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Theorem tgsegconeq 28004
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.)
Hypotheses
Ref Expression
tkgeom.p 𝑃 = (Baseβ€˜πΊ)
tkgeom.d βˆ’ = (distβ€˜πΊ)
tkgeom.i 𝐼 = (Itvβ€˜πΊ)
tkgeom.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
tgcgrextend.a (πœ‘ β†’ 𝐴 ∈ 𝑃)
tgcgrextend.b (πœ‘ β†’ 𝐡 ∈ 𝑃)
tgcgrextend.c (πœ‘ β†’ 𝐢 ∈ 𝑃)
tgcgrextend.d (πœ‘ β†’ 𝐷 ∈ 𝑃)
tgcgrextend.e (πœ‘ β†’ 𝐸 ∈ 𝑃)
tgcgrextend.f (πœ‘ β†’ 𝐹 ∈ 𝑃)
tgsegconeq.1 (πœ‘ β†’ 𝐷 β‰  𝐴)
tgsegconeq.2 (πœ‘ β†’ 𝐴 ∈ (𝐷𝐼𝐸))
tgsegconeq.3 (πœ‘ β†’ 𝐴 ∈ (𝐷𝐼𝐹))
tgsegconeq.4 (πœ‘ β†’ (𝐴 βˆ’ 𝐸) = (𝐡 βˆ’ 𝐢))
tgsegconeq.5 (πœ‘ β†’ (𝐴 βˆ’ 𝐹) = (𝐡 βˆ’ 𝐢))
Assertion
Ref Expression
tgsegconeq (πœ‘ β†’ 𝐸 = 𝐹)

Proof of Theorem tgsegconeq
StepHypRef Expression
1 tkgeom.p . 2 𝑃 = (Baseβ€˜πΊ)
2 tkgeom.d . 2 βˆ’ = (distβ€˜πΊ)
3 tkgeom.i . 2 𝐼 = (Itvβ€˜πΊ)
4 tkgeom.g . 2 (πœ‘ β†’ 𝐺 ∈ TarskiG)
5 tgcgrextend.e . 2 (πœ‘ β†’ 𝐸 ∈ 𝑃)
6 tgcgrextend.f . 2 (πœ‘ β†’ 𝐹 ∈ 𝑃)
7 tgcgrextend.d . . . 4 (πœ‘ β†’ 𝐷 ∈ 𝑃)
8 tgcgrextend.a . . . 4 (πœ‘ β†’ 𝐴 ∈ 𝑃)
9 tgsegconeq.1 . . . 4 (πœ‘ β†’ 𝐷 β‰  𝐴)
10 tgsegconeq.2 . . . 4 (πœ‘ β†’ 𝐴 ∈ (𝐷𝐼𝐸))
11 eqidd 2731 . . . 4 (πœ‘ β†’ (𝐷 βˆ’ 𝐴) = (𝐷 βˆ’ 𝐴))
12 eqidd 2731 . . . 4 (πœ‘ β†’ (𝐴 βˆ’ 𝐸) = (𝐴 βˆ’ 𝐸))
13 tgsegconeq.3 . . . . 5 (πœ‘ β†’ 𝐴 ∈ (𝐷𝐼𝐹))
14 tgsegconeq.4 . . . . . 6 (πœ‘ β†’ (𝐴 βˆ’ 𝐸) = (𝐡 βˆ’ 𝐢))
15 tgsegconeq.5 . . . . . 6 (πœ‘ β†’ (𝐴 βˆ’ 𝐹) = (𝐡 βˆ’ 𝐢))
1614, 15eqtr4d 2773 . . . . 5 (πœ‘ β†’ (𝐴 βˆ’ 𝐸) = (𝐴 βˆ’ 𝐹))
171, 2, 3, 4, 7, 8, 5, 7, 8, 6, 10, 13, 11, 16tgcgrextend 28003 . . . 4 (πœ‘ β†’ (𝐷 βˆ’ 𝐸) = (𝐷 βˆ’ 𝐹))
181, 2, 3, 4, 7, 8, 5, 7, 8, 5, 5, 6, 9, 10, 10, 11, 12, 17, 16axtg5seg 27983 . . 3 (πœ‘ β†’ (𝐸 βˆ’ 𝐸) = (𝐸 βˆ’ 𝐹))
1918eqcomd 2736 . 2 (πœ‘ β†’ (𝐸 βˆ’ 𝐹) = (𝐸 βˆ’ 𝐸))
201, 2, 3, 4, 5, 6, 5, 19axtgcgrid 27981 1 (πœ‘ β†’ 𝐸 = 𝐹)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  β€˜cfv 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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