Theorem tgbtwnouttr2 25814

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 787	. . 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 717	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝐺 ∈ TarskiG)
7		tgbtwnintr.3	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
8	7	ad2antrr 717	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝐶 ∈ 𝑃)
9		tgbtwnintr.4	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
10	9	ad2antrr 717	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝐷 ∈ 𝑃)
11		tgbtwnintr.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
12	11	ad2antrr 717	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝐵 ∈ 𝑃)
13		simplr 785	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝑥 ∈ 𝑃)
14		tgbtwnouttr2.1	. . . . . 6 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
15	14	ad2antrr 717	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝐵 ≠ 𝐶)
16		tgbtwnintr.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
17	16	ad2antrr 717	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝐴 ∈ 𝑃)
18		tgbtwnouttr2.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
19	18	ad2antrr 717	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝐵 ∈ (𝐴𝐼𝐶))
20	2, 3, 4, 6, 17, 12, 8, 13, 19, 1	tgbtwnexch3 25813	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝐶 ∈ (𝐵𝐼𝑥))
21		tgbtwnouttr2.3	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))
22	21	ad2antrr 717	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝐶 ∈ (𝐵𝐼𝐷))
23		simprr 789	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → (𝐶 − 𝑥) = (𝐶 − 𝐷))
24		eqidd 2826	. . . . 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 25805	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝑥 = 𝐷)
26	25	oveq2d 6926	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → (𝐴𝐼𝑥) = (𝐴𝐼𝐷))
27	1, 26	eleqtrd 2908	. 2 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝐶 ∈ (𝐴𝐼𝐷))
28	2, 3, 4, 5, 16, 7, 7, 9	axtgsegcon 25783	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷)))
29	27, 28	r19.29a 3288	1 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷))

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1656   ∈ wcel 2164   ≠ wne 2999  ‘cfv 6127  (class class class)co 6910  Basecbs 16229  distcds 16321  TarskiGcstrkg 25749  Itvcitv 25755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-nul 5015

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-iota 6090  df-fv 6135  df-ov 6913  df-trkgc 25767  df-trkgb 25768  df-trkgcb 25769  df-trkg 25772

This theorem is referenced by:  tgbtwnexch2  25815  tgbtwnouttr  25816  tgbtwnconn22  25898  tglineeltr  25950  mirconn  25997  footex  26037