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Theorem dfcgra2 26622
 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 26600 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 2824 . . . . 5 (hlG‘𝐺) = (hlG‘𝐺)
4 dfcgra2.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
54adantr 484 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐺 ∈ TarskiG)
6 dfcgra2.a . . . . . 6 (𝜑𝐴𝑃)
76adantr 484 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐴𝑃)
8 dfcgra2.b . . . . . 6 (𝜑𝐵𝑃)
98adantr 484 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐵𝑃)
10 dfcgra2.c . . . . . 6 (𝜑𝐶𝑃)
1110adantr 484 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐶𝑃)
12 dfcgra2.d . . . . . 6 (𝜑𝐷𝑃)
1312adantr 484 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐷𝑃)
14 dfcgra2.e . . . . . 6 (𝜑𝐸𝑃)
1514adantr 484 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐸𝑃)
16 dfcgra2.f . . . . . 6 (𝜑𝐹𝑃)
1716adantr 484 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐹𝑃)
18 simpr 488 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
191, 2, 3, 5, 7, 9, 11, 13, 15, 17, 18cgrane1 26604 . . . 4 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐴𝐵)
201, 2, 3, 5, 7, 9, 11, 13, 15, 17, 18cgrane2 26605 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐵𝐶)
2120necomd 3069 . . . 4 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐶𝐵)
2219, 21jca 515 . . 3 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → (𝐴𝐵𝐶𝐵))
231, 2, 3, 5, 7, 9, 11, 13, 15, 17, 18cgrane3 26606 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐸𝐷)
2423necomd 3069 . . . 4 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐷𝐸)
251, 2, 3, 5, 7, 9, 11, 13, 15, 17, 18cgrane4 26607 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐸𝐹)
2625necomd 3069 . . . 4 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐹𝐸)
2724, 26jca 515 . . 3 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → (𝐷𝐸𝐹𝐸))
28 simprl 770 . . . . . . . . 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 772 . . . . . . . . 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 735 . . . . . . . . . 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 785 . . . . . . . . . 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 735 . . . . . . . . . 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 783 . . . . . . . . . 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 775 . . . . . . . . . 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 735 . . . . . . . . . 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 768 . . . . . . . . . 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 735 . . . . . . . . . . 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 735 . . . . . . . . . . . 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 735 . . . . . . . . . . . . . 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 735 . . . . . . . . . . . . . . 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 787 . . . . . . . . . . . . . . . 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 26614 . . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . . . 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 733 . . . . . . . . . . . . . . . . . 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 26282 . . . . . . . . . . . . . . . . 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 26404 . . . . . . . . . . . . . . . 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 26396 . . . . . . . . . . . . . . 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 26608 . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . . 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 733 . . . . . . . . . . . . . . . . 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 26282 . . . . . . . . . . . . . . . 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 26404 . . . . . . . . . . . . . . 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 26396 . . . . . . . . . . . . . 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 26609 . . . . . . . . . . . . 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 26614 . . . . . . . . . . . 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 767 . . . . . . . . . . . . . 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 733 . . . . . . . . . . . . . . 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 26282 . . . . . . . . . . . . . 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 26404 . . . . . . . . . . . . 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 26396 . . . . . . . . . . . 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 26608 . . . . . . . . . . 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 771 . . . . . . . . . . . . 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 733 . . . . . . . . . . . . . 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 26282 . . . . . . . . . . . . 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 26404 . . . . . . . . . . . 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 26396 . . . . . . . . . . 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 26609 . . . . . . . . . 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 3069 . . . . . . . . . . 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 26401 . . . . . . . . . 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 3069 . . . . . . . . . . 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 26401 . . . . . . . . . 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 26280 . . . . . . . . . . . 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 769 . . . . . . . . . . . . 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 26271 . . . . . . . . . . . 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 769 . . . . . . . . . . . . . 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 2830 . . . . . . . . . . . . 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 26271 . . . . . . . . . . . 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 26277 . . . . . . . . . . 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 26271 . . . . . . . . . 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 26280 . . . . . . . . . . . 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 773 . . . . . . . . . . . . 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 26271 . . . . . . . . . . . 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 773 . . . . . . . . . . . . . 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 2830 . . . . . . . . . . . . 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 26271 . . . . . . . . . . . 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 26277 . . . . . . . . . . 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 26271 . . . . . . . . . 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 26610 . . . . . . . . 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 1125 . . . . . . . 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 416 . . . . . . 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 3267 . . . . . 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 3267 . . . . 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 410 . . . 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 26256 . . . . . . . 8 (𝜑 → ∃𝑎𝑃 (𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)))
961, 44, 2, 4, 8, 10, 14, 16axtgsegcon 26256 . . . . . . . 8 (𝜑 → ∃𝑐𝑃 (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹)))
97 reeanv 3359 . . . . . . . 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 586 . . . . . . 7 (𝜑 → ∃𝑎𝑃𝑐𝑃 ((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))))
991, 44, 2, 4, 14, 12, 8, 6axtgsegcon 26256 . . . . . . . 8 (𝜑 → ∃𝑑𝑃 (𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)))
1001, 44, 2, 4, 14, 16, 8, 10axtgsegcon 26256 . . . . . . . 8 (𝜑 → ∃𝑓𝑃 (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶)))
101 reeanv 3359 . . . . . . . 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 586 . . . . . . 7 (𝜑 → ∃𝑑𝑃𝑓𝑃 ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))
10398, 102jca 515 . . . . . 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 3341 . . . . . . . . 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 464 . . . . . . . . . 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 3243 . . . . . . . . 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 464 . . . . . . . . 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 304 . . . . . . . 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 3243 . . . . . . 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 3341 . . . . . . 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 279 . . . . . 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 221 . . . . 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 484 . . . 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 3271 . . 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 1125 . 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 1086 . . 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 735 . . . . . . . . 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 735 . . . . . . . . 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 735 . . . . . . . . 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 735 . . . . . . . . 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 735 . . . . . . . . 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 735 . . . . . . . . 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 735 . . . . . . . . 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 783 . . . . . . . . . 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 785 . . . . . . . . . . 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 775 . . . . . . . . . . . . 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 768 . . . . . . . . . . . . 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 2824 . . . . . . . . . . . . . 14 (cgrG‘𝐺) = (cgrG‘𝐺)
129 simpr1 1191 . . . . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . . 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 1192 . . . . . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . . . 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 26280 . . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . . . 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 2830 . . . . . . . . . . . . . . . . 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 26270 . . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . . 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 26270 . . . . . . . . . . . . . . . 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 26277 . . . . . . . . . . . . . . 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 26271 . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . . . 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 26280 . . . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . . 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 773 . . . . . . . . . . . . . . . . 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 26271 . . . . . . . . . . . . . . . 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 773 . . . . . . . . . . . . . . . . . 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 2830 . . . . . . . . . . . . . . . . 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 26271 . . . . . . . . . . . . . . . 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 26277 . . . . . . . . . . . . . . 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 26271 . . . . . . . . . . . . . 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 1193 . . . . . . . . . . . . . . 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 26272 . . . . . . . . . . . . . 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 26308 . . . . . . . . . . . . 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 787 . . . . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . . 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 26282 . . . . . . . . . . . . . . 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 26404 . . . . . . . . . . . . . 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 26396 . . . . . . . . . . . . 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 773 . . . . . . . . . . . . . . . 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 26282 . . . . . . . . . . . . . . 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 26404 . . . . . . . . . . . . . 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 26396 . . . . . . . . . . . . 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 26603 . . . . . . . . . . . 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 26614 . . . . . . . . . . 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 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))) → 𝐴𝐵)
1661, 44, 2, 117, 122, 121, 125, 130, 165tgbtwnne 26282 . . . . . . . . . . . 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 26404 . . . . . . . . . . 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 26608 . . . . . . . . . 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 769 . . . . . . . . . . . 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 26282 . . . . . . . . . . 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 26404 . . . . . . . . . 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 26609 . . . . . . . . 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 26614 . . . . . . . 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 749 . . . . . . 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 488 . . . . . . . 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 7154 . . . . . . . . . . . . 13 (𝑑 = 𝑧 → (𝐸𝐼𝑑) = (𝐸𝐼𝑧))
177176eleq2d 2901 . . . . . . . . . . . 12 (𝑑 = 𝑧 → (𝐷 ∈ (𝐸𝐼𝑑) ↔ 𝐷 ∈ (𝐸𝐼𝑧)))
178 oveq2 7154 . . . . . . . . . . . . 13 (𝑑 = 𝑧 → (𝐷 𝑑) = (𝐷 𝑧))
179178eqeq1d 2826 . . . . . . . . . . . 12 (𝑑 = 𝑧 → ((𝐷 𝑑) = (𝐵 𝐴) ↔ (𝐷 𝑧) = (𝐵 𝐴)))
180177, 179anbi12d 633 . . . . . . . . . . 11 (𝑑 = 𝑧 → ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ↔ (𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴))))
181180anbi1d 632 . . . . . . . . . 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 7153 . . . . . . . . . . 11 (𝑑 = 𝑧 → (𝑑 𝑓) = (𝑧 𝑓))
183182eqeq2d 2835 . . . . . . . . . 10 (𝑑 = 𝑧 → ((𝑥 𝑦) = (𝑑 𝑓) ↔ (𝑥 𝑦) = (𝑧 𝑓)))
184181, 1833anbi23d 1436 . . . . . . . . 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 7154 . . . . . . . . . . . . 13 (𝑓 = 𝑡 → (𝐸𝐼𝑓) = (𝐸𝐼𝑡))
186185eleq2d 2901 . . . . . . . . . . . 12 (𝑓 = 𝑡 → (𝐹 ∈ (𝐸𝐼𝑓) ↔ 𝐹 ∈ (𝐸𝐼𝑡)))
187 oveq2 7154 . . . . . . . . . . . . 13 (𝑓 = 𝑡 → (𝐹 𝑓) = (𝐹 𝑡))
188187eqeq1d 2826 . . . . . . . . . . . 12 (𝑓 = 𝑡 → ((𝐹 𝑓) = (𝐵 𝐶) ↔ (𝐹 𝑡) = (𝐵 𝐶)))
189186, 188anbi12d 633 . . . . . . . . . . 11 (𝑓 = 𝑡 → ((𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶)) ↔ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))))
190189anbi2d 631 . . . . . . . . . 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 7154 . . . . . . . . . . 11 (𝑓 = 𝑡 → (𝑧 𝑓) = (𝑧 𝑡))
192191eqeq2d 2835 . . . . . . . . . 10 (𝑓 = 𝑡 → ((𝑥 𝑦) = (𝑧 𝑓) ↔ (𝑥 𝑦) = (𝑧 𝑡)))
193190, 1923anbi23d 1436 . . . . . . . . 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 3448 . . . . . . . 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 221 . . . . . . 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 3328 . . . . . 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 749 . . . . 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 488 . . . . . 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 7154 . . . . . . . . . . . 12 (𝑎 = 𝑥 → (𝐵𝐼𝑎) = (𝐵𝐼𝑥))
200199eleq2d 2901 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝐴 ∈ (𝐵𝐼𝑎) ↔ 𝐴 ∈ (𝐵𝐼𝑥)))
201 oveq2 7154 . . . . . . . . . . . 12 (𝑎 = 𝑥 → (𝐴 𝑎) = (𝐴 𝑥))
202201eqeq1d 2826 . . . . . . . . . . 11 (𝑎 = 𝑥 → ((𝐴 𝑎) = (𝐸 𝐷) ↔ (𝐴 𝑥) = (𝐸 𝐷)))
203200, 202anbi12d 633 . . . . . . . . . 10 (𝑎 = 𝑥 → ((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ↔ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷))))
204203anbi1d 632 . . . . . . . . 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 7153 . . . . . . . . . 10 (𝑎 = 𝑥 → (𝑎 𝑐) = (𝑥 𝑐))
206205eqeq1d 2826 . . . . . . . . 9 (𝑎 = 𝑥 → ((𝑎 𝑐) = (𝑑 𝑓) ↔ (𝑥 𝑐) = (𝑑 𝑓)))
207204, 2063anbi13d 1435 . . . . . . . 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 3293 . . . . . . 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 7154 . . . . . . . . . . . 12 (𝑐 = 𝑦 → (𝐵𝐼𝑐) = (𝐵𝐼𝑦))
210209eleq2d 2901 . . . . . . . . . . 11 (𝑐 = 𝑦 → (𝐶 ∈ (𝐵𝐼𝑐) ↔ 𝐶 ∈ (𝐵𝐼𝑦)))
211 oveq2 7154 . . . . . . . . . . . 12 (𝑐 = 𝑦 → (𝐶 𝑐) = (𝐶 𝑦))
212211eqeq1d 2826 . . . . . . . . . . 11 (𝑐 = 𝑦 → ((𝐶 𝑐) = (𝐸 𝐹) ↔ (𝐶 𝑦) = (𝐸 𝐹)))
213210, 212anbi12d 633 . . . . . . . . . 10 (𝑐 = 𝑦 → ((𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹)) ↔ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))))
214213anbi2d 631 . . . . . . . . 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 7154 . . . . . . . . . 10 (𝑐 = 𝑦 → (𝑥 𝑐) = (𝑥 𝑦))
216215eqeq1d 2826 . . . . . . . . 9 (𝑐 = 𝑦 → ((𝑥 𝑐) = (𝑑 𝑓) ↔ (𝑥 𝑦) = (𝑑 𝑓)))
217214, 2163anbi13d 1435 . . . . . . . 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 3293 . . . . . . 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 3448 . . . . . 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 221 . . . . 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 3328 . . . 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 470 . . 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 596 . 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 800 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115   ≠ wne 3014  ∃wrex 3134   class class class wbr 5053  ‘cfv 6344  (class class class)co 7146  ⟨“cs3 14202  Basecbs 16481  distcds 16572  TarskiGcstrkg 26222  Itvcitv 26228  cgrGccgrg 26302  hlGchlg 26392  cgrAccgra 26599 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4826  df-int 4864  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6136  df-ord 6182  df-on 6183  df-lim 6184  df-suc 6185  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-riota 7104  df-ov 7149  df-oprab 7150  df-mpo 7151  df-om 7572  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-er 8281  df-map 8400  df-pm 8401  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-dju 9323  df-card 9361  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11695  df-3 11696  df-n0 11893  df-xnn0 11963  df-z 11977  df-uz 12239  df-fz 12893  df-fzo 13036  df-hash 13694  df-word 13865  df-concat 13921  df-s1 13948  df-s2 14208  df-s3 14209  df-trkgc 26240  df-trkgb 26241  df-trkgcb 26242  df-trkg 26245  df-cgrg 26303  df-leg 26375  df-hlg 26393  df-cgra 26600 This theorem is referenced by: (None)
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