Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > tgbtwntriv1 | Structured version Visualization version GIF version |
Description: Betweenness always holds for the first endpoint. Theorem 3.3 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 15-Mar-2019.) |
Ref | Expression |
---|---|
tkgeom.p | ⊢ 𝑃 = (Base‘𝐺) |
tkgeom.d | ⊢ − = (dist‘𝐺) |
tkgeom.i | ⊢ 𝐼 = (Itv‘𝐺) |
tkgeom.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
tgbtwntriv2.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
tgbtwntriv2.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
Ref | Expression |
---|---|
tgbtwntriv1 | ⊢ (𝜑 → 𝐴 ∈ (𝐴𝐼𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tkgeom.p | . 2 ⊢ 𝑃 = (Base‘𝐺) | |
2 | tkgeom.d | . 2 ⊢ − = (dist‘𝐺) | |
3 | tkgeom.i | . 2 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | tkgeom.g | . 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | tgbtwntriv2.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
6 | tgbtwntriv2.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
7 | 1, 2, 3, 4, 5, 6 | tgbtwntriv2 26425 | . 2 ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐼𝐴)) |
8 | 1, 2, 3, 4, 5, 6, 6, 7 | tgbtwncom 26426 | 1 ⊢ (𝜑 → 𝐴 ∈ (𝐴𝐼𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2113 ‘cfv 6333 (class class class)co 7164 Basecbs 16579 distcds 16670 TarskiGcstrkg 26368 Itvcitv 26374 |
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 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-nul 5171 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-sbc 3680 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-iota 6291 df-fv 6341 df-ov 7167 df-trkgc 26386 df-trkgb 26387 df-trkgcb 26388 df-trkg 26391 |
This theorem is referenced by: tgldim0itv 26442 legtri3 26528 leg0 26530 legbtwn 26532 ncolne1 26563 tglnne 26566 tglinerflx1 26571 mirinv 26604 miriso 26608 colmid 26626 krippenlem 26628 colperpex 26671 outpasch 26693 hlpasch 26694 |
Copyright terms: Public domain | W3C validator |