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Theorem dfcgra2 28515
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 28493 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 2731 . . . . 5 (hlG‘𝐺) = (hlG‘𝐺)
4 dfcgra2.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
54adantr 480 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐺 ∈ TarskiG)
6 dfcgra2.a . . . . . 6 (𝜑𝐴𝑃)
76adantr 480 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐴𝑃)
8 dfcgra2.b . . . . . 6 (𝜑𝐵𝑃)
98adantr 480 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐵𝑃)
10 dfcgra2.c . . . . . 6 (𝜑𝐶𝑃)
1110adantr 480 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐶𝑃)
12 dfcgra2.d . . . . . 6 (𝜑𝐷𝑃)
1312adantr 480 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐷𝑃)
14 dfcgra2.e . . . . . 6 (𝜑𝐸𝑃)
1514adantr 480 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐸𝑃)
16 dfcgra2.f . . . . . 6 (𝜑𝐹𝑃)
1716adantr 480 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐹𝑃)
18 simpr 484 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
191, 2, 3, 5, 7, 9, 11, 13, 15, 17, 18cgrane1 28497 . . . 4 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐴𝐵)
201, 2, 3, 5, 7, 9, 11, 13, 15, 17, 18cgrane2 28498 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐵𝐶)
2120necomd 2995 . . . 4 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐶𝐵)
2219, 21jca 511 . . 3 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → (𝐴𝐵𝐶𝐵))
231, 2, 3, 5, 7, 9, 11, 13, 15, 17, 18cgrane3 28499 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐸𝐷)
2423necomd 2995 . . . 4 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐷𝐸)
251, 2, 3, 5, 7, 9, 11, 13, 15, 17, 18cgrane4 28500 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐸𝐹)
2625necomd 2995 . . . 4 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐹𝐸)
2724, 26jca 511 . . 3 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) → (𝐷𝐸𝐹𝐸))
28 simprl 768 . . . . . . . . 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 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))))) → ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))
304ad6antr 733 . . . . . . . . . 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 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))))) → 𝑎𝑃)
328ad6antr 733 . . . . . . . . . 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 781 . . . . . . . . . 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 773 . . . . . . . . . 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 733 . . . . . . . . . 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 766 . . . . . . . . . 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 733 . . . . . . . . . . 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 733 . . . . . . . . . . . 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 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))))) → 𝐶𝑃)
406ad6antr 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))))) → 𝐴𝑃)
41 simp-6r 785 . . . . . . . . . . . . . . . 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 28507 . . . . . . . . . . . . . . 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 765 . . . . . . . . . . . . . . . . 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 731 . . . . . . . . . . . . . . . . . 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 28175 . . . . . . . . . . . . . . . . 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 28297 . . . . . . . . . . . . . . . 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 28289 . . . . . . . . . . . . . . 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 28501 . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . 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 731 . . . . . . . . . . . . . . . . 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 28175 . . . . . . . . . . . . . . . 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 28297 . . . . . . . . . . . . . . 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 28289 . . . . . . . . . . . . . 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 28502 . . . . . . . . . . . . 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 28507 . . . . . . . . . . . 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 765 . . . . . . . . . . . . . 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 731 . . . . . . . . . . . . . . 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 28175 . . . . . . . . . . . . . 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 28297 . . . . . . . . . . . . 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 28289 . . . . . . . . . . . 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 28501 . . . . . . . . . . 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 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))))) → 𝐹 ∈ (𝐸𝐼𝑓))
6426ad5antr 731 . . . . . . . . . . . . . 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 28175 . . . . . . . . . . . . 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 28297 . . . . . . . . . . . 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 28289 . . . . . . . . . . 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 28502 . . . . . . . . . 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 2995 . . . . . . . . . . 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 28294 . . . . . . . . . 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 2995 . . . . . . . . . . 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 28294 . . . . . . . . . 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 28173 . . . . . . . . . . . 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 767 . . . . . . . . . . . . 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 28164 . . . . . . . . . . . 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 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 𝑑) = (𝐵 𝐴))
7776eqcomd 2737 . . . . . . . . . . . . 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 28164 . . . . . . . . . . . 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 28170 . . . . . . . . . . 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 28164 . . . . . . . . . 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 28173 . . . . . . . . . . . 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 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))))) → (𝐶 𝑐) = (𝐸 𝐹))
831, 44, 2, 30, 39, 33, 35, 37, 82tgcgrcoml 28164 . . . . . . . . . . . 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 771 . . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . 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 28164 . . . . . . . . . . . 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 28170 . . . . . . . . . . 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 28164 . . . . . . . . . 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 28503 . . . . . . . . 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 1127 . . . . . . . 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 412 . . . . . . 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 3167 . . . . . 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 3167 . . . . 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 406 . . . 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 28149 . . . . . . . 8 (𝜑 → ∃𝑎𝑃 (𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)))
961, 44, 2, 4, 8, 10, 14, 16axtgsegcon 28149 . . . . . . . 8 (𝜑 → ∃𝑐𝑃 (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹)))
97 reeanv 3225 . . . . . . . 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 582 . . . . . . 7 (𝜑 → ∃𝑎𝑃𝑐𝑃 ((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))))
991, 44, 2, 4, 14, 12, 8, 6axtgsegcon 28149 . . . . . . . 8 (𝜑 → ∃𝑑𝑃 (𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)))
1001, 44, 2, 4, 14, 16, 8, 10axtgsegcon 28149 . . . . . . . 8 (𝜑 → ∃𝑓𝑃 (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶)))
101 reeanv 3225 . . . . . . . 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 582 . . . . . . 7 (𝜑 → ∃𝑑𝑃𝑓𝑃 ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))))
10398, 102jca 511 . . . . . 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 3223 . . . . . . . . 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 460 . . . . . . . . . 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 460 . . . . . . . . 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 301 . . . . . . . 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 3223 . . . . . . 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 276 . . . . . 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 480 . . . 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 3211 . . 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 1127 . 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 1088 . . 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 733 . . . . . . . . 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 733 . . . . . . . . 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 733 . . . . . . . . 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 733 . . . . . . . . 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 733 . . . . . . . . 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 733 . . . . . . . . 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 733 . . . . . . . . 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 781 . . . . . . . . . 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 783 . . . . . . . . . . 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 773 . . . . . . . . . . . . 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 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))) → 𝑡𝑃)
128 eqid 2731 . . . . . . . . . . . . . 14 (cgrG‘𝐺) = (cgrG‘𝐺)
129 simpr1 1193 . . . . . . . . . . . . . . . . 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 765 . . . . . . . . . . . . . . . 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 1194 . . . . . . . . . . . . . . . . . 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 765 . . . . . . . . . . . . . . . . 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 28173 . . . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . . . 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 28163 . . . . . . . . . . . . . . . 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 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))) → (𝐴 𝑥) = (𝐸 𝐷))
1381, 44, 2, 117, 121, 125, 119, 118, 137tgcgrcomr 28163 . . . . . . . . . . . . . . . 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 28170 . . . . . . . . . . . . . . 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 28164 . . . . . . . . . . . . . 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 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))) → 𝐶 ∈ (𝐵𝐼𝑦))
1421, 44, 2, 117, 122, 123, 124, 141tgbtwncom 28173 . . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . 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 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 𝑦) = (𝐸 𝐹))
1451, 44, 2, 117, 123, 124, 119, 120, 144tgcgrcoml 28164 . . . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . . . 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 28164 . . . . . . . . . . . . . . . 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 28170 . . . . . . . . . . . . . . 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 28164 . . . . . . . . . . . . . 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 1195 . . . . . . . . . . . . . . 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 28165 . . . . . . . . . . . . . 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 28201 . . . . . . . . . . . . 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 785 . . . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . 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 28175 . . . . . . . . . . . . . . 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 28297 . . . . . . . . . . . . . 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 28289 . . . . . . . . . . . . 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 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))) → 𝐹𝐸)
1601, 44, 2, 117, 119, 120, 127, 143, 159tgbtwnne 28175 . . . . . . . . . . . . . . 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 28297 . . . . . . . . . . . . . 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 28289 . . . . . . . . . . . . 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 28496 . . . . . . . . . . . 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 28507 . . . . . . . . . . 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 765 . . . . . . . . . . . . 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 28175 . . . . . . . . . . . 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 28297 . . . . . . . . . . 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 28501 . . . . . . . . . 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 767 . . . . . . . . . . . 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 28175 . . . . . . . . . . 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 28297 . . . . . . . . . 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 28502 . . . . . . . . 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 28507 . . . . . . . 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 747 . . . . . . 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 484 . . . . . . . 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 7420 . . . . . . . . . . . . 13 (𝑑 = 𝑧 → (𝐸𝐼𝑑) = (𝐸𝐼𝑧))
177176eleq2d 2818 . . . . . . . . . . . 12 (𝑑 = 𝑧 → (𝐷 ∈ (𝐸𝐼𝑑) ↔ 𝐷 ∈ (𝐸𝐼𝑧)))
178 oveq2 7420 . . . . . . . . . . . . 13 (𝑑 = 𝑧 → (𝐷 𝑑) = (𝐷 𝑧))
179178eqeq1d 2733 . . . . . . . . . . . 12 (𝑑 = 𝑧 → ((𝐷 𝑑) = (𝐵 𝐴) ↔ (𝐷 𝑧) = (𝐵 𝐴)))
180177, 179anbi12d 630 . . . . . . . . . . 11 (𝑑 = 𝑧 → ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ↔ (𝐷 ∈ (𝐸𝐼𝑧) ∧ (𝐷 𝑧) = (𝐵 𝐴))))
181180anbi1d 629 . . . . . . . . . 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 7419 . . . . . . . . . . 11 (𝑑 = 𝑧 → (𝑑 𝑓) = (𝑧 𝑓))
183182eqeq2d 2742 . . . . . . . . . 10 (𝑑 = 𝑧 → ((𝑥 𝑦) = (𝑑 𝑓) ↔ (𝑥 𝑦) = (𝑧 𝑓)))
184181, 1833anbi23d 1438 . . . . . . . . 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 7420 . . . . . . . . . . . . 13 (𝑓 = 𝑡 → (𝐸𝐼𝑓) = (𝐸𝐼𝑡))
186185eleq2d 2818 . . . . . . . . . . . 12 (𝑓 = 𝑡 → (𝐹 ∈ (𝐸𝐼𝑓) ↔ 𝐹 ∈ (𝐸𝐼𝑡)))
187 oveq2 7420 . . . . . . . . . . . . 13 (𝑓 = 𝑡 → (𝐹 𝑓) = (𝐹 𝑡))
188187eqeq1d 2733 . . . . . . . . . . . 12 (𝑓 = 𝑡 → ((𝐹 𝑓) = (𝐵 𝐶) ↔ (𝐹 𝑡) = (𝐵 𝐶)))
189186, 188anbi12d 630 . . . . . . . . . . 11 (𝑓 = 𝑡 → ((𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶)) ↔ (𝐹 ∈ (𝐸𝐼𝑡) ∧ (𝐹 𝑡) = (𝐵 𝐶))))
190189anbi2d 628 . . . . . . . . . 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 7420 . . . . . . . . . . 11 (𝑓 = 𝑡 → (𝑧 𝑓) = (𝑧 𝑡))
192191eqeq2d 2742 . . . . . . . . . 10 (𝑓 = 𝑡 → ((𝑥 𝑦) = (𝑧 𝑓) ↔ (𝑥 𝑦) = (𝑧 𝑡)))
193190, 1923anbi23d 1438 . . . . . . . . 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 3238 . . . . . . . 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 3212 . . . . . 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 747 . . . . 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 484 . . . . . 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 7420 . . . . . . . . . . . 12 (𝑎 = 𝑥 → (𝐵𝐼𝑎) = (𝐵𝐼𝑥))
200199eleq2d 2818 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝐴 ∈ (𝐵𝐼𝑎) ↔ 𝐴 ∈ (𝐵𝐼𝑥)))
201 oveq2 7420 . . . . . . . . . . . 12 (𝑎 = 𝑥 → (𝐴 𝑎) = (𝐴 𝑥))
202201eqeq1d 2733 . . . . . . . . . . 11 (𝑎 = 𝑥 → ((𝐴 𝑎) = (𝐸 𝐷) ↔ (𝐴 𝑥) = (𝐸 𝐷)))
203200, 202anbi12d 630 . . . . . . . . . 10 (𝑎 = 𝑥 → ((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ↔ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐸 𝐷))))
204203anbi1d 629 . . . . . . . . 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 7419 . . . . . . . . . 10 (𝑎 = 𝑥 → (𝑎 𝑐) = (𝑥 𝑐))
206205eqeq1d 2733 . . . . . . . . 9 (𝑎 = 𝑥 → ((𝑎 𝑐) = (𝑑 𝑓) ↔ (𝑥 𝑐) = (𝑑 𝑓)))
207204, 2063anbi13d 1437 . . . . . . . 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 3218 . . . . . . 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 7420 . . . . . . . . . . . 12 (𝑐 = 𝑦 → (𝐵𝐼𝑐) = (𝐵𝐼𝑦))
210209eleq2d 2818 . . . . . . . . . . 11 (𝑐 = 𝑦 → (𝐶 ∈ (𝐵𝐼𝑐) ↔ 𝐶 ∈ (𝐵𝐼𝑦)))
211 oveq2 7420 . . . . . . . . . . . 12 (𝑐 = 𝑦 → (𝐶 𝑐) = (𝐶 𝑦))
212211eqeq1d 2733 . . . . . . . . . . 11 (𝑐 = 𝑦 → ((𝐶 𝑐) = (𝐸 𝐹) ↔ (𝐶 𝑦) = (𝐸 𝐹)))
213210, 212anbi12d 630 . . . . . . . . . 10 (𝑐 = 𝑦 → ((𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹)) ↔ (𝐶 ∈ (𝐵𝐼𝑦) ∧ (𝐶 𝑦) = (𝐸 𝐹))))
214213anbi2d 628 . . . . . . . . 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 7420 . . . . . . . . . 10 (𝑐 = 𝑦 → (𝑥 𝑐) = (𝑥 𝑦))
216215eqeq1d 2733 . . . . . . . . 9 (𝑐 = 𝑦 → ((𝑥 𝑐) = (𝑑 𝑓) ↔ (𝑥 𝑦) = (𝑑 𝑓)))
217214, 2163anbi13d 1437 . . . . . . . 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 3218 . . . . . . 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 3238 . . . . . 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 3212 . . . 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 466 . . 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 593 . 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 798 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 395  w3a 1086   = wceq 1540  wcel 2105  wne 2939  wrex 3069   class class class wbr 5148  cfv 6543  (class class class)co 7412  ⟨“cs3 14800  Basecbs 17151  distcds 17213  TarskiGcstrkg 28112  Itvcitv 28118  cgrGccgrg 28195  hlGchlg 28285  cgrAccgra 28492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-oadd 8476  df-er 8709  df-map 8828  df-pm 8829  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-dju 9902  df-card 9940  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-nn 12220  df-2 12282  df-3 12283  df-n0 12480  df-xnn0 12552  df-z 12566  df-uz 12830  df-fz 13492  df-fzo 13635  df-hash 14298  df-word 14472  df-concat 14528  df-s1 14553  df-s2 14806  df-s3 14807  df-trkgc 28133  df-trkgb 28134  df-trkgcb 28135  df-trkg 28138  df-cgrg 28196  df-leg 28268  df-hlg 28286  df-cgra 28493
This theorem is referenced by: (None)
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