Theorem tgbtwnconn1 28628

Description: Connectivity law for betweenness. Theorem 5.1 of [Schwabhauser] p. 39-41. In earlier presentations of Tarski's axioms, this theorem appeared as an additional axiom. It was derived from the other axioms by Gupta, 1965. (Contributed by Thierry Arnoux, 30-Apr-2019.)

Hypotheses

Ref	Expression
tgbtwnconn1.p	⊢ 𝑃 = (Base‘𝐺)
tgbtwnconn1.i	⊢ 𝐼 = (Itv‘𝐺)
tgbtwnconn1.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tgbtwnconn1.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
tgbtwnconn1.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
tgbtwnconn1.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
tgbtwnconn1.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
tgbtwnconn1.1	⊢ (𝜑 → 𝐴 ≠ 𝐵)
tgbtwnconn1.2	⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
tgbtwnconn1.3	⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷))

Assertion

Ref	Expression
tgbtwnconn1	⊢ (𝜑 → (𝐶 ∈ (𝐴𝐼𝐷) ∨ 𝐷 ∈ (𝐴𝐼𝐶)))

Proof of Theorem tgbtwnconn1

Dummy variables 𝑒 𝑓 ℎ 𝑗 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpllr 776	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶)))
2	1	simpld 494	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → 𝐷 ∈ (𝐴𝐼𝑒))
3	2	adantr 480	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 = 𝑒) → 𝐷 ∈ (𝐴𝐼𝑒))
4		simpr 484	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 = 𝑒) → 𝐶 = 𝑒)
5	4	oveq2d 7374	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 = 𝑒) → (𝐴𝐼𝐶) = (𝐴𝐼𝑒))
6	3, 5	eleqtrrd 2838	. . . . 5 ⊢ ((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 = 𝑒) → 𝐷 ∈ (𝐴𝐼𝐶))
7	6	olcd 875	. . . 4 ⊢ ((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 = 𝑒) → (𝐶 ∈ (𝐴𝐼𝐷) ∨ 𝐷 ∈ (𝐴𝐼𝐶)))
8		simprl 771	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → 𝐶 ∈ (𝐴𝐼𝑓))
9	8	adantr 480	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐷 = 𝑓) → 𝐶 ∈ (𝐴𝐼𝑓))
10		simpr 484	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐷 = 𝑓) → 𝐷 = 𝑓)
11	10	oveq2d 7374	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐷 = 𝑓) → (𝐴𝐼𝐷) = (𝐴𝐼𝑓))
12	9, 11	eleqtrrd 2838	. . . . 5 ⊢ ((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐷 = 𝑓) → 𝐶 ∈ (𝐴𝐼𝐷))
13	12	orcd 874	. . . 4 ⊢ ((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐷 = 𝑓) → (𝐶 ∈ (𝐴𝐼𝐷) ∨ 𝐷 ∈ (𝐴𝐼𝐶)))
14		df-ne 2932	. . . . . 6 ⊢ (𝐶 ≠ 𝑒 ↔ ¬ 𝐶 = 𝑒)
15		tgbtwnconn1.p	. . . . . . . . . . 11 ⊢ 𝑃 = (Base‘𝐺)
16		tgbtwnconn1.i	. . . . . . . . . . 11 ⊢ 𝐼 = (Itv‘𝐺)
17		tgbtwnconn1.g	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
18	17	ad4antr 733	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → 𝐺 ∈ TarskiG)
19	18	ad7antr 739	. . . . . . . . . . 11 ⊢ ((((((((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) ∧ ℎ ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗 ∈ 𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝐺 ∈ TarskiG)
20		tgbtwnconn1.a	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
21	20	ad4antr 733	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → 𝐴 ∈ 𝑃)
22	21	ad7antr 739	. . . . . . . . . . 11 ⊢ ((((((((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) ∧ ℎ ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗 ∈ 𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝐴 ∈ 𝑃)
23		tgbtwnconn1.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
24	23	ad4antr 733	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → 𝐵 ∈ 𝑃)
25	24	ad7antr 739	. . . . . . . . . . 11 ⊢ ((((((((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) ∧ ℎ ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗 ∈ 𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝐵 ∈ 𝑃)
26		tgbtwnconn1.c	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
27	26	ad4antr 733	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → 𝐶 ∈ 𝑃)
28	27	ad7antr 739	. . . . . . . . . . 11 ⊢ ((((((((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) ∧ ℎ ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗 ∈ 𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝐶 ∈ 𝑃)
29		tgbtwnconn1.d	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
30	29	ad4antr 733	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → 𝐷 ∈ 𝑃)
31	30	ad7antr 739	. . . . . . . . . . 11 ⊢ ((((((((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) ∧ ℎ ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗 ∈ 𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝐷 ∈ 𝑃)
32		simp-11l 797	. . . . . . . . . . . 12 ⊢ ((((((((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) ∧ ℎ ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗 ∈ 𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝜑)
33		tgbtwnconn1.1	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
34	32, 33	syl 17	. . . . . . . . . . 11 ⊢ ((((((((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) ∧ ℎ ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗 ∈ 𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝐴 ≠ 𝐵)
35		tgbtwnconn1.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
36	32, 35	syl 17	. . . . . . . . . . 11 ⊢ ((((((((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) ∧ ℎ ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗 ∈ 𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝐵 ∈ (𝐴𝐼𝐶))
37		tgbtwnconn1.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷))
38	32, 37	syl 17	. . . . . . . . . . 11 ⊢ ((((((((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) ∧ ℎ ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗 ∈ 𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝐵 ∈ (𝐴𝐼𝐷))
39		eqid 2735	. . . . . . . . . . 11 ⊢ (dist‘𝐺) = (dist‘𝐺)
40		simp-4r 784	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → 𝑒 ∈ 𝑃)
41	40	ad7antr 739	. . . . . . . . . . 11 ⊢ ((((((((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) ∧ ℎ ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗 ∈ 𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝑒 ∈ 𝑃)
42		simplr 769	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → 𝑓 ∈ 𝑃)
43	42	ad7antr 739	. . . . . . . . . . 11 ⊢ ((((((((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) ∧ ℎ ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗 ∈ 𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝑓 ∈ 𝑃)
44		simp-6r 788	. . . . . . . . . . 11 ⊢ ((((((((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) ∧ ℎ ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗 ∈ 𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → ℎ ∈ 𝑃)
45		simp-4r 784	. . . . . . . . . . 11 ⊢ ((((((((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) ∧ ℎ ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗 ∈ 𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝑗 ∈ 𝑃)
46	2	ad7antr 739	. . . . . . . . . . 11 ⊢ ((((((((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) ∧ ℎ ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗 ∈ 𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝐷 ∈ (𝐴𝐼𝑒))
47	8	ad7antr 739	. . . . . . . . . . 11 ⊢ ((((((((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) ∧ ℎ ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗 ∈ 𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝐶 ∈ (𝐴𝐼𝑓))
48		simp-5r 786	. . . . . . . . . . . 12 ⊢ ((((((((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) ∧ ℎ ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗 ∈ 𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶)))
49	48	simpld 494	. . . . . . . . . . 11 ⊢ ((((((((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) ∧ ℎ ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗 ∈ 𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝑒 ∈ (𝐴𝐼ℎ))
50		simpllr 776	. . . . . . . . . . . 12 ⊢ ((((((((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) ∧ ℎ ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗 ∈ 𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷)))
51	50	simpld 494	. . . . . . . . . . 11 ⊢ ((((((((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) ∧ ℎ ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗 ∈ 𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝑓 ∈ (𝐴𝐼𝑗))
52	1	simprd 495	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))
53	52	ad7antr 739	. . . . . . . . . . . 12 ⊢ ((((((((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) ∧ ℎ ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗 ∈ 𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))
54	15, 39, 16, 19, 31, 41, 31, 28, 53	tgcgrcomlr 28533	. . . . . . . . . . 11 ⊢ ((((((((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) ∧ ℎ ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗 ∈ 𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → (𝑒(dist‘𝐺)𝐷) = (𝐶(dist‘𝐺)𝐷))
55		simprr 773	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))
56	55	ad7antr 739	. . . . . . . . . . 11 ⊢ ((((((((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) ∧ ℎ ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗 ∈ 𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))
57	48	simprd 495	. . . . . . . . . . 11 ⊢ ((((((((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) ∧ ℎ ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗 ∈ 𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶))
58	50	simprd 495	. . . . . . . . . . 11 ⊢ ((((((((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) ∧ ℎ ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗 ∈ 𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))
59		simplr 769	. . . . . . . . . . 11 ⊢ ((((((((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) ∧ ℎ ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗 ∈ 𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝑥 ∈ 𝑃)
60		simprl 771	. . . . . . . . . . 11 ⊢ ((((((((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) ∧ ℎ ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗 ∈ 𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝑥 ∈ (𝐶𝐼𝑒))
61		simprr 773	. . . . . . . . . . 11 ⊢ ((((((((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) ∧ ℎ ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗 ∈ 𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝑥 ∈ (𝐷𝐼𝑓))
62		simp-7r 790	. . . . . . . . . . 11 ⊢ ((((((((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) ∧ ℎ ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗 ∈ 𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝐶 ≠ 𝑒)
63	15, 16, 19, 22, 25, 28, 31, 34, 36, 38, 39, 41, 43, 44, 45, 46, 47, 49, 51, 54, 56, 57, 58, 59, 60, 61, 62	tgbtwnconn1lem3 28627	. . . . . . . . . 10 ⊢ ((((((((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) ∧ ℎ ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗 ∈ 𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝐷 = 𝑓)
64	15, 39, 16, 18, 21, 27, 42, 8	tgbtwncom 28541	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → 𝐶 ∈ (𝑓𝐼𝐴))
65	15, 39, 16, 18, 21, 30, 40, 2	tgbtwncom 28541	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → 𝐷 ∈ (𝑒𝐼𝐴))
66	15, 39, 16, 18, 42, 40, 21, 27, 30, 64, 65	axtgpasch 28520	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓)))
67	66	ad5antr 735	. . . . . . . . . 10 ⊢ ((((((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) ∧ ℎ ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗 ∈ 𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓)))
68	63, 67	r19.29a 3143	. . . . . . . . 9 ⊢ ((((((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) ∧ ℎ ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗 ∈ 𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) → 𝐷 = 𝑓)
69	15, 39, 16, 18, 21, 42, 24, 30	axtgsegcon 28517	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → ∃𝑗 ∈ 𝑃 (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷)))
70	69	ad3antrrr 731	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) ∧ ℎ ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶))) → ∃𝑗 ∈ 𝑃 (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷)))
71	68, 70	r19.29a 3143	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) ∧ ℎ ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶))) → 𝐷 = 𝑓)
72	15, 39, 16, 18, 21, 40, 24, 27	axtgsegcon 28517	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → ∃ℎ ∈ 𝑃 (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶)))
73	72	adantr 480	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) → ∃ℎ ∈ 𝑃 (𝑒 ∈ (𝐴𝐼ℎ) ∧ (𝑒(dist‘𝐺)ℎ) = (𝐵(dist‘𝐺)𝐶)))
74	71, 73	r19.29a 3143	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 ≠ 𝑒) → 𝐷 = 𝑓)
75	74	ex 412	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → (𝐶 ≠ 𝑒 → 𝐷 = 𝑓))
76	14, 75	biimtrrid 243	. . . . 5 ⊢ (((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → (¬ 𝐶 = 𝑒 → 𝐷 = 𝑓))
77	76	orrd 864	. . . 4 ⊢ (((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → (𝐶 = 𝑒 ∨ 𝐷 = 𝑓))
78	7, 13, 77	mpjaodan 961	. . 3 ⊢ (((((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → (𝐶 ∈ (𝐴𝐼𝐷) ∨ 𝐷 ∈ (𝐴𝐼𝐶)))
79	15, 39, 16, 17, 20, 26, 26, 29	axtgsegcon 28517	. . . 4 ⊢ (𝜑 → ∃𝑓 ∈ 𝑃 (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷)))
80	79	ad2antrr 727	. . 3 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) → ∃𝑓 ∈ 𝑃 (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷)))
81	78, 80	r19.29a 3143	. 2 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) → (𝐶 ∈ (𝐴𝐼𝐷) ∨ 𝐷 ∈ (𝐴𝐼𝐶)))
82	15, 39, 16, 17, 20, 29, 29, 26	axtgsegcon 28517	. 2 ⊢ (𝜑 → ∃𝑒 ∈ 𝑃 (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶)))
83	81, 82	r19.29a 3143	1 ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐼𝐷) ∨ 𝐷 ∈ (𝐴𝐼𝐶)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2931  ∃wrex 3059  ‘cfv 6491  (class class class)co 7358  Basecbs 17138  distcds 17188  TarskiGcstrkg 28480  Itvcitv 28486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4863  df-int 4902  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-oadd 8401  df-er 8635  df-pm 8768  df-en 8886  df-dom 8887  df-sdom 8888  df-fin 8889  df-dju 9815  df-card 9853  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-2 12210  df-3 12211  df-n0 12404  df-xnn0 12477  df-z 12491  df-uz 12754  df-fz 13426  df-fzo 13573  df-hash 14256  df-word 14439  df-concat 14496  df-s1 14522  df-s2 14773  df-s3 14774  df-trkgc 28501  df-trkgb 28502  df-trkgcb 28503  df-trkg 28506  df-cgrg 28564

This theorem is referenced by:  tgbtwnconn2  28629  tgbtwnconnln1  28633  hltr  28663  hlbtwn  28664