Theorem tgcgrtriv 28568

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 727	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → 𝐺 ∈ TarskiG)
6		tgcgrtriv.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7	6	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → 𝐴 ∈ 𝑃)
8		simplr 769	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → 𝑥 ∈ 𝑃)
9		tgcgrtriv.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
10	9	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → 𝐵 ∈ 𝑃)
11		simprr 773	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → (𝐴 − 𝑥) = (𝐵 − 𝐵))
12	1, 2, 3, 5, 7, 8, 10, 11	axtgcgrid 28547	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → 𝐴 = 𝑥)
13	12	oveq2d 7384	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → (𝐴 − 𝐴) = (𝐴 − 𝑥))
14	13, 11	eqtrd 2772	. 2 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → (𝐴 − 𝐴) = (𝐵 − 𝐵))
15	1, 2, 3, 4, 9, 6, 9, 9	axtgsegcon 28548	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵)))
16	14, 15	r19.29a 3146	1 ⊢ (𝜑 → (𝐴 − 𝐴) = (𝐵 − 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  distcds 17198  TarskiGcstrkg 28511  Itvcitv 28517

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5253

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-ov 7371  df-trkgc 28532  df-trkgcb 28534  df-trkg 28537

This theorem is referenced by:  tgcgrextend  28569  tgcgrsub  28593  iscgrglt  28598  trgcgrg  28599  tgbtwnconn1lem3  28658  leg0  28676