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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 1048 | . . . 4 ⊢ ((𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑋 ∈ (𝑍𝐼𝑌))) | |
3 | btwnlng1.p | . . . . . 6 ⊢ 𝑃 = (Base‘𝐺) | |
4 | eqid 2622 | . . . . . 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 25384 | . . . . 5 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ↔ 𝑍 ∈ (𝑌𝐼𝑋))) |
11 | 3, 4, 5, 6, 7, 9, 8 | tgbtwncomb 25384 | . . . . 5 ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐼𝑍) ↔ 𝑌 ∈ (𝑍𝐼𝑋))) |
12 | 3, 4, 5, 6, 8, 7, 9 | tgbtwncomb 25384 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ (𝑍𝐼𝑌) ↔ 𝑋 ∈ (𝑌𝐼𝑍))) |
13 | 10, 11, 12 | 3orbi123d 1398 | . . . 4 ⊢ (𝜑 → ((𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑋 ∈ (𝑍𝐼𝑌)) ↔ (𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑋 ∈ (𝑌𝐼𝑍)))) |
14 | 2, 13 | syl5bb 272 | . . 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 25448 | . . 3 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))) |
18 | 16 | necomd 2849 | . . . 4 ⊢ (𝜑 → 𝑌 ≠ 𝑋) |
19 | 3, 15, 5, 6, 9, 7, 18, 8 | tgellng 25448 | . . 3 ⊢ (𝜑 → (𝑍 ∈ (𝑌𝐿𝑋) ↔ (𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑋 ∈ (𝑌𝐼𝑍)))) |
20 | 14, 17, 19 | 3bitr4d 300 | . 2 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ 𝑍 ∈ (𝑌𝐿𝑋))) |
21 | 1, 20 | mpbird 247 | 1 ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1036 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 distcds 15950 TarskiGcstrkg 25329 Itvcitv 25335 LineGclng 25336 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-trkgc 25347 df-trkgb 25348 df-trkgcb 25349 df-trkg 25352 |
This theorem is referenced by: tglineelsb2 25527 tglinecom 25530 ncolncol 25541 coltr 25542 midexlem 25587 footex 25613 opphllem1 25639 opphllem2 25640 outpasch 25647 hlpasch 25648 trgcopy 25696 trgcopyeulem 25697 cgracgr 25710 tgasa1 25739 |
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