Theorem tgcgrtriv 29003

Description: Degenerate segments are congruent. Theorem 2.8 of [Schwabhauser] p. 28. (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)
tgcgrtriv.1	⊢ (𝜑 → 𝐴 ∈ 𝑃)
tgcgrtriv.2	⊢ (𝜑 → 𝐵 ∈ 𝑃)

Assertion

Ref	Expression
tgcgrtriv	⊢ (𝜑 → (𝐴 − 𝐴) = (𝐵 − 𝐵))

Proof of Theorem tgcgrtriv

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tkgeom.p	. . . . 5 ⊢ 𝑃 = (Base‘𝐺)
2		tkgeom.d	. . . . 5 ⊢ − = (dist‘𝐺)
3		tkgeom.i	. . . . 5 ⊢ 𝐼 = (Itv‘𝐺)
4		tkgeom.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → 𝐺 ∈ TarskiG)
6		tgcgrtriv.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7	6	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → 𝐴 ∈ 𝑃)
8		simplr 781	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → 𝑥 ∈ 𝑃)
9		tgcgrtriv.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
10	9	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → 𝐵 ∈ 𝑃)
11		simprr 785	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → (𝐴 − 𝑥) = (𝐵 − 𝐵))
12	1, 2, 3, 5, 7, 8, 10, 11	axtgcgrid 28982	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → 𝐴 = 𝑥)
13	12	oveq2d 7467	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → (𝐴 − 𝐴) = (𝐴 − 𝑥))
14	13, 11	eqtrd 2798	. 2 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → (𝐴 − 𝐴) = (𝐵 − 𝐵))
15	1, 2, 3, 4, 9, 6, 9, 9	axtgsegcon 28983	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵)))
16	14, 15	r19.29a 3174	1 ⊢ (𝜑 → (𝐴 − 𝐴) = (𝐵 − 𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ‘cfv 6566  (class class class)co 7451  Basecbs 17431  distcds 17481  TarskiGcstrkg 28946  Itvcitv 28952

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-nul 5266

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-iota 6520  df-fv 6574  df-ov 7454  df-trkgc 28967  df-trkgcb 28969  df-trkg 28972

This theorem is used by:  tgcgrextend  29004  tgcgrsub  29029  iscgrglt  29034  trgcgrg  29035  tgbtwnconn1lem3  29094  leg0  29112