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Theorem tgbtwnconn1 28583
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 776 . . . . . . . 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 494 . . . . . . 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 480 . . . . . 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 484 . . . . . . 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 7447 . . . . . 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 2844 . . . . 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 875 . . . 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 771 . . . . . . 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 480 . . . . . 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 484 . . . . . . 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 7447 . . . . . 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 2844 . . . . 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 874 . . . 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 2941 . . . . . 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 732 . . . . . . . . . . . 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 738 . . . . . . . . . . 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 732 . . . . . . . . . . . 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 738 . . . . . . . . . . 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 732 . . . . . . . . . . . 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 738 . . . . . . . . . . 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 732 . . . . . . . . . . . 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 738 . . . . . . . . . . 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 732 . . . . . . . . . . . 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 738 . . . . . . . . . . 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 797 . . . . . . . . . . . 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 2737 . . . . . . . . . . 11 (dist‘𝐺) = (dist‘𝐺)
40 simp-4r 784 . . . . . . . . . . . 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 738 . . . . . . . . . . 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 769 . . . . . . . . . . . 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 738 . . . . . . . . . . 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 788 . . . . . . . . . . 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 784 . . . . . . . . . . 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 738 . . . . . . . . . . 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 738 . . . . . . . . . . 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 786 . . . . . . . . . . . 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 494 . . . . . . . . . . 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 776 . . . . . . . . . . . 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 494 . . . . . . . . . . 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 495 . . . . . . . . . . . . 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 738 . . . . . . . . . . . 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 28488 . . . . . . . . . . 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 773 . . . . . . . . . . . 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 738 . . . . . . . . . . 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 495 . . . . . . . . . . 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 495 . . . . . . . . . . 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 769 . . . . . . . . . . 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 771 . . . . . . . . . . 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 773 . . . . . . . . . . 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 790 . . . . . . . . . . 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 28582 . . . . . . . . . 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 28496 . . . . . . . . . . . 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 28496 . . . . . . . . . . . 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 28475 . . . . . . . . . . 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 734 . . . . . . . . . 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 3162 . . . . . . . . 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 28472 . . . . . . . . . 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 730 . . . . . . . . 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 3162 . . . . . . . 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 28472 . . . . . . . . 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 480 . . . . . . . 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 3162 . . . . . . 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 412 . . . . . 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, 75biimtrrid 243 . . . . 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 864 . . . 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 961 . . 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 28472 . . . 4 (𝜑 → ∃𝑓𝑃 (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷)))
8079ad2antrr 726 . . 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 3162 . 2 (((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) → (𝐶 ∈ (𝐴𝐼𝐷) ∨ 𝐷 ∈ (𝐴𝐼𝐶)))
8215, 39, 16, 17, 20, 29, 29, 26axtgsegcon 28472 . 2 (𝜑 → ∃𝑒𝑃 (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶)))
8381, 82r19.29a 3162 1 (𝜑 → (𝐶 ∈ (𝐴𝐼𝐷) ∨ 𝐷 ∈ (𝐴𝐼𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 848   = wceq 1540  wcel 2108  wne 2940  wrex 3070  cfv 6561  (class class class)co 7431  Basecbs 17247  distcds 17306  TarskiGcstrkg 28435  Itvcitv 28441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-er 8745  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-hash 14370  df-word 14553  df-concat 14609  df-s1 14634  df-s2 14887  df-s3 14888  df-trkgc 28456  df-trkgb 28457  df-trkgcb 28458  df-trkg 28461  df-cgrg 28519
This theorem is referenced by:  tgbtwnconn2  28584  tgbtwnconnln1  28588  hltr  28618  hlbtwn  28619
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