Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 1095 | . . . 4 ⊢ ((𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑋 ∈ (𝑍𝐼𝑌))) | |
3 | tglngval.p | . . . . . 6 ⊢ 𝑃 = (Base‘𝐺) | |
4 | eqid 2739 | . . . . . 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 26438 | . . . . 5 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ↔ 𝑍 ∈ (𝑌𝐼𝑋))) |
11 | 3, 4, 5, 6, 7, 9, 8 | tgbtwncomb 26438 | . . . . 5 ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐼𝑍) ↔ 𝑌 ∈ (𝑍𝐼𝑋))) |
12 | 3, 4, 5, 6, 8, 7, 9 | tgbtwncomb 26438 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ (𝑍𝐼𝑌) ↔ 𝑋 ∈ (𝑌𝐼𝑍))) |
13 | 10, 11, 12 | 3orbi123d 1436 | . . . 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 | tglngval.l | . . . 4 ⊢ 𝐿 = (LineG‘𝐺) | |
16 | 3, 15, 5, 6, 7, 9, 8 | tgcolg 26503 | . . 3 ⊢ (𝜑 → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))) |
17 | 3, 15, 5, 6, 9, 7, 8 | tgcolg 26503 | . . 3 ⊢ (𝜑 → ((𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋) ↔ (𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑋 ∈ (𝑌𝐼𝑍)))) |
18 | 14, 16, 17 | 3bitr4d 314 | . 2 ⊢ (𝜑 → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋))) |
19 | 1, 18 | mpbid 235 | 1 ⊢ (𝜑 → (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 846 ∨ w3o 1087 = wceq 1542 ∈ wcel 2114 ‘cfv 6340 (class class class)co 7173 Basecbs 16589 distcds 16680 TarskiGcstrkg 26379 Itvcitv 26385 LineGclng 26386 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5168 ax-nul 5175 ax-pr 5297 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3401 df-sbc 3682 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-br 5032 df-opab 5094 df-id 5430 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-iota 6298 df-fun 6342 df-fv 6348 df-ov 7176 df-oprab 7177 df-mpo 7178 df-trkgc 26397 df-trkgb 26398 df-trkgcb 26399 df-trkg 26402 |
This theorem is referenced by: ncolcom 26510 tglineeltr 26580 mirtrcgr 26632 symquadlem 26638 midexlem 26641 colperpexlem1 26679 mideulem2 26683 opphllem 26684 hlpasch 26705 colhp 26719 trgcopy 26753 cgrg3col4 26802 tgasa1 26807 |
Copyright terms: Public domain | W3C validator |