Theorem tgcgrtriv 28512

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 725	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → 𝐺 ∈ TarskiG)
6		tgcgrtriv.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7	6	ad2antrr 725	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → 𝐴 ∈ 𝑃)
8		simplr 768	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → 𝑥 ∈ 𝑃)
9		tgcgrtriv.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
10	9	ad2antrr 725	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → 𝐵 ∈ 𝑃)
11		simprr 772	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → (𝐴 − 𝑥) = (𝐵 − 𝐵))
12	1, 2, 3, 5, 7, 8, 10, 11	axtgcgrid 28491	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → 𝐴 = 𝑥)
13	12	oveq2d 7466	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → (𝐴 − 𝐴) = (𝐴 − 𝑥))
14	13, 11	eqtrd 2780	. 2 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → (𝐴 − 𝐴) = (𝐵 − 𝐵))
15	1, 2, 3, 4, 9, 6, 9, 9	axtgsegcon 28492	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵)))
16	14, 15	r19.29a 3168	1 ⊢ (𝜑 → (𝐴 − 𝐴) = (𝐵 − 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ‘cfv 6575  (class class class)co 7450  Basecbs 17260  distcds 17322  TarskiGcstrkg 28455  Itvcitv 28461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-nul 5324

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6527  df-fv 6583  df-ov 7453  df-trkgc 28476  df-trkgcb 28478  df-trkg 28481

This theorem is referenced by:  tgcgrextend  28513  tgcgrsub  28537  iscgrglt  28542  trgcgrg  28543  tgbtwnconn1lem3  28602  leg0  28620