Theorem tgbtwnconn1 26363

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 774	. . . . . . . 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 497	. . . . . . 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 483	. . . . . 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 487	. . . . . . 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 7174	. . . . . 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 2918	. . . . 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 870	. . . 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 769	. . . . . . 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 483	. . . . . 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 487	. . . . . . 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 7174	. . . . . 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 2918	. . . . 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 869	. . . 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 3019	. . . . . 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 730	. . . . . . . . . . . 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 736	. . . . . . . . . . 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 730	. . . . . . . . . . . 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 736	. . . . . . . . . . 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 730	. . . . . . . . . . . 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 736	. . . . . . . . . . 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 730	. . . . . . . . . . . 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 736	. . . . . . . . . . 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 730	. . . . . . . . . . . 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 736	. . . . . . . . . . 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 795	. . . . . . . . . . . 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 2823	. . . . . . . . . . 11 ⊢ (dist‘𝐺) = (dist‘𝐺)
40		simp-4r 782	. . . . . . . . . . . 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 736	. . . . . . . . . . 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 767	. . . . . . . . . . . 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 736	. . . . . . . . . . 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 786	. . . . . . . . . . 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 782	. . . . . . . . . . 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 736	. . . . . . . . . . 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 736	. . . . . . . . . . 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 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))) ∧ 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 497	. . . . . . . . . . 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 774	. . . . . . . . . . . 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 497	. . . . . . . . . . 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 498	. . . . . . . . . . . . 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 736	. . . . . . . . . . . 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 26268	. . . . . . . . . . 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 771	. . . . . . . . . . . 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 736	. . . . . . . . . . 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 498	. . . . . . . . . . 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 498	. . . . . . . . . . 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 767	. . . . . . . . . . 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 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 ∈ (𝐶𝐼𝑒))
61		simprr 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 ∈ (𝐷𝐼𝑓))
62		simp-7r 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))) → 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 26362	. . . . . . . . . 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 26276	. . . . . . . . . . . 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 26276	. . . . . . . . . . . 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 26255	. . . . . . . . . . 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 732	. . . . . . . . . 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 3291	. . . . . . . . 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 26252	. . . . . . . . . 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 728	. . . . . . . . 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 3291	. . . . . . . 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 26252	. . . . . . . . 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 483	. . . . . . . 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 3291	. . . . . . 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 415	. . . . . 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	syl5bir 245	. . . . 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 859	. . . 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 955	. . 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 26252	. . . 4 ⊢ (𝜑 → ∃𝑓 ∈ 𝑃 (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷)))
80	79	ad2antrr 724	. . 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 3291	. 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 26252	. 2 ⊢ (𝜑 → ∃𝑒 ∈ 𝑃 (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶)))
83	81, 82	r19.29a 3291	1 ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐼𝐷) ∨ 𝐷 ∈ (𝐴𝐼𝐶)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∨ wo 843   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  ∃wrex 3141  ‘cfv 6357  (class class class)co 7158  Basecbs 16485  distcds 16576  TarskiGcstrkg 26218  Itvcitv 26224

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-oadd 8108  df-er 8291  df-pm 8411  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-dju 9332  df-card 9370  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-nn 11641  df-2 11703  df-3 11704  df-n0 11901  df-xnn0 11971  df-z 11985  df-uz 12247  df-fz 12896  df-fzo 13037  df-hash 13694  df-word 13865  df-concat 13925  df-s1 13952  df-s2 14212  df-s3 14213  df-trkgc 26236  df-trkgb 26237  df-trkgcb 26238  df-trkg 26241  df-cgrg 26299

This theorem is referenced by:  tgbtwnconn2  26364  tgbtwnconnln1  26368  hltr  26398  hlbtwn  26399