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Theorem erclwwlktr 26106
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 Alexander van der Vekens, 11-Jun-2018.)
Hypothesis
Ref Expression
erclwwlk.r = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑤 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}
Assertion
Ref Expression
erclwwlktr ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)
Distinct variable groups:   𝑛,𝐸,𝑢,𝑤   𝑛,𝑉,𝑢,𝑤   𝑥,𝑛,𝑢,𝑤,𝑦   𝑧,𝑛,𝑢,𝑤,𝑥
Allowed substitution hints:   (𝑥,𝑦,𝑧,𝑤,𝑢,𝑛)   𝐸(𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem erclwwlktr
Dummy variables 𝑚 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3172 . 2 𝑥 ∈ V
2 vex 3172 . 2 𝑦 ∈ V
3 vex 3172 . 2 𝑧 ∈ V
4 erclwwlk.r . . . . . 6 = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑤 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}
54erclwwlkeqlen 26103 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦 → (#‘𝑥) = (#‘𝑦)))
653adant3 1073 . . . 4 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑦 → (#‘𝑥) = (#‘𝑦)))
74erclwwlkeqlen 26103 . . . . . . 7 ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 𝑧 → (#‘𝑦) = (#‘𝑧)))
873adant1 1071 . . . . . 6 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 𝑧 → (#‘𝑦) = (#‘𝑧)))
94erclwwlkeq 26102 . . . . . . . 8 ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 𝑧 ↔ (𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))))
1093adant1 1071 . . . . . . 7 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 𝑧 ↔ (𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))))
114erclwwlkeq 26102 . . . . . . . . . 10 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦 ↔ (𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛))))
12113adant3 1073 . . . . . . . . 9 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑦 ↔ (𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛))))
13 simpr1 1059 . . . . . . . . . . . . . . 15 (((((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) ∧ (𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛))) → 𝑥 ∈ (𝑉 ClWWalks 𝐸))
14 simplr2 1096 . . . . . . . . . . . . . . 15 (((((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) ∧ (𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛))) → 𝑧 ∈ (𝑉 ClWWalks 𝐸))
15 oveq2 6532 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = 𝑚 → (𝑦 cyclShift 𝑛) = (𝑦 cyclShift 𝑚))
1615eqeq2d 2616 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 𝑚 → (𝑥 = (𝑦 cyclShift 𝑛) ↔ 𝑥 = (𝑦 cyclShift 𝑚)))
1716cbvrexv 3144 . . . . . . . . . . . . . . . . . . . . . . . 24 (∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑚))
18 oveq2 6532 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 = 𝑘 → (𝑧 cyclShift 𝑛) = (𝑧 cyclShift 𝑘))
1918eqeq2d 2616 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = 𝑘 → (𝑦 = (𝑧 cyclShift 𝑛) ↔ 𝑦 = (𝑧 cyclShift 𝑘)))
2019cbvrexv 3144 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛) ↔ ∃𝑘 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑘))
21 clwwlkprop 26061 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑧 ∈ (𝑉 ClWWalks 𝐸) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑧 ∈ Word 𝑉))
2221simp3d 1067 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑧 ∈ (𝑉 ClWWalks 𝐸) → 𝑧 ∈ Word 𝑉)
2322ad2antlr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸)) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → 𝑧 ∈ Word 𝑉)
24 simpr 475 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸)) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)))
2523, 24cshwcsh2id 13368 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸)) ∧ ((#‘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)))
2625expdcom 453 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → ((𝑘 ∈ (0...(#‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘)) → ((((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸)) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))
2726ancoms 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑥 = (𝑦 cyclShift 𝑚) ∧ 𝑚 ∈ (0...(#‘𝑦))) → ((𝑘 ∈ (0...(#‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘)) → ((((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸)) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))
2827expdcom 453 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑘 ∈ (0...(#‘𝑧)) → (𝑦 = (𝑧 cyclShift 𝑘) → ((𝑥 = (𝑦 cyclShift 𝑚) ∧ 𝑚 ∈ (0...(#‘𝑦))) → ((((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸)) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))))
2928com4t 90 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑥 = (𝑦 cyclShift 𝑚) ∧ 𝑚 ∈ (0...(#‘𝑦))) → ((((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸)) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → (𝑘 ∈ (0...(#‘𝑧)) → (𝑦 = (𝑧 cyclShift 𝑘) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))))
3029ex 448 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 = (𝑦 cyclShift 𝑚) → (𝑚 ∈ (0...(#‘𝑦)) → ((((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸)) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → (𝑘 ∈ (0...(#‘𝑧)) → (𝑦 = (𝑧 cyclShift 𝑘) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))))
3130com13 85 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸)) ∧ ((#‘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))))))
3231imp41 616 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸)) ∧ ((#‘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)))
3332rexlimdva 3009 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸)) ∧ ((#‘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)))
3433ex 448 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸)) ∧ ((#‘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))))
3534rexlimdva 3009 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸)) ∧ ((#‘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))))
3620, 35syl7bi 243 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸)) ∧ ((#‘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))))
3717, 36syl5bi 230 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸)) ∧ ((#‘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))))
3837exp31 627 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) → (𝑧 ∈ (𝑉 ClWWalks 𝐸) → (((#‘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))))))
3938com15 98 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛) → (𝑧 ∈ (𝑉 ClWWalks 𝐸) → (((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) → (∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → ((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))))
4039impcom 444 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛)) → (((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) → (∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → ((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))))
41403adant1 1071 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛)) → (((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) → (∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → ((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))))
4241impcom 444 . . . . . . . . . . . . . . . . . 18 ((((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) ∧ (𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))) → (∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → ((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))
4342com13 85 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) → (∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → ((((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) ∧ (𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))
44433impia 1252 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) → ((((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) ∧ (𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))
4544impcom 444 . . . . . . . . . . . . . . 15 (((((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) ∧ (𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛))) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))
4613, 14, 453jca 1234 . . . . . . . . . . . . . 14 (((((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) ∧ (𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛))) → (𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))
474erclwwlkeq 26102 . . . . . . . . . . . . . . 15 ((𝑥 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑧 ↔ (𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))
48473adant2 1072 . . . . . . . . . . . . . 14 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑧 ↔ (𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))
4946, 48syl5ibrcom 235 . . . . . . . . . . . . 13 (((((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) ∧ (𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛))) → ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → 𝑥 𝑧))
5049exp31 627 . . . . . . . . . . . 12 (((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) → ((𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛)) → ((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) → ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → 𝑥 𝑧))))
5150com24 92 . . . . . . . . . . 11 (((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) → ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) → ((𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛)) → 𝑥 𝑧))))
5251ex 448 . . . . . . . . . 10 ((#‘𝑦) = (#‘𝑧) → ((#‘𝑥) = (#‘𝑦) → ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) → ((𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛)) → 𝑥 𝑧)))))
5352com4t 90 . . . . . . . . 9 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) → ((#‘𝑦) = (#‘𝑧) → ((#‘𝑥) = (#‘𝑦) → ((𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛)) → 𝑥 𝑧)))))
5412, 53sylbid 228 . . . . . . . 8 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑦 → ((#‘𝑦) = (#‘𝑧) → ((#‘𝑥) = (#‘𝑦) → ((𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛)) → 𝑥 𝑧)))))
5554com25 96 . . . . . . 7 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛)) → ((#‘𝑦) = (#‘𝑧) → ((#‘𝑥) = (#‘𝑦) → (𝑥 𝑦𝑥 𝑧)))))
5610, 55sylbid 228 . . . . . 6 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 𝑧 → ((#‘𝑦) = (#‘𝑧) → ((#‘𝑥) = (#‘𝑦) → (𝑥 𝑦𝑥 𝑧)))))
578, 56mpdd 41 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 𝑧 → ((#‘𝑥) = (#‘𝑦) → (𝑥 𝑦𝑥 𝑧))))
5857com24 92 . . . 4 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑦 → ((#‘𝑥) = (#‘𝑦) → (𝑦 𝑧𝑥 𝑧))))
596, 58mpdd 41 . . 3 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑦 → (𝑦 𝑧𝑥 𝑧)))
6059impd 445 . 2 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))
611, 2, 3, 60mp3an 1415 1 ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wrex 2893  Vcvv 3169   class class class wbr 4574  {copab 4633  cfv 5787  (class class class)co 6524  0cc0 9789  ...cfz 12149  #chash 12931  Word cword 13089   cyclShift ccsh 13328   ClWWalks cclwwlk 26039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-cnex 9845  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-mulcom 9853  ax-addass 9854  ax-mulass 9855  ax-distr 9856  ax-i2m1 9857  ax-1ne0 9858  ax-1rid 9859  ax-rnegex 9860  ax-rrecex 9861  ax-cnre 9862  ax-pre-lttri 9863  ax-pre-lttrn 9864  ax-pre-ltadd 9865  ax-pre-mulgt0 9866  ax-pre-sup 9867
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rmo 2900  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-om 6932  df-1st 7033  df-2nd 7034  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-1o 7421  df-oadd 7425  df-er 7603  df-map 7720  df-pm 7721  df-en 7816  df-dom 7817  df-sdom 7818  df-fin 7819  df-sup 8205  df-inf 8206  df-card 8622  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-sub 10116  df-neg 10117  df-div 10531  df-nn 10865  df-2 10923  df-n0 11137  df-z 11208  df-uz 11517  df-rp 11662  df-fz 12150  df-fzo 12287  df-fl 12407  df-mod 12483  df-hash 12932  df-word 13097  df-concat 13099  df-substr 13101  df-csh 13329  df-clwwlk 26042
This theorem is referenced by:  erclwwlk  26107
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