Theorem tgbtwnouttr2 28521

Description: Outer transitivity law for betweenness. Left-hand side of Theorem 3.7 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)
tgbtwnouttr2.1	⊢ (𝜑 → 𝐵 ≠ 𝐶)
tgbtwnouttr2.2	⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
tgbtwnouttr2.3	⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))

Assertion

Ref	Expression
tgbtwnouttr2	⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷))

Proof of Theorem tgbtwnouttr2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprl 770	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝐶 ∈ (𝐴𝐼𝑥))
2		tkgeom.p	. . . . 5 ⊢ 𝑃 = (Base‘𝐺)
3		tkgeom.d	. . . . 5 ⊢ − = (dist‘𝐺)
4		tkgeom.i	. . . . 5 ⊢ 𝐼 = (Itv‘𝐺)
5		tkgeom.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
6	5	ad2antrr 725	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝐺 ∈ TarskiG)
7		tgbtwnintr.3	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
8	7	ad2antrr 725	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝐶 ∈ 𝑃)
9		tgbtwnintr.4	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
10	9	ad2antrr 725	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝐷 ∈ 𝑃)
11		tgbtwnintr.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
12	11	ad2antrr 725	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝐵 ∈ 𝑃)
13		simplr 768	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝑥 ∈ 𝑃)
14		tgbtwnouttr2.1	. . . . . 6 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
15	14	ad2antrr 725	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝐵 ≠ 𝐶)
16		tgbtwnintr.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
17	16	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝐴 ∈ 𝑃)
18		tgbtwnouttr2.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
19	18	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝐵 ∈ (𝐴𝐼𝐶))
20	2, 3, 4, 6, 17, 12, 8, 13, 19, 1	tgbtwnexch3 28520	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝐶 ∈ (𝐵𝐼𝑥))
21		tgbtwnouttr2.3	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))
22	21	ad2antrr 725	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝐶 ∈ (𝐵𝐼𝐷))
23		simprr 772	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → (𝐶 − 𝑥) = (𝐶 − 𝐷))
24		eqidd 2741	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → (𝐶 − 𝐷) = (𝐶 − 𝐷))
25	2, 3, 4, 6, 8, 8, 10, 12, 13, 10, 15, 20, 22, 23, 24	tgsegconeq 28512	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝑥 = 𝐷)
26	25	oveq2d 7464	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → (𝐴𝐼𝑥) = (𝐴𝐼𝐷))
27	1, 26	eleqtrd 2846	. 2 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝐶 ∈ (𝐴𝐼𝐷))
28	2, 3, 4, 5, 16, 7, 7, 9	axtgsegcon 28490	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷)))
29	27, 28	r19.29a 3168	1 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  distcds 17320  TarskiGcstrkg 28453  Itvcitv 28459

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-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-ov 7451  df-trkgc 28474  df-trkgb 28475  df-trkgcb 28476  df-trkg 28479

This theorem is referenced by:  tgbtwnexch2  28522  tgbtwnouttr  28523  tgbtwnconn22  28605  tglineeltr  28657  mirconn  28704  footexALT  28744  footexlem1  28745  footexlem2  28746