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Mirrors > Home > MPE Home > Th. List > tgbtwnouttr2 | Structured version Visualization version Unicode version |
Description: Outer transitivity law for betweenness. Left-hand side of Theorem 3.7 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 18-Mar-2019.) |
Ref | Expression |
---|---|
tkgeom.p | |
tkgeom.d | |
tkgeom.i | Itv |
tkgeom.g | TarskiG |
tgbtwnintr.1 | |
tgbtwnintr.2 | |
tgbtwnintr.3 | |
tgbtwnintr.4 | |
tgbtwnouttr2.1 | |
tgbtwnouttr2.2 | |
tgbtwnouttr2.3 |
Ref | Expression |
---|---|
tgbtwnouttr2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprl 794 | . . 3 | |
2 | tkgeom.p | . . . . 5 | |
3 | tkgeom.d | . . . . 5 | |
4 | tkgeom.i | . . . . 5 Itv | |
5 | tkgeom.g | . . . . . 6 TarskiG | |
6 | 5 | ad2antrr 762 | . . . . 5 TarskiG |
7 | tgbtwnintr.3 | . . . . . 6 | |
8 | 7 | ad2antrr 762 | . . . . 5 |
9 | tgbtwnintr.4 | . . . . . 6 | |
10 | 9 | ad2antrr 762 | . . . . 5 |
11 | tgbtwnintr.2 | . . . . . 6 | |
12 | 11 | ad2antrr 762 | . . . . 5 |
13 | simplr 792 | . . . . 5 | |
14 | tgbtwnouttr2.1 | . . . . . 6 | |
15 | 14 | ad2antrr 762 | . . . . 5 |
16 | tgbtwnintr.1 | . . . . . . 7 | |
17 | 16 | ad2antrr 762 | . . . . . 6 |
18 | tgbtwnouttr2.2 | . . . . . . 7 | |
19 | 18 | ad2antrr 762 | . . . . . 6 |
20 | 2, 3, 4, 6, 17, 12, 8, 13, 19, 1 | tgbtwnexch3 25389 | . . . . 5 |
21 | tgbtwnouttr2.3 | . . . . . 6 | |
22 | 21 | ad2antrr 762 | . . . . 5 |
23 | simprr 796 | . . . . 5 | |
24 | eqidd 2623 | . . . . 5 | |
25 | 2, 3, 4, 6, 8, 8, 10, 12, 13, 10, 15, 20, 22, 23, 24 | tgsegconeq 25381 | . . . 4 |
26 | 25 | oveq2d 6666 | . . 3 |
27 | 1, 26 | eleqtrd 2703 | . 2 |
28 | 2, 3, 4, 5, 16, 7, 7, 9 | axtgsegcon 25363 | . 2 |
29 | 27, 28 | r19.29a 3078 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 wne 2794 cfv 5888 (class class class)co 6650 cbs 15857 cds 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-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-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: tgbtwnexch2 25391 tgbtwnouttr 25392 tgbtwnconn22 25474 tglineeltr 25526 mirconn 25573 footex 25613 |
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