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	⊢ (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 2759	. . . 4 ⊢ (𝜑 → (𝐷 − 𝐴) = (𝐷 − 𝐴))
12		eqidd 2759	. . . 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 2796	. . . . 5 ⊢ (𝜑 → (𝐴 − 𝐸) = (𝐴 − 𝐹))
17	1, 2, 3, 4, 7, 8, 5, 7, 8, 6, 10, 13, 11, 16	tgcgrextend 26378	. . . 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 26358	. . 3 ⊢ (𝜑 → (𝐸 − 𝐸) = (𝐸 − 𝐹))
19	18	eqcomd 2764	. 2 ⊢ (𝜑 → (𝐸 − 𝐹) = (𝐸 − 𝐸))
20	1, 2, 3, 4, 5, 6, 5, 19	axtgcgrid 26356	1 ⊢ (𝜑 → 𝐸 = 𝐹)

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