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Theorem tgbtwnconn1 25387
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 798 . . . . . . . 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 475 . . . . . . 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 481 . . . . . 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 477 . . . . . . 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 6626 . . . . . 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 2701 . . . . 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 408 . . . 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 793 . . . . . . 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 481 . . . . . 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 477 . . . . . . 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 6626 . . . . . 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 2701 . . . . 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 407 . . . 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 2791 . . . . . 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 767 . . . . . . . . . . . 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 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))) → 𝐺 ∈ TarskiG)
20 tgbtwnconn1.a . . . . . . . . . . . . 13 (𝜑𝐴𝑃)
2120ad4antr 767 . . . . . . . . . . . 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 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))) → 𝐴𝑃)
23 tgbtwnconn1.b . . . . . . . . . . . . 13 (𝜑𝐵𝑃)
2423ad4antr 767 . . . . . . . . . . . 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 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))) → 𝐵𝑃)
26 tgbtwnconn1.c . . . . . . . . . . . . 13 (𝜑𝐶𝑃)
2726ad4antr 767 . . . . . . . . . . . 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 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))) → 𝐶𝑃)
29 tgbtwnconn1.d . . . . . . . . . . . . 13 (𝜑𝐷𝑃)
3029ad4antr 767 . . . . . . . . . . . 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 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))) → 𝐷𝑃)
32 simp-11l 819 . . . . . . . . . . . 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 2621 . . . . . . . . . . 11 (dist‘𝐺) = (dist‘𝐺)
40 simp-4r 806 . . . . . . . . . . . 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 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))) → 𝑒𝑃)
42 simplr 791 . . . . . . . . . . . 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 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))) → 𝑓𝑃)
44 simp-6r 810 . . . . . . . . . . 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 806 . . . . . . . . . . 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 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))) → 𝐷 ∈ (𝐴𝐼𝑒))
478ad7antr 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))) → 𝐶 ∈ (𝐴𝐼𝑓))
48 simp-5r 808 . . . . . . . . . . . 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 475 . . . . . . . . . . 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 798 . . . . . . . . . . . 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 475 . . . . . . . . . . 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 479 . . . . . . . . . . . . 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 773 . . . . . . . . . . . 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 25292 . . . . . . . . . . 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 795 . . . . . . . . . . . 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 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))) → (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷))
5748simprd 479 . . . . . . . . . . 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 479 . . . . . . . . . . 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 791 . . . . . . . . . . 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 793 . . . . . . . . . . 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 795 . . . . . . . . . . 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 812 . . . . . . . . . . 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 25386 . . . . . . . . . 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 25300 . . . . . . . . . . . 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 25300 . . . . . . . . . . . 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 25283 . . . . . . . . . . 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 769 . . . . . . . . . 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 3072 . . . . . . . . 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 25280 . . . . . . . . . 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 765 . . . . . . . . 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 3072 . . . . . . . 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 25280 . . . . . . . . 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 481 . . . . . . . 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 3072 . . . . . . 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 450 . . . . . 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 233 . . . . 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 393 . . . 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 826 . . 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 25280 . . . 4 (𝜑 → ∃𝑓𝑃 (𝐶 ∈ (𝐴𝐼𝑓) ∧ (𝐶(dist‘𝐺)𝑓) = (𝐶(dist‘𝐺)𝐷)))
8079ad2antrr 761 . . 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 3072 . 2 (((𝜑𝑒𝑃) ∧ (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶))) → (𝐶 ∈ (𝐴𝐼𝐷) ∨ 𝐷 ∈ (𝐴𝐼𝐶)))
8215, 39, 16, 17, 20, 29, 29, 26axtgsegcon 25280 . 2 (𝜑 → ∃𝑒𝑃 (𝐷 ∈ (𝐴𝐼𝑒) ∧ (𝐷(dist‘𝐺)𝑒) = (𝐷(dist‘𝐺)𝐶)))
8381, 82r19.29a 3072 1 (𝜑 → (𝐶 ∈ (𝐴𝐼𝐷) ∨ 𝐷 ∈ (𝐴𝐼𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 383  wa 384   = wceq 1480  wcel 1987  wne 2790  wrex 2908  cfv 5852  (class class class)co 6610  Basecbs 15792  distcds 15882  TarskiGcstrkg 25246  Itvcitv 25252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-pm 7812  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-card 8717  df-cda 8942  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-3 11032  df-n0 11245  df-xnn0 11316  df-z 11330  df-uz 11640  df-fz 12277  df-fzo 12415  df-hash 13066  df-word 13246  df-concat 13248  df-s1 13249  df-s2 13538  df-s3 13539  df-trkgc 25264  df-trkgb 25265  df-trkgcb 25266  df-trkg 25269  df-cgrg 25323
This theorem is referenced by:  tgbtwnconn2  25388  tgbtwnconnln1  25392  hltr  25422  hlbtwn  25423
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