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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 1091 | . . . 4 ⊢ ((𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑋 ∈ (𝑍𝐼𝑌))) | |
3 | btwnlng1.p | . . . . . 6 ⊢ 𝑃 = (Base‘𝐺) | |
4 | eqid 2798 | . . . . . 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 26283 | . . . . 5 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ↔ 𝑍 ∈ (𝑌𝐼𝑋))) |
11 | 3, 4, 5, 6, 7, 9, 8 | tgbtwncomb 26283 | . . . . 5 ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐼𝑍) ↔ 𝑌 ∈ (𝑍𝐼𝑋))) |
12 | 3, 4, 5, 6, 8, 7, 9 | tgbtwncomb 26283 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ (𝑍𝐼𝑌) ↔ 𝑋 ∈ (𝑌𝐼𝑍))) |
13 | 10, 11, 12 | 3orbi123d 1432 | . . . 4 ⊢ (𝜑 → ((𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑋 ∈ (𝑍𝐼𝑌)) ↔ (𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑋 ∈ (𝑌𝐼𝑍)))) |
14 | 2, 13 | syl5bb 286 | . . 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 26347 | . . 3 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))) |
18 | 16 | necomd 3042 | . . . 4 ⊢ (𝜑 → 𝑌 ≠ 𝑋) |
19 | 3, 15, 5, 6, 9, 7, 18, 8 | tgellng 26347 | . . 3 ⊢ (𝜑 → (𝑍 ∈ (𝑌𝐿𝑋) ↔ (𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑋 ∈ (𝑌𝐼𝑍)))) |
20 | 14, 17, 19 | 3bitr4d 314 | . 2 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ 𝑍 ∈ (𝑌𝐿𝑋))) |
21 | 1, 20 | mpbird 260 | 1 ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1083 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 distcds 16566 TarskiGcstrkg 26224 Itvcitv 26230 LineGclng 26231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-trkgc 26242 df-trkgb 26243 df-trkgcb 26244 df-trkg 26247 |
This theorem is referenced by: tglineelsb2 26426 tglinecom 26429 ncolncol 26440 coltr 26441 midexlem 26486 footexALT 26512 footexlem1 26513 footexlem2 26514 opphllem1 26541 opphllem2 26542 outpasch 26549 hlpasch 26550 trgcopy 26598 trgcopyeulem 26599 cgracgr 26612 tgasa1 26652 |
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