Theorem tgbtwnouttr2 26275

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 769	. . 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 724	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝐺 ∈ TarskiG)
7		tgbtwnintr.3	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
8	7	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝐶 ∈ 𝑃)
9		tgbtwnintr.4	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
10	9	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝐷 ∈ 𝑃)
11		tgbtwnintr.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
12	11	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝐵 ∈ 𝑃)
13		simplr 767	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝑥 ∈ 𝑃)
14		tgbtwnouttr2.1	. . . . . 6 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
15	14	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝐵 ≠ 𝐶)
16		tgbtwnintr.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
17	16	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝐴 ∈ 𝑃)
18		tgbtwnouttr2.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
19	18	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝐵 ∈ (𝐴𝐼𝐶))
20	2, 3, 4, 6, 17, 12, 8, 13, 19, 1	tgbtwnexch3 26274	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝐶 ∈ (𝐵𝐼𝑥))
21		tgbtwnouttr2.3	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))
22	21	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝐶 ∈ (𝐵𝐼𝐷))
23		simprr 771	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → (𝐶 − 𝑥) = (𝐶 − 𝐷))
24		eqidd 2822	. . . . 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 26266	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝑥 = 𝐷)
26	25	oveq2d 7166	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → (𝐴𝐼𝑥) = (𝐴𝐼𝐷))
27	1, 26	eleqtrd 2915	. 2 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷))) → 𝐶 ∈ (𝐴𝐼𝐷))
28	2, 3, 4, 5, 16, 7, 7, 9	axtgsegcon 26244	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 − 𝑥) = (𝐶 − 𝐷)))
29	27, 28	r19.29a 3289	1 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ‘cfv 6349  (class class class)co 7150  Basecbs 16477  distcds 16568  TarskiGcstrkg 26210  Itvcitv 26216

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-nul 5202

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-iota 6308  df-fv 6357  df-ov 7153  df-trkgc 26228  df-trkgb 26229  df-trkgcb 26230  df-trkg 26233

This theorem is referenced by:  tgbtwnexch2  26276  tgbtwnouttr  26277  tgbtwnconn22  26359  tglineeltr  26411  mirconn  26458  footexALT  26498  footexlem1  26499  footexlem2  26500