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Theorem tgsegconeq 25602
Description: Two points that satisfy the conclusion of axtgsegcon 25584 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 2772 . . . 4 (𝜑 → (𝐷 𝐴) = (𝐷 𝐴))
12 eqidd 2772 . . . 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 2808 . . . . 5 (𝜑 → (𝐴 𝐸) = (𝐴 𝐹))
171, 2, 3, 4, 7, 8, 5, 7, 8, 6, 10, 13, 11, 16tgcgrextend 25601 . . . 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 25585 . . 3 (𝜑 → (𝐸 𝐸) = (𝐸 𝐹))
1918eqcomd 2777 . 2 (𝜑 → (𝐸 𝐹) = (𝐸 𝐸))
201, 2, 3, 4, 5, 6, 5, 19axtgcgrid 25583 1 (𝜑𝐸 = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  wne 2943  cfv 6031  (class class class)co 6793  Basecbs 16064  distcds 16158  TarskiGcstrkg 25550  Itvcitv 25556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4923
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-ov 6796  df-trkgc 25568  df-trkgcb 25570  df-trkg 25573
This theorem is referenced by:  tgbtwnouttr2  25611  tgcgrxfr  25634  tgbtwnconn1lem1  25688  hlcgreulem  25733  mirreu3  25770
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