| tgbtwnswapid.4 | ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐼𝐶)) | 
| tgbtwnswapid.5 | ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) | 
| Ref | Expression | 
|---|---|
| tgbtwnswapid | ⊢ (𝜑 → 𝐴 = 𝐵) | 
| 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 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐺 ∈ TarskiG) | 
| 6 | tgbtwnswapid.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
| 7 | 6 | ad2antrr 726 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐴 ∈ 𝑃) | 
| 8 | simplr 768 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥 ∈ 𝑃) | |
| 9 | simprl 770 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥 ∈ (𝐴𝐼𝐴)) | |
| 10 | 1, 2, 3, 5, 7, 8, 9 | axtgbtwnid 28475 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐴 = 𝑥) | 
| 11 | tgbtwnswapid.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
| 12 | 11 | ad2antrr 726 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐵 ∈ 𝑃) | 
| 13 | simprr 772 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥 ∈ (𝐵𝐼𝐵)) | |
| 14 | 1, 2, 3, 5, 12, 8, 13 | axtgbtwnid 28475 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐵 = 𝑥) | 
| 15 | 10, 14 | eqtr4d 2779 | . 2 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐴 = 𝐵) | 
| 16 | tgbtwnswapid.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
| 17 | tgbtwnswapid.4 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐼𝐶)) | |
| 18 | tgbtwnswapid.5 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) | |
| 19 | 1, 2, 3, 4, 11, 6, 16, 6, 11, 17, 18 | axtgpasch 28476 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) | 
| 20 | 15, 19 | r19.29a 3161 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ‘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-trkgb 28458 df-trkg 28462 | 
| This theorem is referenced by: legtri3 28599 | 
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