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Theorem tgsegconeq 25181
 Description: Two points that satisfy the conclusion of axtgsegcon 25163 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 2611 . . . 4 (𝜑 → (𝐷 𝐴) = (𝐷 𝐴))
12 eqidd 2611 . . . 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 2647 . . . . 5 (𝜑 → (𝐴 𝐸) = (𝐴 𝐹))
171, 2, 3, 4, 7, 8, 5, 7, 8, 6, 10, 13, 11, 16tgcgrextend 25180 . . . 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 25164 . . 3 (𝜑 → (𝐸 𝐸) = (𝐸 𝐹))
1918eqcomd 2616 . 2 (𝜑 → (𝐸 𝐹) = (𝐸 𝐸))
201, 2, 3, 4, 5, 6, 5, 19axtgcgrid 25162 1 (𝜑𝐸 = 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  distcds 15777  TarskiGcstrkg 25129  Itvcitv 25135 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-trkgc 25147  df-trkgcb 25149  df-trkg 25152 This theorem is referenced by:  tgbtwnouttr2  25190  tgcgrxfr  25213  tgbtwnconn1lem1  25267  hlcgreulem  25312  mirreu3  25349
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