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Mirrors > Home > MPE Home > Th. List > tgbtwncomb | Structured version Visualization version GIF version |
Description: Betweenness commutes, biconditional version. (Contributed by Thierry Arnoux, 3-Apr-2019.) |
Ref | Expression |
---|---|
tkgeom.p | β’ π = (BaseβπΊ) |
tkgeom.d | β’ β = (distβπΊ) |
tkgeom.i | β’ πΌ = (ItvβπΊ) |
tkgeom.g | β’ (π β πΊ β TarskiG) |
tgbtwntriv2.1 | β’ (π β π΄ β π) |
tgbtwntriv2.2 | β’ (π β π΅ β π) |
tgbtwncomb.3 | β’ (π β πΆ β π) |
Ref | Expression |
---|---|
tgbtwncomb | β’ (π β (π΅ β (π΄πΌπΆ) β π΅ β (πΆπΌπ΄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tkgeom.p | . . 3 β’ π = (BaseβπΊ) | |
2 | tkgeom.d | . . 3 β’ β = (distβπΊ) | |
3 | tkgeom.i | . . 3 β’ πΌ = (ItvβπΊ) | |
4 | tkgeom.g | . . . 4 β’ (π β πΊ β TarskiG) | |
5 | 4 | adantr 480 | . . 3 β’ ((π β§ π΅ β (π΄πΌπΆ)) β πΊ β TarskiG) |
6 | tgbtwntriv2.1 | . . . 4 β’ (π β π΄ β π) | |
7 | 6 | adantr 480 | . . 3 β’ ((π β§ π΅ β (π΄πΌπΆ)) β π΄ β π) |
8 | tgbtwntriv2.2 | . . . 4 β’ (π β π΅ β π) | |
9 | 8 | adantr 480 | . . 3 β’ ((π β§ π΅ β (π΄πΌπΆ)) β π΅ β π) |
10 | tgbtwncomb.3 | . . . 4 β’ (π β πΆ β π) | |
11 | 10 | adantr 480 | . . 3 β’ ((π β§ π΅ β (π΄πΌπΆ)) β πΆ β π) |
12 | simpr 484 | . . 3 β’ ((π β§ π΅ β (π΄πΌπΆ)) β π΅ β (π΄πΌπΆ)) | |
13 | 1, 2, 3, 5, 7, 9, 11, 12 | tgbtwncom 28007 | . 2 β’ ((π β§ π΅ β (π΄πΌπΆ)) β π΅ β (πΆπΌπ΄)) |
14 | 4 | adantr 480 | . . 3 β’ ((π β§ π΅ β (πΆπΌπ΄)) β πΊ β TarskiG) |
15 | 10 | adantr 480 | . . 3 β’ ((π β§ π΅ β (πΆπΌπ΄)) β πΆ β π) |
16 | 8 | adantr 480 | . . 3 β’ ((π β§ π΅ β (πΆπΌπ΄)) β π΅ β π) |
17 | 6 | adantr 480 | . . 3 β’ ((π β§ π΅ β (πΆπΌπ΄)) β π΄ β π) |
18 | simpr 484 | . . 3 β’ ((π β§ π΅ β (πΆπΌπ΄)) β π΅ β (πΆπΌπ΄)) | |
19 | 1, 2, 3, 14, 15, 16, 17, 18 | tgbtwncom 28007 | . 2 β’ ((π β§ π΅ β (πΆπΌπ΄)) β π΅ β (π΄πΌπΆ)) |
20 | 13, 19 | impbida 798 | 1 β’ (π β (π΅ β (π΄πΌπΆ) β π΅ β (πΆπΌπ΄))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1540 β wcel 2105 βcfv 6543 (class class class)co 7412 Basecbs 17149 distcds 17211 TarskiGcstrkg 27946 Itvcitv 27952 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-nul 5306 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-iota 6495 df-fv 6551 df-ov 7415 df-trkgc 27967 df-trkgb 27968 df-trkgcb 27969 df-trkg 27972 |
This theorem is referenced by: colcom 28077 colrot1 28078 lnhl 28134 lncom 28141 lnrot1 28142 lnrot2 28143 mirreu3 28173 |
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