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Theorem dfcgra2 27772
Description: This is the full statement of definition 11.2 of [Schwabhauser] p. 95. This proof serves to confirm that the definition we have chosen, df-cgra 27750 is indeed equivalent to the textbook's definition. (Contributed by Thierry Arnoux, 2-Aug-2020.)
Hypotheses
Ref Expression
dfcgra2.p 𝑃 = (Base‘𝐺)
dfcgra2.i 𝐼 = (Itv‘𝐺)
dfcgra2.m = (dist‘𝐺)
dfcgra2.g (𝜑𝐺 ∈ TarskiG)
dfcgra2.a (𝜑𝐴𝑃)
dfcgra2.b (𝜑𝐵𝑃)
dfcgra2.c (𝜑𝐶𝑃)
dfcgra2.d (𝜑𝐷𝑃)
dfcgra2.e (𝜑𝐸𝑃)
dfcgra2.f (𝜑𝐹𝑃)
Assertion
Ref Expression
dfcgra2 (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸) ∧ ∃𝑎𝑃𝑐𝑃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑎 𝑐) = (𝑑 𝑓)))))
Distinct variable groups:   ,𝑎,𝑐,𝑑,𝑓   𝐴,𝑎,𝑐,𝑑,𝑓   𝐵,𝑎,𝑐,𝑑,𝑓   𝐶,𝑎,𝑐,𝑑,𝑓   𝐷,𝑎,𝑐,𝑑,𝑓   𝐸,𝑎,𝑐,𝑑,𝑓   𝐹,𝑎,𝑐,𝑑,𝑓   𝐺,𝑎,𝑐,𝑑,𝑓   𝐼,𝑎,𝑐,𝑑,𝑓   𝑃,𝑎,𝑐,𝑑,𝑓   𝜑,𝑎,𝑐,𝑑,𝑓

Proof of Theorem dfcgra2
Dummy variables 𝑡 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfcgra2.p . . . . 5 𝑃 = (Base‘𝐺)
2 dfcgra2.i . . . . 5 𝐼 = (Itv‘𝐺)
3 eqid 2736 . . . . 5 (hlG‘𝐺) = (hlG‘𝐺)
4 dfcgra2.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
54adantr 481 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐺 ∈ TarskiG)
6 dfcgra2.a . . . . . 6 (𝜑𝐴𝑃)
76adantr 481 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐴𝑃)
8 dfcgra2.b . . . . . 6 (𝜑𝐵𝑃)
98adantr 481 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐵𝑃)
10 dfcgra2.c . . . . . 6 (𝜑𝐶𝑃)
1110adantr 481 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐶𝑃)
12 dfcgra2.d . . . . . 6 (𝜑𝐷𝑃)
1312adantr 481 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐷𝑃)
14 dfcgra2.e . . . . . 6 (𝜑𝐸𝑃)
1514adantr 481 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐸𝑃)
16 dfcgra2.f . . . . . 6 (𝜑𝐹𝑃)
1716adantr 481 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐹𝑃)
18 simpr 485 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
191, 2, 3, 5, 7, 9, 11, 13, 15, 17, 18cgrane1 27754 . . . 4 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐴𝐵)
201, 2, 3, 5, 7, 9, 11, 13, 15, 17, 18cgrane2 27755 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐵𝐶)
2120necomd 2999 . . . 4 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐶𝐵)
2219, 21jca 512 . . 3 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → (𝐴𝐵𝐶𝐵))
231, 2, 3, 5, 7, 9, 11, 13, 15, 17, 18cgrane3 27756 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐸𝐷)
2423necomd 2999 . . . 4 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐷𝐸)
251, 2, 3, 5, 7, 9, 11, 13, 15, 17, 18cgrane4 27757 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐸𝐹)
2625necomd 2999 . . . 4 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐹𝐸)
2724, 26jca 512 . . 3 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → (𝐷𝐸𝐹𝐸))
28 simprl 769 . . . . . . . . 9 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → ((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))))
29 simprr 771 . . . . . . . . 9 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))
304ad6antr 734 . . . . . . . . . 10 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝐺 ∈ TarskiG)
31 simp-5r 784 . . . . . . . . . 10 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝑎𝑃)
328ad6antr 734 . . . . . . . . . 10 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝐵𝑃)
33 simp-4r 782 . . . . . . . . . 10 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝑐𝑃)
34 simpllr 774 . . . . . . . . . 10 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝑑𝑃)
3514ad6antr 734 . . . . . . . . . 10 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝐸𝑃)
36 simplr 767 . . . . . . . . . 10 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝑓𝑃)
3716ad6antr 734 . . . . . . . . . . 11 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝐹𝑃)
3812ad6antr 734 . . . . . . . . . . . 12 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝐷𝑃)
3910ad6antr 734 . . . . . . . . . . . . . 14 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝐶𝑃)
406ad6antr 734 . . . . . . . . . . . . . . 15 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝐴𝑃)
41 simp-6r 786 . . . . . . . . . . . . . . . 16 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
421, 2, 30, 3, 40, 32, 39, 38, 35, 37, 41cgracom 27764 . . . . . . . . . . . . . . 15 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
4328simplld 766 . . . . . . . . . . . . . . . . 17 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝐴 ∈ (𝐵𝐼𝑎))
44 dfcgra2.m . . . . . . . . . . . . . . . . . 18 = (dist‘𝐺)
4519ad5antr 732 . . . . . . . . . . . . . . . . . 18 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝐴𝐵)
461, 44, 2, 30, 32, 40, 31, 43, 45tgbtwnne 27432 . . . . . . . . . . . . . . . . 17 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝐵𝑎)
471, 2, 3, 32, 31, 40, 30, 40, 43, 46, 45btwnhl1 27554 . . . . . . . . . . . . . . . 16 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝐴((hlG‘𝐺)‘𝐵)𝑎)
481, 2, 3, 40, 31, 32, 30, 47hlcomd 27546 . . . . . . . . . . . . . . 15 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝑎((hlG‘𝐺)‘𝐵)𝐴)
491, 2, 3, 30, 38, 35, 37, 40, 32, 39, 42, 31, 48cgrahl1 27758 . . . . . . . . . . . . . 14 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝑎𝐵𝐶”⟩)
5028simprld 770 . . . . . . . . . . . . . . . 16 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝐶 ∈ (𝐵𝐼𝑐))
5121ad5antr 732 . . . . . . . . . . . . . . . . 17 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝐶𝐵)
521, 44, 2, 30, 32, 39, 33, 50, 51tgbtwnne 27432 . . . . . . . . . . . . . . . 16 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝐵𝑐)
531, 2, 3, 32, 33, 39, 30, 40, 50, 52, 51btwnhl1 27554 . . . . . . . . . . . . . . 15 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝐶((hlG‘𝐺)‘𝐵)𝑐)
541, 2, 3, 39, 33, 32, 30, 53hlcomd 27546 . . . . . . . . . . . . . 14 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝑐((hlG‘𝐺)‘𝐵)𝐶)
551, 2, 3, 30, 38, 35, 37, 31, 32, 39, 49, 33, 54cgrahl2 27759 . . . . . . . . . . . . 13 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝑎𝐵𝑐”⟩)
561, 2, 30, 3, 38, 35, 37, 31, 32, 33, 55cgracom 27764 . . . . . . . . . . . 12 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → ⟨“𝑎𝐵𝑐”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
5729simplld 766 . . . . . . . . . . . . . 14 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝐷 ∈ (𝐸𝐼𝑑))
5824ad5antr 732 . . . . . . . . . . . . . . 15 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝐷𝐸)
591, 44, 2, 30, 35, 38, 34, 57, 58tgbtwnne 27432 . . . . . . . . . . . . . 14 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝐸𝑑)
601, 2, 3, 35, 34, 38, 30, 40, 57, 59, 58btwnhl1 27554 . . . . . . . . . . . . 13 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝐷((hlG‘𝐺)‘𝐸)𝑑)
611, 2, 3, 38, 34, 35, 30, 60hlcomd 27546 . . . . . . . . . . . 12 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝑑((hlG‘𝐺)‘𝐸)𝐷)
621, 2, 3, 30, 31, 32, 33, 38, 35, 37, 56, 34, 61cgrahl1 27758 . . . . . . . . . . 11 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → ⟨“𝑎𝐵𝑐”⟩(cgrA‘𝐺)⟨“𝑑𝐸𝐹”⟩)
6329simprld 770 . . . . . . . . . . . . 13 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝐹 ∈ (𝐸𝐼𝑓))
6426ad5antr 732 . . . . . . . . . . . . . 14 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝐹𝐸)
651, 44, 2, 30, 35, 37, 36, 63, 64tgbtwnne 27432 . . . . . . . . . . . . 13 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝐸𝑓)
661, 2, 3, 35, 36, 37, 30, 40, 63, 65, 64btwnhl1 27554 . . . . . . . . . . . 12 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝐹((hlG‘𝐺)‘𝐸)𝑓)
671, 2, 3, 37, 36, 35, 30, 66hlcomd 27546 . . . . . . . . . . 11 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝑓((hlG‘𝐺)‘𝐸)𝐹)
681, 2, 3, 30, 31, 32, 33, 34, 35, 37, 62, 36, 67cgrahl2 27759 . . . . . . . . . 10 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → ⟨“𝑎𝐵𝑐”⟩(cgrA‘𝐺)⟨“𝑑𝐸𝑓”⟩)
6946necomd 2999 . . . . . . . . . . 11 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝑎𝐵)
701, 2, 3, 31, 40, 32, 30, 69hlid 27551 . . . . . . . . . 10 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝑎((hlG‘𝐺)‘𝐵)𝑎)
7152necomd 2999 . . . . . . . . . . 11 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝑐𝐵)
721, 2, 3, 33, 40, 32, 30, 71hlid 27551 . . . . . . . . . 10 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝑐((hlG‘𝐺)‘𝐵)𝑐)
731, 44, 2, 30, 32, 40, 31, 43tgbtwncom 27430 . . . . . . . . . . . 12 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝐴 ∈ (𝑎𝐼𝐵))
7428simplrd 768 . . . . . . . . . . . . 13 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → (𝐴 𝑎) = (𝐸 𝐷))
751, 44, 2, 30, 40, 31, 35, 38, 74tgcgrcoml 27421 . . . . . . . . . . . 12 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → (𝑎 𝐴) = (𝐸 𝐷))
7629simplrd 768 . . . . . . . . . . . . . 14 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → (𝐷 𝑑) = (𝐵 𝐴))
7776eqcomd 2742 . . . . . . . . . . . . 13 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → (𝐵 𝐴) = (𝐷 𝑑))
781, 44, 2, 30, 32, 40, 38, 34, 77tgcgrcoml 27421 . . . . . . . . . . . 12 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → (𝐴 𝐵) = (𝐷 𝑑))
791, 44, 2, 30, 31, 40, 32, 35, 38, 34, 73, 57, 75, 78tgcgrextend 27427 . . . . . . . . . . 11 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → (𝑎 𝐵) = (𝐸 𝑑))
801, 44, 2, 30, 31, 32, 35, 34, 79tgcgrcoml 27421 . . . . . . . . . 10 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → (𝐵 𝑎) = (𝐸 𝑑))
811, 44, 2, 30, 32, 39, 33, 50tgbtwncom 27430 . . . . . . . . . . . 12 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → 𝐶 ∈ (𝑐𝐼𝐵))
8228simprrd 772 . . . . . . . . . . . . 13 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → (𝐶 𝑐) = (𝐸 𝐹))
831, 44, 2, 30, 39, 33, 35, 37, 82tgcgrcoml 27421 . . . . . . . . . . . 12 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → (𝑐 𝐶) = (𝐸 𝐹))
8429simprrd 772 . . . . . . . . . . . . . 14 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → (𝐹 𝑓) = (𝐵 𝐶))
8584eqcomd 2742 . . . . . . . . . . . . 13 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → (𝐵 𝐶) = (𝐹 𝑓))
861, 44, 2, 30, 32, 39, 37, 36, 85tgcgrcoml 27421 . . . . . . . . . . . 12 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → (𝐶 𝐵) = (𝐹 𝑓))
871, 44, 2, 30, 33, 39, 32, 35, 37, 36, 81, 63, 83, 86tgcgrextend 27427 . . . . . . . . . . 11 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → (𝑐 𝐵) = (𝐸 𝑓))
881, 44, 2, 30, 33, 32, 35, 36, 87tgcgrcoml 27421 . . . . . . . . . 10 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → (𝐵 𝑐) = (𝐸 𝑓))
891, 2, 3, 30, 31, 32, 33, 34, 35, 36, 68, 31, 44, 33, 70, 72, 80, 88cgracgr 27760 . . . . . . . . 9 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → (𝑎 𝑐) = (𝑑 𝑓))
9028, 29, 893jca 1128 . . . . . . . 8 (((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑎 𝑐) = (𝑑 𝑓)))
9190ex 413 . . . . . . 7 ((((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) ∧ 𝑓𝑃) → ((((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶)))) → (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑎 𝑐) = (𝑑 𝑓))))
9291reximdva 3165 . . . . . 6 (((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ 𝑑𝑃) → (∃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶)))) → ∃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑎 𝑐) = (𝑑 𝑓))))
9392reximdva 3165 . . . . 5 ((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) → (∃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶)))) → ∃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑎 𝑐) = (𝑑 𝑓))))
9493imp 407 . . . 4 (((((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) ∧ 𝑎𝑃) ∧ 𝑐𝑃) ∧ ∃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))) → ∃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑎 𝑐) = (𝑑 𝑓)))
951, 44, 2, 4, 8, 6, 14, 12axtgsegcon 27406 . . . . . . . 8 (𝜑 → ∃𝑎𝑃 (𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)))
961, 44, 2, 4, 8, 10, 14, 16axtgsegcon 27406 . . . . . . . 8 (𝜑 → ∃𝑐𝑃 (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹)))
97 reeanv 3217 . . . . . . . 8 (∃𝑎𝑃𝑐𝑃 ((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ↔ (∃𝑎𝑃 (𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ ∃𝑐𝑃 (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))))
9895, 96, 97sylanbrc 583 . . . . . . 7 (𝜑 → ∃𝑎𝑃𝑐𝑃 ((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))))
991, 44, 2, 4, 14, 12, 8, 6axtgsegcon 27406 . . . . . . . 8 (𝜑 → ∃𝑑𝑃 (𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)))
1001, 44, 2, 4, 14, 16, 8, 10axtgsegcon 27406 . . . . . . . 8 (𝜑 → ∃𝑓𝑃 (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶)))
101 reeanv 3217 . . . . . . . 8 (∃𝑑𝑃𝑓𝑃 ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ↔ (∃𝑑𝑃 (𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ ∃𝑓𝑃 (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))
10299, 100, 101sylanbrc 583 . . . . . . 7 (𝜑 → ∃𝑑𝑃𝑓𝑃 ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))
10398, 102jca 512 . . . . . 6 (𝜑 → (∃𝑎𝑃𝑐𝑃 ((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ∃𝑑𝑃𝑓𝑃 ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶)))))
104 r19.41vv 3215 . . . . . . . . 9 (∃𝑑𝑃𝑓𝑃 (((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ ((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹)))) ↔ (∃𝑑𝑃𝑓𝑃 ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ ((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹)))))
105 ancom 461 . . . . . . . . . 10 ((((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ ((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹)))) ↔ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶)))))
1061052rexbii 3128 . . . . . . . . 9 (∃𝑑𝑃𝑓𝑃 (((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ ((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹)))) ↔ ∃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶)))))
107 ancom 461 . . . . . . . . 9 ((∃𝑑𝑃𝑓𝑃 ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ ((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹)))) ↔ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ∃𝑑𝑃𝑓𝑃 ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶)))))
108104, 106, 1073bitr3i 300 . . . . . . . 8 (∃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶)))) ↔ (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ∃𝑑𝑃𝑓𝑃 ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶)))))
1091082rexbii 3128 . . . . . . 7 (∃𝑎𝑃𝑐𝑃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶)))) ↔ ∃𝑎𝑃𝑐𝑃 (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ∃𝑑𝑃𝑓𝑃 ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶)))))
110 r19.41vv 3215 . . . . . . 7 (∃𝑎𝑃𝑐𝑃 (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ∃𝑑𝑃𝑓𝑃 ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶)))) ↔ (∃𝑎𝑃𝑐𝑃 ((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ∃𝑑𝑃𝑓𝑃 ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶)))))
111109, 110bitr2i 275 . . . . . 6 ((∃𝑎𝑃𝑐𝑃 ((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ∃𝑑𝑃𝑓𝑃 ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶)))) ↔ ∃𝑎𝑃𝑐𝑃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶)))))
112103, 111sylib 217 . . . . 5 (𝜑 → ∃𝑎𝑃𝑐𝑃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶)))))
113112adantr 481 . . . 4 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → ∃𝑎𝑃𝑐𝑃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶)))))
11494, 113reximddv2 3206 . . 3 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → ∃𝑎𝑃𝑐𝑃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑎 𝑐) = (𝑑 𝑓)))
11522, 27, 1143jca 1128 . 2 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸) ∧ ∃𝑎𝑃𝑐𝑃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑎 𝑐) = (𝑑 𝑓))))
116 df-3an 1089 . . 3 (((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸) ∧ ∃𝑎𝑃𝑐𝑃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑎 𝑐) = (𝑑 𝑓))) ↔ (((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸)) ∧ ∃𝑎𝑃𝑐𝑃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑎 𝑐) = (𝑑 𝑓))))
1174ad6antr 734 . . . . . . . . 9 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → 𝐺 ∈ TarskiG)
11812ad6antr 734 . . . . . . . . 9 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → 𝐷𝑃)
11914ad6antr 734 . . . . . . . . 9 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → 𝐸𝑃)
12016ad6antr 734 . . . . . . . . 9 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → 𝐹𝑃)
1216ad6antr 734 . . . . . . . . 9 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → 𝐴𝑃)
1228ad6antr 734 . . . . . . . . 9 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → 𝐵𝑃)
12310ad6antr 734 . . . . . . . . 9 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → 𝐶𝑃)
124 simp-4r 782 . . . . . . . . . 10 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → 𝑦𝑃)
125 simp-5r 784 . . . . . . . . . . 11 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → 𝑥𝑃)
126 simpllr 774 . . . . . . . . . . . . 13 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → 𝑧𝑃)
127 simplr 767 . . . . . . . . . . . . 13 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → 𝑡𝑃)
128 eqid 2736 . . . . . . . . . . . . . 14 (cgrG‘𝐺) = (cgrG‘𝐺)
129 simpr1 1194 . . . . . . . . . . . . . . . . 17 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → ((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))))
130129simplld 766 . . . . . . . . . . . . . . . 16 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → 𝐴 ∈ (𝐵𝐼𝑥))
131 simpr2 1195 . . . . . . . . . . . . . . . . . 18 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))))
132131simplld 766 . . . . . . . . . . . . . . . . 17 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → 𝐷 ∈ (𝐸𝐼𝑧))
1331, 44, 2, 117, 119, 118, 126, 132tgbtwncom 27430 . . . . . . . . . . . . . . . 16 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → 𝐷 ∈ (𝑧𝐼𝐸))
134131simplrd 768 . . . . . . . . . . . . . . . . . 18 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → (𝐷 𝑧) = (𝐵 𝐴))
135134eqcomd 2742 . . . . . . . . . . . . . . . . 17 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → (𝐵 𝐴) = (𝐷 𝑧))
1361, 44, 2, 117, 122, 121, 118, 126, 135tgcgrcomr 27420 . . . . . . . . . . . . . . . 16 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → (𝐵 𝐴) = (𝑧 𝐷))
137129simplrd 768 . . . . . . . . . . . . . . . . 17 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → (𝐴 𝑥) = (𝐸 𝐷))
1381, 44, 2, 117, 121, 125, 119, 118, 137tgcgrcomr 27420 . . . . . . . . . . . . . . . 16 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → (𝐴 𝑥) = (𝐷 𝐸))
1391, 44, 2, 117, 122, 121, 125, 126, 118, 119, 130, 133, 136, 138tgcgrextend 27427 . . . . . . . . . . . . . . 15 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → (𝐵 𝑥) = (𝑧 𝐸))
1401, 44, 2, 117, 122, 125, 126, 119, 139tgcgrcoml 27421 . . . . . . . . . . . . . 14 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → (𝑥 𝐵) = (𝑧 𝐸))
141129simprld 770 . . . . . . . . . . . . . . . . 17 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → 𝐶 ∈ (𝐵𝐼𝑦))
1421, 44, 2, 117, 122, 123, 124, 141tgbtwncom 27430 . . . . . . . . . . . . . . . 16 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → 𝐶 ∈ (𝑦𝐼𝐵))
143131simprld 770 . . . . . . . . . . . . . . . 16 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → 𝐹 ∈ (𝐸𝐼𝑡))
144129simprrd 772 . . . . . . . . . . . . . . . . 17 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → (𝐶 𝑦) = (𝐸 𝐹))
1451, 44, 2, 117, 123, 124, 119, 120, 144tgcgrcoml 27421 . . . . . . . . . . . . . . . 16 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → (𝑦 𝐶) = (𝐸 𝐹))
146131simprrd 772 . . . . . . . . . . . . . . . . . 18 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → (𝐹 𝑡) = (𝐵 𝐶))
147146eqcomd 2742 . . . . . . . . . . . . . . . . 17 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → (𝐵 𝐶) = (𝐹 𝑡))
1481, 44, 2, 117, 122, 123, 120, 127, 147tgcgrcoml 27421 . . . . . . . . . . . . . . . 16 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → (𝐶 𝐵) = (𝐹 𝑡))
1491, 44, 2, 117, 124, 123, 122, 119, 120, 127, 142, 143, 145, 148tgcgrextend 27427 . . . . . . . . . . . . . . 15 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → (𝑦 𝐵) = (𝐸 𝑡))
1501, 44, 2, 117, 124, 122, 119, 127, 149tgcgrcoml 27421 . . . . . . . . . . . . . 14 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → (𝐵 𝑦) = (𝐸 𝑡))
151 simpr3 1196 . . . . . . . . . . . . . . 15 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → (𝑥 𝑦) = (𝑧 𝑡))
1521, 44, 2, 117, 125, 124, 126, 127, 151tgcgrcomlr 27422 . . . . . . . . . . . . . 14 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → (𝑦 𝑥) = (𝑡 𝑧))
1531, 44, 128, 117, 125, 122, 124, 126, 119, 127, 140, 150, 152trgcgr 27458 . . . . . . . . . . . . 13 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → ⟨“𝑥𝐵𝑦”⟩(cgrG‘𝐺)⟨“𝑧𝐸𝑡”⟩)
154 simp-6r 786 . . . . . . . . . . . . . . . . 17 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸)))
155154simprld 770 . . . . . . . . . . . . . . . 16 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → 𝐷𝐸)
1561, 44, 2, 117, 119, 118, 126, 132, 155tgbtwnne 27432 . . . . . . . . . . . . . . 15 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → 𝐸𝑧)
1571, 2, 3, 119, 126, 118, 117, 122, 132, 156, 155btwnhl1 27554 . . . . . . . . . . . . . 14 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → 𝐷((hlG‘𝐺)‘𝐸)𝑧)
1581, 2, 3, 118, 126, 119, 117, 157hlcomd 27546 . . . . . . . . . . . . 13 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → 𝑧((hlG‘𝐺)‘𝐸)𝐷)
159154simprrd 772 . . . . . . . . . . . . . . . 16 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → 𝐹𝐸)
1601, 44, 2, 117, 119, 120, 127, 143, 159tgbtwnne 27432 . . . . . . . . . . . . . . 15 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → 𝐸𝑡)
1611, 2, 3, 119, 127, 120, 117, 122, 143, 160, 159btwnhl1 27554 . . . . . . . . . . . . . 14 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → 𝐹((hlG‘𝐺)‘𝐸)𝑡)
1621, 2, 3, 120, 127, 119, 117, 161hlcomd 27546 . . . . . . . . . . . . 13 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → 𝑡((hlG‘𝐺)‘𝐸)𝐹)
1631, 2, 3, 117, 125, 122, 124, 118, 119, 120, 126, 127, 153, 158, 162iscgrad 27753 . . . . . . . . . . . 12 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → ⟨“𝑥𝐵𝑦”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
1641, 2, 117, 3, 125, 122, 124, 118, 119, 120, 163cgracom 27764 . . . . . . . . . . 11 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝑥𝐵𝑦”⟩)
165154simplld 766 . . . . . . . . . . . . 13 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → 𝐴𝐵)
1661, 44, 2, 117, 122, 121, 125, 130, 165tgbtwnne 27432 . . . . . . . . . . . 12 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → 𝐵𝑥)
1671, 2, 3, 122, 125, 121, 117, 121, 130, 166, 165btwnhl1 27554 . . . . . . . . . . 11 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → 𝐴((hlG‘𝐺)‘𝐵)𝑥)
1681, 2, 3, 117, 118, 119, 120, 125, 122, 124, 164, 121, 167cgrahl1 27758 . . . . . . . . . 10 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝑦”⟩)
169154simplrd 768 . . . . . . . . . . . 12 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → 𝐶𝐵)
1701, 44, 2, 117, 122, 123, 124, 141, 169tgbtwnne 27432 . . . . . . . . . . 11 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → 𝐵𝑦)
1711, 2, 3, 122, 124, 123, 117, 121, 141, 170, 169btwnhl1 27554 . . . . . . . . . 10 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → 𝐶((hlG‘𝐺)‘𝐵)𝑦)
1721, 2, 3, 117, 118, 119, 120, 121, 122, 124, 168, 123, 171cgrahl2 27759 . . . . . . . . 9 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
1731, 2, 117, 3, 118, 119, 120, 121, 122, 123, 172cgracom 27764 . . . . . . . 8 (((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
174173adantl3r 748 . . . . . . 7 ((((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ∃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑑 𝑓))) ∧ 𝑧𝑃) ∧ 𝑡𝑃) ∧ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
175 simpr 485 . . . . . . . 8 (((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ∃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑑 𝑓))) → ∃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑑 𝑓)))
176 oveq2 7365 . . . . . . . . . . . . 13 (𝑑 = 𝑧 → (𝐸𝐼𝑑) = (𝐸𝐼𝑧))
177176eleq2d 2823 . . . . . . . . . . . 12 (𝑑 = 𝑧 → (𝐷 ∈ (𝐸𝐼𝑑) ↔ 𝐷 ∈ (𝐸𝐼𝑧)))
178 oveq2 7365 . . . . . . . . . . . . 13 (𝑑 = 𝑧 → (𝐷 𝑑) = (𝐷 𝑧))
179178eqeq1d 2738 . . . . . . . . . . . 12 (𝑑 = 𝑧 → ((𝐷 𝑑) = (𝐵 𝐴) ↔ (𝐷 𝑧) = (𝐵 𝐴)))
180177, 179anbi12d 631 . . . . . . . . . . 11 (𝑑 = 𝑧 → ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ↔ (𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴))))
181180anbi1d 630 . . . . . . . . . 10 (𝑑 = 𝑧 → (((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ↔ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶)))))
182 oveq1 7364 . . . . . . . . . . 11 (𝑑 = 𝑧 → (𝑑 𝑓) = (𝑧 𝑓))
183182eqeq2d 2747 . . . . . . . . . 10 (𝑑 = 𝑧 → ((𝑥 𝑦) = (𝑑 𝑓) ↔ (𝑥 𝑦) = (𝑧 𝑓)))
184181, 1833anbi23d 1439 . . . . . . . . 9 (𝑑 = 𝑧 → ((((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑑 𝑓)) ↔ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑓))))
185 oveq2 7365 . . . . . . . . . . . . 13 (𝑓 = 𝑡 → (𝐸𝐼𝑓) = (𝐸𝐼𝑡))
186185eleq2d 2823 . . . . . . . . . . . 12 (𝑓 = 𝑡 → (𝐹 ∈ (𝐸𝐼𝑓) ↔ 𝐹 ∈ (𝐸𝐼𝑡)))
187 oveq2 7365 . . . . . . . . . . . . 13 (𝑓 = 𝑡 → (𝐹 𝑓) = (𝐹 𝑡))
188187eqeq1d 2738 . . . . . . . . . . . 12 (𝑓 = 𝑡 → ((𝐹 𝑓) = (𝐵 𝐶) ↔ (𝐹 𝑡) = (𝐵 𝐶)))
189186, 188anbi12d 631 . . . . . . . . . . 11 (𝑓 = 𝑡 → ((𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶)) ↔ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))))
190189anbi2d 629 . . . . . . . . . 10 (𝑓 = 𝑡 → (((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ↔ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶)))))
191 oveq2 7365 . . . . . . . . . . 11 (𝑓 = 𝑡 → (𝑧 𝑓) = (𝑧 𝑡))
192191eqeq2d 2747 . . . . . . . . . 10 (𝑓 = 𝑡 → ((𝑥 𝑦) = (𝑧 𝑓) ↔ (𝑥 𝑦) = (𝑧 𝑡)))
193190, 1923anbi23d 1439 . . . . . . . . 9 (𝑓 = 𝑡 → ((((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑓)) ↔ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡))))
194184, 193cbvrex2vw 3228 . . . . . . . 8 (∃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑑 𝑓)) ↔ ∃𝑧𝑃𝑡𝑃 (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡)))
195175, 194sylib 217 . . . . . . 7 (((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ∃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑑 𝑓))) → ∃𝑧𝑃𝑡𝑃 (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑧 𝑡)))
196174, 195r19.29vva 3207 . . . . . 6 (((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ∃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑑 𝑓))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
197196adantl3r 748 . . . . 5 ((((((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ ∃𝑎𝑃𝑐𝑃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑎 𝑐) = (𝑑 𝑓))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ∃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑑 𝑓))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
198 simpr 485 . . . . . 6 (((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ ∃𝑎𝑃𝑐𝑃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑎 𝑐) = (𝑑 𝑓))) → ∃𝑎𝑃𝑐𝑃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑎 𝑐) = (𝑑 𝑓)))
199 oveq2 7365 . . . . . . . . . . . 12 (𝑎 = 𝑥 → (𝐵𝐼𝑎) = (𝐵𝐼𝑥))
200199eleq2d 2823 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝐴 ∈ (𝐵𝐼𝑎) ↔ 𝐴 ∈ (𝐵𝐼𝑥)))
201 oveq2 7365 . . . . . . . . . . . 12 (𝑎 = 𝑥 → (𝐴 𝑎) = (𝐴 𝑥))
202201eqeq1d 2738 . . . . . . . . . . 11 (𝑎 = 𝑥 → ((𝐴 𝑎) = (𝐸 𝐷) ↔ (𝐴 𝑥) = (𝐸 𝐷)))
203200, 202anbi12d 631 . . . . . . . . . 10 (𝑎 = 𝑥 → ((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ↔ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷))))
204203anbi1d 630 . . . . . . . . 9 (𝑎 = 𝑥 → (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ↔ ((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹)))))
205 oveq1 7364 . . . . . . . . . 10 (𝑎 = 𝑥 → (𝑎 𝑐) = (𝑥 𝑐))
206205eqeq1d 2738 . . . . . . . . 9 (𝑎 = 𝑥 → ((𝑎 𝑐) = (𝑑 𝑓) ↔ (𝑥 𝑐) = (𝑑 𝑓)))
207204, 2063anbi13d 1438 . . . . . . . 8 (𝑎 = 𝑥 → ((((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑎 𝑐) = (𝑑 𝑓)) ↔ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑥 𝑐) = (𝑑 𝑓))))
2082072rexbidv 3213 . . . . . . 7 (𝑎 = 𝑥 → (∃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑎 𝑐) = (𝑑 𝑓)) ↔ ∃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑥 𝑐) = (𝑑 𝑓))))
209 oveq2 7365 . . . . . . . . . . . 12 (𝑐 = 𝑦 → (𝐵𝐼𝑐) = (𝐵𝐼𝑦))
210209eleq2d 2823 . . . . . . . . . . 11 (𝑐 = 𝑦 → (𝐶 ∈ (𝐵𝐼𝑐) ↔ 𝐶 ∈ (𝐵𝐼𝑦)))
211 oveq2 7365 . . . . . . . . . . . 12 (𝑐 = 𝑦 → (𝐶 𝑐) = (𝐶 𝑦))
212211eqeq1d 2738 . . . . . . . . . . 11 (𝑐 = 𝑦 → ((𝐶 𝑐) = (𝐸 𝐹) ↔ (𝐶 𝑦) = (𝐸 𝐹)))
213210, 212anbi12d 631 . . . . . . . . . 10 (𝑐 = 𝑦 → ((𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹)) ↔ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))))
214213anbi2d 629 . . . . . . . . 9 (𝑐 = 𝑦 → (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ↔ ((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹)))))
215 oveq2 7365 . . . . . . . . . 10 (𝑐 = 𝑦 → (𝑥 𝑐) = (𝑥 𝑦))
216215eqeq1d 2738 . . . . . . . . 9 (𝑐 = 𝑦 → ((𝑥 𝑐) = (𝑑 𝑓) ↔ (𝑥 𝑦) = (𝑑 𝑓)))
217214, 2163anbi13d 1438 . . . . . . . 8 (𝑐 = 𝑦 → ((((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑥 𝑐) = (𝑑 𝑓)) ↔ (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑑 𝑓))))
2182172rexbidv 3213 . . . . . . 7 (𝑐 = 𝑦 → (∃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑥 𝑐) = (𝑑 𝑓)) ↔ ∃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑑 𝑓))))
219208, 218cbvrex2vw 3228 . . . . . 6 (∃𝑎𝑃𝑐𝑃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑎 𝑐) = (𝑑 𝑓)) ↔ ∃𝑥𝑃𝑦𝑃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑑 𝑓)))
220198, 219sylib 217 . . . . 5 (((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ ∃𝑎𝑃𝑐𝑃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑎 𝑐) = (𝑑 𝑓))) → ∃𝑥𝑃𝑦𝑃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑥 𝑦) = (𝑑 𝑓)))
221197, 220r19.29vva 3207 . . . 4 (((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸))) ∧ ∃𝑎𝑃𝑐𝑃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑎 𝑐) = (𝑑 𝑓))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
222221anasss 467 . . 3 ((𝜑 ∧ (((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸)) ∧ ∃𝑎𝑃𝑐𝑃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑎 𝑐) = (𝑑 𝑓)))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
223116, 222sylan2b 594 . 2 ((𝜑 ∧ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸) ∧ ∃𝑎𝑃𝑐𝑃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑎 𝑐) = (𝑑 𝑓)))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
224115, 223impbida 799 1 (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸) ∧ ∃𝑎𝑃𝑐𝑃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑎 𝑐) = (𝑑 𝑓)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wrex 3073   class class class wbr 5105  cfv 6496  (class class class)co 7357  ⟨“cs3 14731  Basecbs 17083  distcds 17142  TarskiGcstrkg 27369  Itvcitv 27375  cgrGccgrg 27452  hlGchlg 27542  cgrAccgra 27749
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-map 8767  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  df-leg 27525  df-hlg 27543  df-cgra 27750
This theorem is referenced by: (None)
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