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Theorem erclwwlkstr 26802
 Description: ∼ is a transitive relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 10-Apr-2018.) (Revised by AV, 30-Apr-2021.)
Hypothesis
Ref Expression
erclwwlks.r = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}
Assertion
Ref Expression
erclwwlkstr ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)
Distinct variable groups:   𝑛,𝐺,𝑢,𝑤   𝑥,𝑛,𝑢,𝑤,𝑦   𝑧,𝑛,𝑢,𝑤,𝑥
Allowed substitution hints:   (𝑥,𝑦,𝑧,𝑤,𝑢,𝑛)   𝐺(𝑥,𝑦,𝑧)

Proof of Theorem erclwwlkstr
Dummy variables 𝑚 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3189 . 2 𝑥 ∈ V
2 vex 3189 . 2 𝑦 ∈ V
3 vex 3189 . 2 𝑧 ∈ V
4 erclwwlks.r . . . . . 6 = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}
54erclwwlkseqlen 26799 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦 → (#‘𝑥) = (#‘𝑦)))
653adant3 1079 . . . 4 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑦 → (#‘𝑥) = (#‘𝑦)))
74erclwwlkseqlen 26799 . . . . . . 7 ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 𝑧 → (#‘𝑦) = (#‘𝑧)))
873adant1 1077 . . . . . 6 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 𝑧 → (#‘𝑦) = (#‘𝑧)))
94erclwwlkseq 26798 . . . . . . . 8 ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 𝑧 ↔ (𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))))
1093adant1 1077 . . . . . . 7 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 𝑧 ↔ (𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))))
114erclwwlkseq 26798 . . . . . . . . . 10 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦 ↔ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛))))
12113adant3 1079 . . . . . . . . 9 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑦 ↔ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛))))
13 simpr1 1065 . . . . . . . . . . . . . . 15 (((((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) ∧ (𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛))) → 𝑥 ∈ (ClWWalks‘𝐺))
14 simplr2 1102 . . . . . . . . . . . . . . 15 (((((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) ∧ (𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛))) → 𝑧 ∈ (ClWWalks‘𝐺))
15 oveq2 6612 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = 𝑚 → (𝑦 cyclShift 𝑛) = (𝑦 cyclShift 𝑚))
1615eqeq2d 2631 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 𝑚 → (𝑥 = (𝑦 cyclShift 𝑛) ↔ 𝑥 = (𝑦 cyclShift 𝑚)))
1716cbvrexv 3160 . . . . . . . . . . . . . . . . . . . . . . . 24 (∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑚))
18 oveq2 6612 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 = 𝑘 → (𝑧 cyclShift 𝑛) = (𝑧 cyclShift 𝑘))
1918eqeq2d 2631 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = 𝑘 → (𝑦 = (𝑧 cyclShift 𝑛) ↔ 𝑦 = (𝑧 cyclShift 𝑘)))
2019cbvrexv 3160 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛) ↔ ∃𝑘 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑘))
21 eqid 2621 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (Vtx‘𝐺) = (Vtx‘𝐺)
2221clwwlkbp 26750 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 ∈ (ClWWalks‘𝐺) → (𝐺 ∈ V ∧ 𝑧 ∈ Word (Vtx‘𝐺) ∧ 𝑧 ≠ ∅))
2322simp2d 1072 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑧 ∈ (ClWWalks‘𝐺) → 𝑧 ∈ Word (Vtx‘𝐺))
2423ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ 𝑧 ∈ (ClWWalks‘𝐺)) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → 𝑧 ∈ Word (Vtx‘𝐺))
25 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ 𝑧 ∈ (ClWWalks‘𝐺)) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)))
2624, 25cshwcsh2id 13511 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ 𝑧 ∈ (ClWWalks‘𝐺)) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ∧ (𝑘 ∈ (0...(#‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘))) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))
2726exp5l 645 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ 𝑧 ∈ (ClWWalks‘𝐺)) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → (𝑚 ∈ (0...(#‘𝑦)) → (𝑥 = (𝑦 cyclShift 𝑚) → (𝑘 ∈ (0...(#‘𝑧)) → (𝑦 = (𝑧 cyclShift 𝑘) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))))
2827imp41 618 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ 𝑧 ∈ (ClWWalks‘𝐺)) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) ∧ 𝑚 ∈ (0...(#‘𝑦))) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → (𝑦 = (𝑧 cyclShift 𝑘) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))
2928rexlimdva 3024 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ 𝑧 ∈ (ClWWalks‘𝐺)) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) ∧ 𝑚 ∈ (0...(#‘𝑦))) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → (∃𝑘 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑘) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))
3029ex 450 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ 𝑧 ∈ (ClWWalks‘𝐺)) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) ∧ 𝑚 ∈ (0...(#‘𝑦))) → (𝑥 = (𝑦 cyclShift 𝑚) → (∃𝑘 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑘) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))
3130rexlimdva 3024 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ 𝑧 ∈ (ClWWalks‘𝐺)) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → (∃𝑚 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑚) → (∃𝑘 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑘) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))
3220, 31syl7bi 245 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ 𝑧 ∈ (ClWWalks‘𝐺)) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → (∃𝑚 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑚) → (∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))
3317, 32syl5bi 232 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ 𝑧 ∈ (ClWWalks‘𝐺)) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → (∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → (∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))
3433exp31 629 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) → (𝑧 ∈ (ClWWalks‘𝐺) → (((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) → (∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → (∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))))
3534com15 101 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛) → (𝑧 ∈ (ClWWalks‘𝐺) → (((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) → (∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))))
3635impcom 446 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛)) → (((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) → (∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))))
37363adant1 1077 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛)) → (((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) → (∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))))
3837impcom 446 . . . . . . . . . . . . . . . . . 18 ((((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) ∧ (𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))) → (∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))
3938com13 88 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) → (∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → ((((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) ∧ (𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))
40393impia 1258 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) → ((((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) ∧ (𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))
4140impcom 446 . . . . . . . . . . . . . . 15 (((((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) ∧ (𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛))) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))
4213, 14, 413jca 1240 . . . . . . . . . . . . . 14 (((((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) ∧ (𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛))) → (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))
434erclwwlkseq 26798 . . . . . . . . . . . . . . 15 ((𝑥 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑧 ↔ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))
44433adant2 1078 . . . . . . . . . . . . . 14 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑧 ↔ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))
4542, 44syl5ibrcom 237 . . . . . . . . . . . . 13 (((((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) ∧ (𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛))) → ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → 𝑥 𝑧))
4645exp31 629 . . . . . . . . . . . 12 (((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) → ((𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛)) → ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) → ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → 𝑥 𝑧))))
4746com24 95 . . . . . . . . . . 11 (((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) → ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) → ((𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛)) → 𝑥 𝑧))))
4847ex 450 . . . . . . . . . 10 ((#‘𝑦) = (#‘𝑧) → ((#‘𝑥) = (#‘𝑦) → ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) → ((𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛)) → 𝑥 𝑧)))))
4948com4t 93 . . . . . . . . 9 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) → ((#‘𝑦) = (#‘𝑧) → ((#‘𝑥) = (#‘𝑦) → ((𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛)) → 𝑥 𝑧)))))
5012, 49sylbid 230 . . . . . . . 8 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑦 → ((#‘𝑦) = (#‘𝑧) → ((#‘𝑥) = (#‘𝑦) → ((𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛)) → 𝑥 𝑧)))))
5150com25 99 . . . . . . 7 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛)) → ((#‘𝑦) = (#‘𝑧) → ((#‘𝑥) = (#‘𝑦) → (𝑥 𝑦𝑥 𝑧)))))
5210, 51sylbid 230 . . . . . 6 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 𝑧 → ((#‘𝑦) = (#‘𝑧) → ((#‘𝑥) = (#‘𝑦) → (𝑥 𝑦𝑥 𝑧)))))
538, 52mpdd 43 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 𝑧 → ((#‘𝑥) = (#‘𝑦) → (𝑥 𝑦𝑥 𝑧))))
5453com24 95 . . . 4 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑦 → ((#‘𝑥) = (#‘𝑦) → (𝑦 𝑧𝑥 𝑧))))
556, 54mpdd 43 . . 3 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑦 → (𝑦 𝑧𝑥 𝑧)))
5655impd 447 . 2 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))
571, 2, 3, 56mp3an 1421 1 ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  ∃wrex 2908  Vcvv 3186  ∅c0 3891   class class class wbr 4613  {copab 4672  ‘cfv 5847  (class class class)co 6604  0cc0 9880  ...cfz 12268  #chash 13057  Word cword 13230   cyclShift ccsh 13471  Vtxcvtx 25774  ClWWalkscclwwlks 26742 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-sup 8292  df-inf 8293  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-n0 11237  df-z 11322  df-uz 11632  df-rp 11777  df-fz 12269  df-fzo 12407  df-fl 12533  df-mod 12609  df-hash 13058  df-word 13238  df-concat 13240  df-substr 13242  df-csh 13472  df-clwwlks 26744 This theorem is referenced by:  erclwwlks  26803
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