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Theorem erclwwlksntr 26831
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.)
Hypotheses
Ref Expression
erclwwlksn.w 𝑊 = (𝑁 ClWWalksN 𝐺)
erclwwlksn.r = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}
Assertion
Ref Expression
erclwwlksntr ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)
Distinct variable groups:   𝑡,𝑊,𝑢   𝑛,𝑁,𝑢,𝑡,𝑥   𝑦,𝑛,𝑡,𝑢,𝑥   𝑛,𝑊   𝑧,𝑛,𝑡,𝑢,𝑦,𝑥
Allowed substitution hints:   (𝑥,𝑦,𝑧,𝑢,𝑡,𝑛)   𝐺(𝑥,𝑦,𝑧,𝑢,𝑡,𝑛)   𝑁(𝑦,𝑧)   𝑊(𝑥,𝑦,𝑧)

Proof of Theorem erclwwlksntr
Dummy variables 𝑚 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3192 . 2 𝑥 ∈ V
2 vex 3192 . 2 𝑦 ∈ V
3 vex 3192 . 2 𝑧 ∈ V
4 erclwwlksn.w . . . . . 6 𝑊 = (𝑁 ClWWalksN 𝐺)
5 erclwwlksn.r . . . . . 6 = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}
64, 5erclwwlksneqlen 26828 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦 → (#‘𝑥) = (#‘𝑦)))
763adant3 1079 . . . 4 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑦 → (#‘𝑥) = (#‘𝑦)))
84, 5erclwwlksneqlen 26828 . . . . . . 7 ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 𝑧 → (#‘𝑦) = (#‘𝑧)))
983adant1 1077 . . . . . 6 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 𝑧 → (#‘𝑦) = (#‘𝑧)))
104, 5erclwwlksneq 26827 . . . . . . . 8 ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 𝑧 ↔ (𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛))))
11103adant1 1077 . . . . . . 7 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 𝑧 ↔ (𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛))))
124, 5erclwwlksneq 26827 . . . . . . . . . 10 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦 ↔ (𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛))))
13123adant3 1079 . . . . . . . . 9 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑦 ↔ (𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛))))
14 simpr1 1065 . . . . . . . . . . . . . . 15 (((((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) ∧ (𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛))) → 𝑥𝑊)
15 simplr2 1102 . . . . . . . . . . . . . . 15 (((((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) ∧ (𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛))) → 𝑧𝑊)
16 oveq2 6618 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = 𝑚 → (𝑦 cyclShift 𝑛) = (𝑦 cyclShift 𝑚))
1716eqeq2d 2631 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 𝑚 → (𝑥 = (𝑦 cyclShift 𝑛) ↔ 𝑥 = (𝑦 cyclShift 𝑚)))
1817cbvrexv 3163 . . . . . . . . . . . . . . . . . . . . . . . 24 (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑚))
19 oveq2 6618 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 = 𝑘 → (𝑧 cyclShift 𝑛) = (𝑧 cyclShift 𝑘))
2019eqeq2d 2631 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = 𝑘 → (𝑦 = (𝑧 cyclShift 𝑛) ↔ 𝑦 = (𝑧 cyclShift 𝑘)))
2120cbvrexv 3163 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛) ↔ ∃𝑘 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑘))
22 eqid 2621 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (Vtx‘𝐺) = (Vtx‘𝐺)
2322clwwlknbp 26769 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑧 ∈ (𝑁 ClWWalksN 𝐺) → (𝑧 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑧) = 𝑁))
24 eqcom 2628 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((#‘𝑧) = 𝑁𝑁 = (#‘𝑧))
2524biimpi 206 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((#‘𝑧) = 𝑁𝑁 = (#‘𝑧))
2623, 25simpl2im 657 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 ∈ (𝑁 ClWWalksN 𝐺) → 𝑁 = (#‘𝑧))
2726, 4eleq2s 2716 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑧𝑊𝑁 = (#‘𝑧))
2827ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → 𝑁 = (#‘𝑧))
2923simpld 475 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑧 ∈ (𝑁 ClWWalksN 𝐺) → 𝑧 ∈ Word (Vtx‘𝐺))
3029, 4eleq2s 2716 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑧𝑊𝑧 ∈ Word (Vtx‘𝐺))
3130ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → 𝑧 ∈ Word (Vtx‘𝐺))
3231adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑁 = (#‘𝑧) ∧ (((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)))) → 𝑧 ∈ Word (Vtx‘𝐺))
33 simprr 795 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑁 = (#‘𝑧) ∧ (((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)))) → ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)))
3432, 33cshwcsh2id 13519 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑁 = (#‘𝑧) ∧ (((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘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)))
35 oveq2 6618 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑁 = (#‘𝑧) → (0...𝑁) = (0...(#‘𝑧)))
36 oveq2 6618 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((#‘𝑧) = (#‘𝑦) → (0...(#‘𝑧)) = (0...(#‘𝑦)))
3736eqcoms 2629 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((#‘𝑦) = (#‘𝑧) → (0...(#‘𝑧)) = (0...(#‘𝑦)))
3837adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) → (0...(#‘𝑧)) = (0...(#‘𝑦)))
3938adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → (0...(#‘𝑧)) = (0...(#‘𝑦)))
4035, 39sylan9eq 2675 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑁 = (#‘𝑧) ∧ (((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)))) → (0...𝑁) = (0...(#‘𝑦)))
4140eleq2d 2684 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑁 = (#‘𝑧) ∧ (((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)))) → (𝑚 ∈ (0...𝑁) ↔ 𝑚 ∈ (0...(#‘𝑦))))
4241anbi1d 740 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑁 = (#‘𝑧) ∧ (((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)))) → ((𝑚 ∈ (0...𝑁) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ↔ (𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))))
4335eleq2d 2684 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑁 = (#‘𝑧) → (𝑘 ∈ (0...𝑁) ↔ 𝑘 ∈ (0...(#‘𝑧))))
4443anbi1d 740 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑁 = (#‘𝑧) → ((𝑘 ∈ (0...𝑁) ∧ 𝑦 = (𝑧 cyclShift 𝑘)) ↔ (𝑘 ∈ (0...(#‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘))))
4544adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑁 = (#‘𝑧) ∧ (((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)))) → ((𝑘 ∈ (0...𝑁) ∧ 𝑦 = (𝑧 cyclShift 𝑘)) ↔ (𝑘 ∈ (0...(#‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘))))
4642, 45anbi12d 746 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑁 = (#‘𝑧) ∧ (((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)))) → (((𝑚 ∈ (0...𝑁) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑦 = (𝑧 cyclShift 𝑘))) ↔ ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ∧ (𝑘 ∈ (0...(#‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘)))))
4735rexeqdv 3137 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑁 = (#‘𝑧) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))
4847adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑁 = (#‘𝑧) ∧ (((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)))) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))
4934, 46, 483imtr4d 283 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑁 = (#‘𝑧) ∧ (((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)))) → (((𝑚 ∈ (0...𝑁) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑦 = (𝑧 cyclShift 𝑘))) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛)))
5028, 49mpancom 702 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → (((𝑚 ∈ (0...𝑁) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑦 = (𝑧 cyclShift 𝑘))) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛)))
5150exp5l 645 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → (𝑚 ∈ (0...𝑁) → (𝑥 = (𝑦 cyclShift 𝑚) → (𝑘 ∈ (0...𝑁) → (𝑦 = (𝑧 cyclShift 𝑘) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))))
5251imp41 618 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ∧ 𝑘 ∈ (0...𝑁)) → (𝑦 = (𝑧 cyclShift 𝑘) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛)))
5352rexlimdva 3025 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → (∃𝑘 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑘) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛)))
5453ex 450 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) ∧ 𝑚 ∈ (0...𝑁)) → (𝑥 = (𝑦 cyclShift 𝑚) → (∃𝑘 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑘) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))
5554rexlimdva 3025 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → (∃𝑚 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑚) → (∃𝑘 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑘) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))
5621, 55syl7bi 245 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → (∃𝑚 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑚) → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))
5718, 56syl5bi 232 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))
5857exp31 629 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥𝑊𝑦𝑊) → (𝑧𝑊 → (((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))))
5958com15 101 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛) → (𝑧𝑊 → (((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ((𝑥𝑊𝑦𝑊) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))))
6059impcom 446 . . . . . . . . . . . . . . . . . . . 20 ((𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛)) → (((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ((𝑥𝑊𝑦𝑊) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛)))))
61603adant1 1077 . . . . . . . . . . . . . . . . . . 19 ((𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛)) → (((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ((𝑥𝑊𝑦𝑊) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛)))))
6261impcom 446 . . . . . . . . . . . . . . . . . 18 ((((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) ∧ (𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛))) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ((𝑥𝑊𝑦𝑊) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))
6362com13 88 . . . . . . . . . . . . . . . . 17 ((𝑥𝑊𝑦𝑊) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ((((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) ∧ (𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛))) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))
64633impia 1258 . . . . . . . . . . . . . . . 16 ((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) → ((((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) ∧ (𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛))) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛)))
6564impcom 446 . . . . . . . . . . . . . . 15 (((((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) ∧ (𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛))) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))
6614, 15, 653jca 1240 . . . . . . . . . . . . . 14 (((((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) ∧ (𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛))) → (𝑥𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛)))
674, 5erclwwlksneq 26827 . . . . . . . . . . . . . . 15 ((𝑥 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑧 ↔ (𝑥𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))
68673adant2 1078 . . . . . . . . . . . . . 14 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑧 ↔ (𝑥𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))
6966, 68syl5ibrcom 237 . . . . . . . . . . . . 13 (((((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) ∧ (𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛))) → ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → 𝑥 𝑧))
7069exp31 629 . . . . . . . . . . . 12 (((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) → ((𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛)) → ((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) → ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → 𝑥 𝑧))))
7170com24 95 . . . . . . . . . . 11 (((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) → ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) → ((𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛)) → 𝑥 𝑧))))
7271ex 450 . . . . . . . . . 10 ((#‘𝑦) = (#‘𝑧) → ((#‘𝑥) = (#‘𝑦) → ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) → ((𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛)) → 𝑥 𝑧)))))
7372com4t 93 . . . . . . . . 9 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) → ((#‘𝑦) = (#‘𝑧) → ((#‘𝑥) = (#‘𝑦) → ((𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛)) → 𝑥 𝑧)))))
7413, 73sylbid 230 . . . . . . . 8 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑦 → ((#‘𝑦) = (#‘𝑧) → ((#‘𝑥) = (#‘𝑦) → ((𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛)) → 𝑥 𝑧)))))
7574com25 99 . . . . . . 7 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛)) → ((#‘𝑦) = (#‘𝑧) → ((#‘𝑥) = (#‘𝑦) → (𝑥 𝑦𝑥 𝑧)))))
7611, 75sylbid 230 . . . . . 6 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 𝑧 → ((#‘𝑦) = (#‘𝑧) → ((#‘𝑥) = (#‘𝑦) → (𝑥 𝑦𝑥 𝑧)))))
779, 76mpdd 43 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 𝑧 → ((#‘𝑥) = (#‘𝑦) → (𝑥 𝑦𝑥 𝑧))))
7877com24 95 . . . 4 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑦 → ((#‘𝑥) = (#‘𝑦) → (𝑦 𝑧𝑥 𝑧))))
797, 78mpdd 43 . . 3 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑦 → (𝑦 𝑧𝑥 𝑧)))
8079impd 447 . 2 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))
811, 2, 3, 80mp3an 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  wrex 2908  Vcvv 3189   class class class wbr 4618  {copab 4677  cfv 5852  (class class class)co 6610  0cc0 9888  ...cfz 12276  #chash 13065  Word cword 13238   cyclShift ccsh 13479  Vtxcvtx 25791   ClWWalksN cclwwlksn 26760
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 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965  ax-pre-sup 9966
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 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-pm 7812  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-sup 8300  df-inf 8301  df-card 8717  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-div 10637  df-nn 10973  df-2 11031  df-n0 11245  df-z 11330  df-uz 11640  df-rp 11785  df-fz 12277  df-fzo 12415  df-fl 12541  df-mod 12617  df-hash 13066  df-word 13246  df-concat 13248  df-substr 13250  df-csh 13480  df-clwwlks 26761  df-clwwlksn 26762
This theorem is referenced by:  erclwwlksn  26832
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