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Mirrors > Home > MPE Home > Th. List > tgsegconeq | Structured version Visualization version GIF version |
Description: Two points that satisfy the conclusion of axtgsegcon 27982 are identical. Uniqueness portion of Theorem 2.12 of [Schwabhauser] p. 29. (Contributed by Thierry Arnoux, 23-Mar-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) |
tgcgrextend.a | β’ (π β π΄ β π) |
tgcgrextend.b | β’ (π β π΅ β π) |
tgcgrextend.c | β’ (π β πΆ β π) |
tgcgrextend.d | β’ (π β π· β π) |
tgcgrextend.e | β’ (π β πΈ β π) |
tgcgrextend.f | β’ (π β πΉ β π) |
tgsegconeq.1 | β’ (π β π· β π΄) |
tgsegconeq.2 | β’ (π β π΄ β (π·πΌπΈ)) |
tgsegconeq.3 | β’ (π β π΄ β (π·πΌπΉ)) |
tgsegconeq.4 | β’ (π β (π΄ β πΈ) = (π΅ β πΆ)) |
tgsegconeq.5 | β’ (π β (π΄ β πΉ) = (π΅ β πΆ)) |
Ref | Expression |
---|---|
tgsegconeq | β’ (π β πΈ = πΉ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tkgeom.p | . 2 β’ π = (BaseβπΊ) | |
2 | tkgeom.d | . 2 β’ β = (distβπΊ) | |
3 | tkgeom.i | . 2 β’ πΌ = (ItvβπΊ) | |
4 | tkgeom.g | . 2 β’ (π β πΊ β TarskiG) | |
5 | tgcgrextend.e | . 2 β’ (π β πΈ β π) | |
6 | tgcgrextend.f | . 2 β’ (π β πΉ β π) | |
7 | tgcgrextend.d | . . . 4 β’ (π β π· β π) | |
8 | tgcgrextend.a | . . . 4 β’ (π β π΄ β π) | |
9 | tgsegconeq.1 | . . . 4 β’ (π β π· β π΄) | |
10 | tgsegconeq.2 | . . . 4 β’ (π β π΄ β (π·πΌπΈ)) | |
11 | eqidd 2731 | . . . 4 β’ (π β (π· β π΄) = (π· β π΄)) | |
12 | eqidd 2731 | . . . 4 β’ (π β (π΄ β πΈ) = (π΄ β πΈ)) | |
13 | tgsegconeq.3 | . . . . 5 β’ (π β π΄ β (π·πΌπΉ)) | |
14 | tgsegconeq.4 | . . . . . 6 β’ (π β (π΄ β πΈ) = (π΅ β πΆ)) | |
15 | tgsegconeq.5 | . . . . . 6 β’ (π β (π΄ β πΉ) = (π΅ β πΆ)) | |
16 | 14, 15 | eqtr4d 2773 | . . . . 5 β’ (π β (π΄ β πΈ) = (π΄ β πΉ)) |
17 | 1, 2, 3, 4, 7, 8, 5, 7, 8, 6, 10, 13, 11, 16 | tgcgrextend 28003 | . . . 4 β’ (π β (π· β πΈ) = (π· β πΉ)) |
18 | 1, 2, 3, 4, 7, 8, 5, 7, 8, 5, 5, 6, 9, 10, 10, 11, 12, 17, 16 | axtg5seg 27983 | . . 3 β’ (π β (πΈ β πΈ) = (πΈ β πΉ)) |
19 | 18 | eqcomd 2736 | . 2 β’ (π β (πΈ β πΉ) = (πΈ β πΈ)) |
20 | 1, 2, 3, 4, 5, 6, 5, 19 | axtgcgrid 27981 | 1 β’ (π β πΈ = πΉ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1539 β wcel 2104 β wne 2938 βcfv 6542 (class class class)co 7411 Basecbs 17148 distcds 17210 TarskiGcstrkg 27945 Itvcitv 27951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 ax-nul 5305 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-iota 6494 df-fv 6550 df-ov 7414 df-trkgc 27966 df-trkgcb 27968 df-trkg 27971 |
This theorem is referenced by: tgbtwnouttr2 28013 tgcgrxfr 28036 tgbtwnconn1lem1 28090 hlcgreulem 28135 mirreu3 28172 |
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