Theorem tgsegconeq 28332

Description: Two points that satisfy the conclusion of axtgsegcon 28310 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	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tgcgrextend.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
tgcgrextend.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
tgcgrextend.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
tgcgrextend.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
tgcgrextend.e	⊢ (𝜑 → 𝐸 ∈ 𝑃)
tgcgrextend.f	⊢ (𝜑 → 𝐹 ∈ 𝑃)
tgsegconeq.1	⊢ (𝜑 → 𝐷 ≠ 𝐴)
tgsegconeq.2	⊢ (𝜑 → 𝐴 ∈ (𝐷𝐼𝐸))
tgsegconeq.3	⊢ (𝜑 → 𝐴 ∈ (𝐷𝐼𝐹))
tgsegconeq.4	⊢ (𝜑 → (𝐴 − 𝐸) = (𝐵 − 𝐶))
tgsegconeq.5	⊢ (𝜑 → (𝐴 − 𝐹) = (𝐵 − 𝐶))

Assertion

Ref	Expression
tgsegconeq	⊢ (𝜑 → 𝐸 = 𝐹)

Proof of Theorem tgsegconeq

Step	Hyp	Ref	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 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5		tgcgrextend.e	. 2 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
6		tgcgrextend.f	. 2 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
7		tgcgrextend.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
8		tgcgrextend.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
9		tgsegconeq.1	. . . 4 ⊢ (𝜑 → 𝐷 ≠ 𝐴)
10		tgsegconeq.2	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝐷𝐼𝐸))
11		eqidd 2726	. . . 4 ⊢ (𝜑 → (𝐷 − 𝐴) = (𝐷 − 𝐴))
12		eqidd 2726	. . . 4 ⊢ (𝜑 → (𝐴 − 𝐸) = (𝐴 − 𝐸))
13		tgsegconeq.3	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐷𝐼𝐹))
14		tgsegconeq.4	. . . . . 6 ⊢ (𝜑 → (𝐴 − 𝐸) = (𝐵 − 𝐶))
15		tgsegconeq.5	. . . . . 6 ⊢ (𝜑 → (𝐴 − 𝐹) = (𝐵 − 𝐶))
16	14, 15	eqtr4d 2768	. . . . 5 ⊢ (𝜑 → (𝐴 − 𝐸) = (𝐴 − 𝐹))
17	1, 2, 3, 4, 7, 8, 5, 7, 8, 6, 10, 13, 11, 16	tgcgrextend 28331	. . . 4 ⊢ (𝜑 → (𝐷 − 𝐸) = (𝐷 − 𝐹))
18	1, 2, 3, 4, 7, 8, 5, 7, 8, 5, 5, 6, 9, 10, 10, 11, 12, 17, 16	axtg5seg 28311	. . 3 ⊢ (𝜑 → (𝐸 − 𝐸) = (𝐸 − 𝐹))
19	18	eqcomd 2731	. 2 ⊢ (𝜑 → (𝐸 − 𝐹) = (𝐸 − 𝐸))
20	1, 2, 3, 4, 5, 6, 5, 19	axtgcgrid 28309	1 ⊢ (𝜑 → 𝐸 = 𝐹)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098   ≠ wne 2930  ‘cfv 6542  (class class class)co 7415  Basecbs 17177  distcds 17239  TarskiGcstrkg 28273  Itvcitv 28279

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-nul 5301

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3770  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-iota 6494  df-fv 6550  df-ov 7418  df-trkgc 28294  df-trkgcb 28296  df-trkg 28299

This theorem is referenced by:  tgbtwnouttr2  28341  tgcgrxfr  28364  tgbtwnconn1lem1  28418  hlcgreulem  28463  mirreu3  28500