| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > lncom | Structured version Visualization version GIF version | ||
| Description: Swapping the points defining a line keeps it unchanged. Part of Theorem 4.11 of [Schwabhauser] p. 34. (Contributed by Thierry Arnoux, 3-Apr-2019.) |
| Ref | Expression |
|---|---|
| btwnlng1.p | ⊢ 𝑃 = (Base‘𝐺) |
| btwnlng1.i | ⊢ 𝐼 = (Itv‘𝐺) |
| btwnlng1.l | ⊢ 𝐿 = (LineG‘𝐺) |
| btwnlng1.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
| btwnlng1.x | ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
| btwnlng1.y | ⊢ (𝜑 → 𝑌 ∈ 𝑃) |
| btwnlng1.z | ⊢ (𝜑 → 𝑍 ∈ 𝑃) |
| btwnlng1.d | ⊢ (𝜑 → 𝑋 ≠ 𝑌) |
| lncom.1 | ⊢ (𝜑 → 𝑍 ∈ (𝑌𝐿𝑋)) |
| Ref | Expression |
|---|---|
| lncom | ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lncom.1 | . 2 ⊢ (𝜑 → 𝑍 ∈ (𝑌𝐿𝑋)) | |
| 2 | 3orcomb 1094 | . . . 4 ⊢ ((𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑋 ∈ (𝑍𝐼𝑌))) | |
| 3 | btwnlng1.p | . . . . . 6 ⊢ 𝑃 = (Base‘𝐺) | |
| 4 | eqid 2737 | . . . . . 6 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
| 5 | btwnlng1.i | . . . . . 6 ⊢ 𝐼 = (Itv‘𝐺) | |
| 6 | btwnlng1.g | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
| 7 | btwnlng1.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
| 8 | btwnlng1.z | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑃) | |
| 9 | btwnlng1.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
| 10 | 3, 4, 5, 6, 7, 8, 9 | tgbtwncomb 28497 | . . . . 5 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ↔ 𝑍 ∈ (𝑌𝐼𝑋))) |
| 11 | 3, 4, 5, 6, 7, 9, 8 | tgbtwncomb 28497 | . . . . 5 ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐼𝑍) ↔ 𝑌 ∈ (𝑍𝐼𝑋))) |
| 12 | 3, 4, 5, 6, 8, 7, 9 | tgbtwncomb 28497 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ (𝑍𝐼𝑌) ↔ 𝑋 ∈ (𝑌𝐼𝑍))) |
| 13 | 10, 11, 12 | 3orbi123d 1437 | . . . 4 ⊢ (𝜑 → ((𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑋 ∈ (𝑍𝐼𝑌)) ↔ (𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑋 ∈ (𝑌𝐼𝑍)))) |
| 14 | 2, 13 | bitrid 283 | . . 3 ⊢ (𝜑 → ((𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)) ↔ (𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑋 ∈ (𝑌𝐼𝑍)))) |
| 15 | btwnlng1.l | . . . 4 ⊢ 𝐿 = (LineG‘𝐺) | |
| 16 | btwnlng1.d | . . . 4 ⊢ (𝜑 → 𝑋 ≠ 𝑌) | |
| 17 | 3, 15, 5, 6, 7, 9, 16, 8 | tgellng 28561 | . . 3 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))) |
| 18 | 16 | necomd 2996 | . . . 4 ⊢ (𝜑 → 𝑌 ≠ 𝑋) |
| 19 | 3, 15, 5, 6, 9, 7, 18, 8 | tgellng 28561 | . . 3 ⊢ (𝜑 → (𝑍 ∈ (𝑌𝐿𝑋) ↔ (𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑋 ∈ (𝑌𝐼𝑍)))) |
| 20 | 14, 17, 19 | 3bitr4d 311 | . 2 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ 𝑍 ∈ (𝑌𝐿𝑋))) |
| 21 | 1, 20 | mpbird 257 | 1 ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∨ w3o 1086 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 distcds 17306 TarskiGcstrkg 28435 Itvcitv 28441 LineGclng 28442 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 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-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-trkgc 28456 df-trkgb 28457 df-trkgcb 28458 df-trkg 28461 |
| This theorem is referenced by: tglineelsb2 28640 tglinecom 28643 ncolncol 28654 coltr 28655 midexlem 28700 footexALT 28726 footexlem1 28727 footexlem2 28728 opphllem1 28755 opphllem2 28756 outpasch 28763 hlpasch 28764 trgcopy 28812 trgcopyeulem 28813 cgracgr 28826 tgasa1 28866 |
| Copyright terms: Public domain | W3C validator |