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| Mirrors > Home > MPE Home > Th. List > colcom | 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 |
|---|---|
| tglngval.p | ⊢ 𝑃 = (Base‘𝐺) |
| tglngval.l | ⊢ 𝐿 = (LineG‘𝐺) |
| tglngval.i | ⊢ 𝐼 = (Itv‘𝐺) |
| tglngval.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
| tglngval.x | ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
| tglngval.y | ⊢ (𝜑 → 𝑌 ∈ 𝑃) |
| tgcolg.z | ⊢ (𝜑 → 𝑍 ∈ 𝑃) |
| colrot | ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) |
| Ref | Expression |
|---|---|
| colcom | ⊢ (𝜑 → (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | colrot | . 2 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) | |
| 2 | 3orcomb 1094 | . . . 4 ⊢ ((𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑋 ∈ (𝑍𝐼𝑌))) | |
| 3 | tglngval.p | . . . . . 6 ⊢ 𝑃 = (Base‘𝐺) | |
| 4 | eqid 2737 | . . . . . 6 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
| 5 | tglngval.i | . . . . . 6 ⊢ 𝐼 = (Itv‘𝐺) | |
| 6 | tglngval.g | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
| 7 | tglngval.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
| 8 | tgcolg.z | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑃) | |
| 9 | tglngval.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 | tglngval.l | . . . 4 ⊢ 𝐿 = (LineG‘𝐺) | |
| 16 | 3, 15, 5, 6, 7, 9, 8 | tgcolg 28562 | . . 3 ⊢ (𝜑 → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))) |
| 17 | 3, 15, 5, 6, 9, 7, 8 | tgcolg 28562 | . . 3 ⊢ (𝜑 → ((𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋) ↔ (𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑋 ∈ (𝑌𝐼𝑍)))) |
| 18 | 14, 16, 17 | 3bitr4d 311 | . 2 ⊢ (𝜑 → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋))) |
| 19 | 1, 18 | mpbid 232 | 1 ⊢ (𝜑 → (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∨ wo 848 ∨ w3o 1086 = wceq 1540 ∈ wcel 2108 ‘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: ncolcom 28569 tglineeltr 28639 mirtrcgr 28691 symquadlem 28697 midexlem 28700 colperpexlem1 28738 mideulem2 28742 opphllem 28743 hlpasch 28764 colhp 28778 trgcopy 28812 cgrg3col4 28861 tgasa1 28866 |
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