Theorem tgbtwnconn1 27517

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 495	. . . . . . 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 481	. . . . . 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 485	. . . . . . 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 7373	. . . . . 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 2841	. . . . 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 872	. . . 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 481	. . . . . 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 485	. . . . . . 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 7373	. . . . . 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 2841	. . . . 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 871	. . . 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 2944	. . . . . 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 2736	. . . . . . . . . . 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 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 ∈ (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 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 ∈ (𝐴𝐼𝑗))
52	1	simprd 496	. . . . . . . . . . . . 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 27422	. . . . . . . . . . 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 496	. . . . . . . . . . 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 496	. . . . . . . . . . 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 27516	. . . . . . . . . 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 27430	. . . . . . . . . . . 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 27430	. . . . . . . . . . . 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 27409	. . . . . . . . . . 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 3159	. . . . . . . . 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 27406	. . . . . . . . . 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 3159	. . . . . . . 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 27406	. . . . . . . . 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 481	. . . . . . . 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 3159	. . . . . . 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 413	. . . . . 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 242	. . . . 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 861	. . . 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 957	. . 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 27406	. . . 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 3159	. 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 27406	. 2 ⊢ (𝜑 → ∃𝑒 ∈ 𝑃 (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶)))
83	81, 82	r19.29a 3159	1 ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐼𝐷) ∨ 𝐷 ∈ (𝐴𝐼𝐶)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ∃wrex 3073  ‘cfv 6496  (class class class)co 7357  Basecbs 17083  distcds 17142  TarskiGcstrkg 27369  Itvcitv 27375

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-er 8648  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-hash 14231  df-word 14403  df-concat 14459  df-s1 14484  df-s2 14737  df-s3 14738  df-trkgc 27390  df-trkgb 27391  df-trkgcb 27392  df-trkg 27395  df-cgrg 27453

This theorem is referenced by:  tgbtwnconn2  27518  tgbtwnconnln1  27522  hltr  27552  hlbtwn  27553