Theorem tgbtwnintr 28516

Description: Inner transitivity law for betweenness. Left-hand side of Theorem 3.5 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 18-Mar-2019.)

Hypotheses

Ref	Expression
tkgeom.p	⊢ 𝑃 = (Base‘𝐺)
tkgeom.d	⊢ − = (dist‘𝐺)
tkgeom.i	⊢ 𝐼 = (Itv‘𝐺)
tkgeom.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tgbtwnintr.1	⊢ (𝜑 → 𝐴 ∈ 𝑃)
tgbtwnintr.2	⊢ (𝜑 → 𝐵 ∈ 𝑃)
tgbtwnintr.3	⊢ (𝜑 → 𝐶 ∈ 𝑃)
tgbtwnintr.4	⊢ (𝜑 → 𝐷 ∈ 𝑃)
tgbtwnintr.5	⊢ (𝜑 → 𝐴 ∈ (𝐵𝐼𝐷))
tgbtwnintr.6	⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐷))

Assertion

Ref	Expression
tgbtwnintr	⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))

Proof of Theorem tgbtwnintr

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tkgeom.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
2		tkgeom.d	. . . 4 ⊢ − = (dist‘𝐺)
3		tkgeom.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
4		tkgeom.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐺 ∈ TarskiG)
6		tgbtwnintr.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
7	6	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐵 ∈ 𝑃)
8		simplr 769	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥 ∈ 𝑃)
9		simprr 773	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥 ∈ (𝐵𝐼𝐵))
10	1, 2, 3, 5, 7, 8, 9	axtgbtwnid 28489	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐵 = 𝑥)
11		simprl 771	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥 ∈ (𝐴𝐼𝐶))
12	10, 11	eqeltrd 2839	. 2 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐵 ∈ (𝐴𝐼𝐶))
13		tgbtwnintr.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
14		tgbtwnintr.4	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
15		tgbtwnintr.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
16		tgbtwnintr.5	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐼𝐷))
17		tgbtwnintr.6	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐷))
18	1, 2, 3, 4, 6, 13, 14, 15, 6, 16, 17	axtgpasch 28490	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵)))
19	12, 18	r19.29a 3160	1 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  distcds 17307  TarskiGcstrkg 28450  Itvcitv 28456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-trkgb 28472  df-trkg 28476

This theorem is referenced by:  tgbtwnexch3  28517  tgbtwnexch2  28519  tgbtwnconn1lem3  28597  tgbtwnconn3  28600  tgbtwnconn22  28602  tglineeltr  28654  mirconn  28701