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Theorem tgbtwnconn1 26373
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 (𝜑𝐺 ∈ TarskiG)
tgbtwnconn1.a (𝜑𝐴𝑃)
tgbtwnconn1.b (𝜑𝐵𝑃)
tgbtwnconn1.c (𝜑𝐶𝑃)
tgbtwnconn1.d (𝜑𝐷𝑃)
tgbtwnconn1.1 (𝜑𝐴𝐵)
tgbtwnconn1.2 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
tgbtwnconn1.3 (𝜑𝐵 ∈ (𝐴𝐼𝐷))
Assertion
Ref Expression
tgbtwnconn1 (𝜑 → (𝐶 ∈ (𝐴𝐼𝐷) ∨ 𝐷 ∈ (𝐴𝐼𝐶)))

Proof of Theorem tgbtwnconn1
Dummy variables 𝑒 𝑓 𝑗 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpllr 775 . . . . . . . 8 (((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶)))
21simpld 498 . . . . . . 7 (((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → 𝐷 ∈ (𝐴𝐼𝑒))
32adantr 484 . . . . . 6 ((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 = 𝑒) → 𝐷 ∈ (𝐴𝐼𝑒))
4 simpr 488 . . . . . . 7 ((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 = 𝑒) → 𝐶 = 𝑒)
54oveq2d 7155 . . . . . 6 ((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 = 𝑒) → (𝐴𝐼𝐶) = (𝐴𝐼𝑒))
63, 5eleqtrrd 2896 . . . . 5 ((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 = 𝑒) → 𝐷 ∈ (𝐴𝐼𝐶))
76olcd 871 . . . 4 ((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶 = 𝑒) → (𝐶 ∈ (𝐴𝐼𝐷) ∨ 𝐷 ∈ (𝐴𝐼𝐶)))
8 simprl 770 . . . . . . 7 (((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → 𝐶 ∈ (𝐴𝐼𝑓))
98adantr 484 . . . . . 6 ((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐷 = 𝑓) → 𝐶 ∈ (𝐴𝐼𝑓))
10 simpr 488 . . . . . . 7 ((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐷 = 𝑓) → 𝐷 = 𝑓)
1110oveq2d 7155 . . . . . 6 ((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐷 = 𝑓) → (𝐴𝐼𝐷) = (𝐴𝐼𝑓))
129, 11eleqtrrd 2896 . . . . 5 ((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐷 = 𝑓) → 𝐶 ∈ (𝐴𝐼𝐷))
1312orcd 870 . . . 4 ((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐷 = 𝑓) → (𝐶 ∈ (𝐴𝐼𝐷) ∨ 𝐷 ∈ (𝐴𝐼𝐶)))
14 df-ne 2991 . . . . . 6 (𝐶𝑒 ↔ ¬ 𝐶 = 𝑒)
15 tgbtwnconn1.p . . . . . . . . . . 11 𝑃 = (Base‘𝐺)
16 tgbtwnconn1.i . . . . . . . . . . 11 𝐼 = (Itv‘𝐺)
17 tgbtwnconn1.g . . . . . . . . . . . . 13 (𝜑𝐺 ∈ TarskiG)
1817ad4antr 731 . . . . . . . . . . . 12 (((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → 𝐺 ∈ TarskiG)
1918ad7antr 737 . . . . . . . . . . 11 ((((((((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) ∧ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝐺 ∈ TarskiG)
20 tgbtwnconn1.a . . . . . . . . . . . . 13 (𝜑𝐴𝑃)
2120ad4antr 731 . . . . . . . . . . . 12 (((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → 𝐴𝑃)
2221ad7antr 737 . . . . . . . . . . 11 ((((((((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) ∧ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝐴𝑃)
23 tgbtwnconn1.b . . . . . . . . . . . . 13 (𝜑𝐵𝑃)
2423ad4antr 731 . . . . . . . . . . . 12 (((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → 𝐵𝑃)
2524ad7antr 737 . . . . . . . . . . 11 ((((((((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) ∧ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝐵𝑃)
26 tgbtwnconn1.c . . . . . . . . . . . . 13 (𝜑𝐶𝑃)
2726ad4antr 731 . . . . . . . . . . . 12 (((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → 𝐶𝑃)
2827ad7antr 737 . . . . . . . . . . 11 ((((((((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) ∧ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝐶𝑃)
29 tgbtwnconn1.d . . . . . . . . . . . . 13 (𝜑𝐷𝑃)
3029ad4antr 731 . . . . . . . . . . . 12 (((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → 𝐷𝑃)
3130ad7antr 737 . . . . . . . . . . 11 ((((((((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) ∧ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝐷𝑃)
32 simp-11l 796 . . . . . . . . . . . 12 ((((((((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) ∧ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝜑)
33 tgbtwnconn1.1 . . . . . . . . . . . 12 (𝜑𝐴𝐵)
3432, 33syl 17 . . . . . . . . . . 11 ((((((((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) ∧ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝐴𝐵)
35 tgbtwnconn1.2 . . . . . . . . . . . 12 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
3632, 35syl 17 . . . . . . . . . . 11 ((((((((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) ∧ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝐵 ∈ (𝐴𝐼𝐶))
37 tgbtwnconn1.3 . . . . . . . . . . . 12 (𝜑𝐵 ∈ (𝐴𝐼𝐷))
3832, 37syl 17 . . . . . . . . . . 11 ((((((((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) ∧ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝐵 ∈ (𝐴𝐼𝐷))
39 eqid 2801 . . . . . . . . . . 11 (dist‘𝐺) = (dist‘𝐺)
40 simp-4r 783 . . . . . . . . . . . 12 (((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → 𝑒𝑃)
4140ad7antr 737 . . . . . . . . . . 11 ((((((((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) ∧ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝑒𝑃)
42 simplr 768 . . . . . . . . . . . 12 (((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → 𝑓𝑃)
4342ad7antr 737 . . . . . . . . . . 11 ((((((((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) ∧ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝑓𝑃)
44 simp-6r 787 . . . . . . . . . . 11 ((((((((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) ∧ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝑃)
45 simp-4r 783 . . . . . . . . . . 11 ((((((((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) ∧ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝑗𝑃)
462ad7antr 737 . . . . . . . . . . 11 ((((((((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) ∧ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝐷 ∈ (𝐴𝐼𝑒))
478ad7antr 737 . . . . . . . . . . 11 ((((((((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) ∧ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝐶 ∈ (𝐴𝐼𝑓))
48 simp-5r 785 . . . . . . . . . . . 12 ((((((((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) ∧ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶)))
4948simpld 498 . . . . . . . . . . 11 ((((((((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) ∧ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝑒 ∈ (𝐴𝐼))
50 simpllr 775 . . . . . . . . . . . 12 ((((((((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) ∧ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷)))
5150simpld 498 . . . . . . . . . . 11 ((((((((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) ∧ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝑓 ∈ (𝐴𝐼𝑗))
521simprd 499 . . . . . . . . . . . . 13 (((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))
5352ad7antr 737 . . . . . . . . . . . 12 ((((((((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) ∧ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))
5415, 39, 16, 19, 31, 41, 31, 28, 53tgcgrcomlr 26278 . . . . . . . . . . 11 ((((((((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) ∧ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → (𝑒(dist‘𝐺)𝐷) = (𝐶(dist‘𝐺)𝐷))
55 simprr 772 . . . . . . . . . . . 12 (((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))
5655ad7antr 737 . . . . . . . . . . 11 ((((((((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) ∧ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))
5748simprd 499 . . . . . . . . . . 11 ((((((((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) ∧ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶))
5850simprd 499 . . . . . . . . . . 11 ((((((((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) ∧ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))
59 simplr 768 . . . . . . . . . . 11 ((((((((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) ∧ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝑥𝑃)
60 simprl 770 . . . . . . . . . . 11 ((((((((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) ∧ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝑥 ∈ (𝐶𝐼𝑒))
61 simprr 772 . . . . . . . . . . 11 ((((((((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) ∧ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝑥 ∈ (𝐷𝐼𝑓))
62 simp-7r 789 . . . . . . . . . . 11 ((((((((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) ∧ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝐶𝑒)
6315, 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, 62tgbtwnconn1lem3 26372 . . . . . . . . . 10 ((((((((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) ∧ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) ∧ 𝑥𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓))) → 𝐷 = 𝑓)
6415, 39, 16, 18, 21, 27, 42, 8tgbtwncom 26286 . . . . . . . . . . . 12 (((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → 𝐶 ∈ (𝑓𝐼𝐴))
6515, 39, 16, 18, 21, 30, 40, 2tgbtwncom 26286 . . . . . . . . . . . 12 (((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → 𝐷 ∈ (𝑒𝐼𝐴))
6615, 39, 16, 18, 42, 40, 21, 27, 30, 64, 65axtgpasch 26265 . . . . . . . . . . 11 (((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → ∃𝑥𝑃 (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓)))
6766ad5antr 733 . . . . . . . . . 10 ((((((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) ∧ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) → ∃𝑥𝑃 (𝑥 ∈ (𝐶𝐼𝑒) ∧ 𝑥 ∈ (𝐷𝐼𝑓)))
6863, 67r19.29a 3251 . . . . . . . . 9 ((((((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) ∧ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶))) ∧ 𝑗𝑃) ∧ (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷))) → 𝐷 = 𝑓)
6915, 39, 16, 18, 21, 42, 24, 30axtgsegcon 26262 . . . . . . . . . 10 (((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → ∃𝑗𝑃 (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷)))
7069ad3antrrr 729 . . . . . . . . 9 ((((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) ∧ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶))) → ∃𝑗𝑃 (𝑓 ∈ (𝐴𝐼𝑗) ∧ (𝑓(dist‘𝐺)𝑗) = (𝐵(dist‘𝐺)𝐷)))
7168, 70r19.29a 3251 . . . . . . . 8 ((((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) ∧ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶))) → 𝐷 = 𝑓)
7215, 39, 16, 18, 21, 40, 24, 27axtgsegcon 26262 . . . . . . . . 9 (((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → ∃𝑃 (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶)))
7372adantr 484 . . . . . . . 8 ((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) → ∃𝑃 (𝑒 ∈ (𝐴𝐼) ∧ (𝑒(dist‘𝐺)) = (𝐵(dist‘𝐺)𝐶)))
7471, 73r19.29a 3251 . . . . . . 7 ((((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) ∧ 𝐶𝑒) → 𝐷 = 𝑓)
7574ex 416 . . . . . 6 (((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → (𝐶𝑒𝐷 = 𝑓))
7614, 75syl5bir 246 . . . . 5 (((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → (¬ 𝐶 = 𝑒𝐷 = 𝑓))
7776orrd 860 . . . 4 (((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → (𝐶 = 𝑒𝐷 = 𝑓))
787, 13, 77mpjaodan 956 . . 3 (((((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) ∧ 𝑓𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))) → (𝐶 ∈ (𝐴𝐼𝐷) ∨ 𝐷 ∈ (𝐴𝐼𝐶)))
7915, 39, 16, 17, 20, 26, 26, 29axtgsegcon 26262 . . . 4 (𝜑 → ∃𝑓𝑃 (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷)))
8079ad2antrr 725 . . 3 (((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) → ∃𝑓𝑃 (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷)))
8178, 80r19.29a 3251 . 2 (((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) → (𝐶 ∈ (𝐴𝐼𝐷) ∨ 𝐷 ∈ (𝐴𝐼𝐶)))
8215, 39, 16, 17, 20, 29, 29, 26axtgsegcon 26262 . 2 (𝜑 → ∃𝑒𝑃 (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶)))
8381, 82r19.29a 3251 1 (𝜑 → (𝐶 ∈ (𝐴𝐼𝐷) ∨ 𝐷 ∈ (𝐴𝐼𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wo 844   = wceq 1538  wcel 2112  wne 2990  wrex 3110  cfv 6328  (class class class)co 7139  Basecbs 16479  distcds 16570  TarskiGcstrkg 26228  Itvcitv 26234
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-pm 8396  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-dju 9318  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-3 11693  df-n0 11890  df-xnn0 11960  df-z 11974  df-uz 12236  df-fz 12890  df-fzo 13033  df-hash 13691  df-word 13862  df-concat 13918  df-s1 13945  df-s2 14205  df-s3 14206  df-trkgc 26246  df-trkgb 26247  df-trkgcb 26248  df-trkg 26251  df-cgrg 26309
This theorem is referenced by:  tgbtwnconn2  26374  tgbtwnconnln1  26378  hltr  26408  hlbtwn  26409
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