Theorem tgbtwnintr 26758

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 722	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐺 ∈ TarskiG)
6		tgbtwnintr.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
7	6	ad2antrr 722	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐵 ∈ 𝑃)
8		simplr 765	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥 ∈ 𝑃)
9		simprr 769	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥 ∈ (𝐵𝐼𝐵))
10	1, 2, 3, 5, 7, 8, 9	axtgbtwnid 26731	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐵 = 𝑥)
11		simprl 767	. . 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 26732	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵)))
19	12, 18	r19.29a 3217	1 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  distcds 16897  TarskiGcstrkg 26693  Itvcitv 26699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-nul 5225

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-ov 7258  df-trkgb 26714  df-trkg 26718

This theorem is referenced by:  tgbtwnexch3  26759  tgbtwnexch2  26761  tgbtwnconn1lem3  26839  tgbtwnconn3  26842  tgbtwnconn22  26844  tglineeltr  26896  mirconn  26943