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Theorem tgsegconeq 26379
 Description: Two points that satisfy the conclusion of axtgsegcon 26357 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 = (Base‘𝐺)
tkgeom.d = (dist‘𝐺)
tkgeom.i 𝐼 = (Itv‘𝐺)
tkgeom.g (𝜑𝐺 ∈ TarskiG)
tgcgrextend.a (𝜑𝐴𝑃)
tgcgrextend.b (𝜑𝐵𝑃)
tgcgrextend.c (𝜑𝐶𝑃)
tgcgrextend.d (𝜑𝐷𝑃)
tgcgrextend.e (𝜑𝐸𝑃)
tgcgrextend.f (𝜑𝐹𝑃)
tgsegconeq.1 (𝜑𝐷𝐴)
tgsegconeq.2 (𝜑𝐴 ∈ (𝐷𝐼𝐸))
tgsegconeq.3 (𝜑𝐴 ∈ (𝐷𝐼𝐹))
tgsegconeq.4 (𝜑 → (𝐴 𝐸) = (𝐵 𝐶))
tgsegconeq.5 (𝜑 → (𝐴 𝐹) = (𝐵 𝐶))
Assertion
Ref Expression
tgsegconeq (𝜑𝐸 = 𝐹)

Proof of Theorem tgsegconeq
StepHypRef 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 (𝜑𝐺 ∈ TarskiG)
5 tgcgrextend.e . 2 (𝜑𝐸𝑃)
6 tgcgrextend.f . 2 (𝜑𝐹𝑃)
7 tgcgrextend.d . . . 4 (𝜑𝐷𝑃)
8 tgcgrextend.a . . . 4 (𝜑𝐴𝑃)
9 tgsegconeq.1 . . . 4 (𝜑𝐷𝐴)
10 tgsegconeq.2 . . . 4 (𝜑𝐴 ∈ (𝐷𝐼𝐸))
11 eqidd 2759 . . . 4 (𝜑 → (𝐷 𝐴) = (𝐷 𝐴))
12 eqidd 2759 . . . 4 (𝜑 → (𝐴 𝐸) = (𝐴 𝐸))
13 tgsegconeq.3 . . . . 5 (𝜑𝐴 ∈ (𝐷𝐼𝐹))
14 tgsegconeq.4 . . . . . 6 (𝜑 → (𝐴 𝐸) = (𝐵 𝐶))
15 tgsegconeq.5 . . . . . 6 (𝜑 → (𝐴 𝐹) = (𝐵 𝐶))
1614, 15eqtr4d 2796 . . . . 5 (𝜑 → (𝐴 𝐸) = (𝐴 𝐹))
171, 2, 3, 4, 7, 8, 5, 7, 8, 6, 10, 13, 11, 16tgcgrextend 26378 . . . 4 (𝜑 → (𝐷 𝐸) = (𝐷 𝐹))
181, 2, 3, 4, 7, 8, 5, 7, 8, 5, 5, 6, 9, 10, 10, 11, 12, 17, 16axtg5seg 26358 . . 3 (𝜑 → (𝐸 𝐸) = (𝐸 𝐹))
1918eqcomd 2764 . 2 (𝜑 → (𝐸 𝐹) = (𝐸 𝐸))
201, 2, 3, 4, 5, 6, 5, 19axtgcgrid 26356 1 (𝜑𝐸 = 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ‘cfv 6335  (class class class)co 7150  Basecbs 16541  distcds 16632  TarskiGcstrkg 26323  Itvcitv 26329 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-nul 5176 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-iota 6294  df-fv 6343  df-ov 7153  df-trkgc 26341  df-trkgcb 26343  df-trkg 26346 This theorem is referenced by:  tgbtwnouttr2  26388  tgcgrxfr  26411  tgbtwnconn1lem1  26465  hlcgreulem  26510  mirreu3  26547
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