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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 25382 | . 2 ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐼𝐴)) |
8 | 1, 2, 3, 4, 5, 6, 6, 7 | tgbtwncom 25383 | 1 ⊢ (𝜑 → 𝐴 ∈ (𝐴𝐼𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 distcds 15950 TarskiGcstrkg 25329 Itvcitv 25335 |
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-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 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-iota 5851 df-fv 5896 df-ov 6653 df-trkgc 25347 df-trkgb 25348 df-trkgcb 25349 df-trkg 25352 |
This theorem is referenced by: tgldim0itv 25399 legtri3 25485 leg0 25487 legbtwn 25489 ncolne1 25520 tglnne 25523 tglinerflx1 25528 mirinv 25561 miriso 25565 colmid 25583 krippenlem 25585 colperpex 25625 outpasch 25647 hlpasch 25648 |
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