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Theorem erclwwlkntr 26355
 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, 14-Jun-2018.)
Hypotheses
Ref Expression
erclwwlkn.w 𝑊 = ((𝑉 ClWWalksN 𝐸)‘𝑁)
erclwwlkn.r = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}
Assertion
Ref Expression
erclwwlkntr ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)
Distinct variable groups:   𝑡,𝐸,𝑢   𝑡,𝑁,𝑢   𝑛,𝑉,𝑡,𝑢   𝑡,𝑊,𝑢   𝑥,𝑛,𝑡,𝑢   𝑛,𝑁   𝑦,𝑛,𝑡,𝑢,𝑥   𝑛,𝑊   𝑧,𝑛,𝑡,𝑢,𝑦,𝑥
Allowed substitution hints:   (𝑥,𝑦,𝑧,𝑢,𝑡,𝑛)   𝐸(𝑥,𝑦,𝑧,𝑛)   𝑁(𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)   𝑊(𝑥,𝑦,𝑧)

Proof of Theorem erclwwlkntr
Dummy variables 𝑚 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3176 . 2 𝑥 ∈ V
2 vex 3176 . 2 𝑦 ∈ V
3 vex 3176 . 2 𝑧 ∈ V
4 erclwwlkn.w . . . . . 6 𝑊 = ((𝑉 ClWWalksN 𝐸)‘𝑁)
5 erclwwlkn.r . . . . . 6 = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}
64, 5erclwwlkneqlen 26352 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦 → (#‘𝑥) = (#‘𝑦)))
763adant3 1074 . . . 4 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑦 → (#‘𝑥) = (#‘𝑦)))
84, 5erclwwlkneqlen 26352 . . . . . . 7 ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 𝑧 → (#‘𝑦) = (#‘𝑧)))
983adant1 1072 . . . . . 6 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 𝑧 → (#‘𝑦) = (#‘𝑧)))
104, 5erclwwlkneq 26351 . . . . . . . 8 ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 𝑧 ↔ (𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛))))
11103adant1 1072 . . . . . . 7 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 𝑧 ↔ (𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛))))
124, 5erclwwlkneq 26351 . . . . . . . . . 10 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦 ↔ (𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛))))
13123adant3 1074 . . . . . . . . 9 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑦 ↔ (𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛))))
14 simpr1 1060 . . . . . . . . . . . . . . 15 (((((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) ∧ (𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛))) → 𝑥𝑊)
15 simplr2 1097 . . . . . . . . . . . . . . 15 (((((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) ∧ (𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛))) → 𝑧𝑊)
16 oveq2 6557 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = 𝑚 → (𝑦 cyclShift 𝑛) = (𝑦 cyclShift 𝑚))
1716eqeq2d 2620 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 𝑚 → (𝑥 = (𝑦 cyclShift 𝑛) ↔ 𝑥 = (𝑦 cyclShift 𝑚)))
1817cbvrexv 3148 . . . . . . . . . . . . . . . . . . . . . . . 24 (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑚))
19 oveq2 6557 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 = 𝑘 → (𝑧 cyclShift 𝑛) = (𝑧 cyclShift 𝑘))
2019eqeq2d 2620 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = 𝑘 → (𝑦 = (𝑧 cyclShift 𝑛) ↔ 𝑦 = (𝑧 cyclShift 𝑘)))
2120cbvrexv 3148 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛) ↔ ∃𝑘 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑘))
22 clwwlknprop 26300 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑧 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑧 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑧) = 𝑁)))
23 eqcom 2617 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((#‘𝑧) = 𝑁𝑁 = (#‘𝑧))
2423biimpi 205 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((#‘𝑧) = 𝑁𝑁 = (#‘𝑧))
2524adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝑁 ∈ ℕ0 ∧ (#‘𝑧) = 𝑁) → 𝑁 = (#‘𝑧))
26253ad2ant3 1077 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑧 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑧) = 𝑁)) → 𝑁 = (#‘𝑧))
2722, 26syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑧 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → 𝑁 = (#‘𝑧))
2827, 4eleq2s 2706 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑧𝑊𝑁 = (#‘𝑧))
2928adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) → 𝑁 = (#‘𝑧))
3029adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → 𝑁 = (#‘𝑧))
3122simp2d 1067 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (𝑧 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → 𝑧 ∈ Word 𝑉)
3231, 4eleq2s 2706 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑧𝑊𝑧 ∈ Word 𝑉)
3332adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) → 𝑧 ∈ Word 𝑉)
3433adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → 𝑧 ∈ Word 𝑉)
3534adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑁 = (#‘𝑧) ∧ (((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)))) → 𝑧 ∈ Word 𝑉)
36 simprr 792 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑁 = (#‘𝑧) ∧ (((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)))) → ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)))
3735, 36cshwcsh2id 13425 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((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)))
38 oveq2 6557 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑁 = (#‘𝑧) → (0...𝑁) = (0...(#‘𝑧)))
39 oveq2 6557 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ((#‘𝑧) = (#‘𝑦) → (0...(#‘𝑧)) = (0...(#‘𝑦)))
4039eqcoms 2618 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((#‘𝑦) = (#‘𝑧) → (0...(#‘𝑧)) = (0...(#‘𝑦)))
4140adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) → (0...(#‘𝑧)) = (0...(#‘𝑦)))
4241adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → (0...(#‘𝑧)) = (0...(#‘𝑦)))
4338, 42sylan9eq 2664 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝑁 = (#‘𝑧) ∧ (((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)))) → (0...𝑁) = (0...(#‘𝑦)))
4443eleq2d 2673 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑁 = (#‘𝑧) ∧ (((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)))) → (𝑚 ∈ (0...𝑁) ↔ 𝑚 ∈ (0...(#‘𝑦))))
4544anbi1d 737 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑁 = (#‘𝑧) ∧ (((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)))) → ((𝑚 ∈ (0...𝑁) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ↔ (𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))))
4638eleq2d 2673 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑁 = (#‘𝑧) → (𝑘 ∈ (0...𝑁) ↔ 𝑘 ∈ (0...(#‘𝑧))))
4746anbi1d 737 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑁 = (#‘𝑧) → ((𝑘 ∈ (0...𝑁) ∧ 𝑦 = (𝑧 cyclShift 𝑘)) ↔ (𝑘 ∈ (0...(#‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘))))
4847adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑁 = (#‘𝑧) ∧ (((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)))) → ((𝑘 ∈ (0...𝑁) ∧ 𝑦 = (𝑧 cyclShift 𝑘)) ↔ (𝑘 ∈ (0...(#‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘))))
4945, 48anbi12d 743 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((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)))))
5038rexeqdv 3122 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑁 = (#‘𝑧) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))
5150adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑁 = (#‘𝑧) ∧ (((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)))) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))
5237, 49, 513imtr4d 282 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((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)))
5330, 52mpancom 700 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((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)))
5453expdcom 454 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑚 ∈ (0...𝑁) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → ((𝑘 ∈ (0...𝑁) ∧ 𝑦 = (𝑧 cyclShift 𝑘)) → ((((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))
5554ancoms 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑥 = (𝑦 cyclShift 𝑚) ∧ 𝑚 ∈ (0...𝑁)) → ((𝑘 ∈ (0...𝑁) ∧ 𝑦 = (𝑧 cyclShift 𝑘)) → ((((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))
5655expdcom 454 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑘 ∈ (0...𝑁) → (𝑦 = (𝑧 cyclShift 𝑘) → ((𝑥 = (𝑦 cyclShift 𝑚) ∧ 𝑚 ∈ (0...𝑁)) → ((((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛)))))
5756com4t 91 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑥 = (𝑦 cyclShift 𝑚) ∧ 𝑚 ∈ (0...𝑁)) → ((((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → (𝑘 ∈ (0...𝑁) → (𝑦 = (𝑧 cyclShift 𝑘) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛)))))
5857ex 449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 = (𝑦 cyclShift 𝑚) → (𝑚 ∈ (0...𝑁) → ((((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → (𝑘 ∈ (0...𝑁) → (𝑦 = (𝑧 cyclShift 𝑘) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))))
5958com13 86 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))))
6059imp41 617 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
6160rexlimdva 3013 . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
6261ex 449 . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
6362rexlimdva 3013 . . . . . . . . . . . . . . . . . . . . . . . . 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))))
6421, 63syl7bi 244 . . . . . . . . . . . . . . . . . . . . . . . 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))))
6518, 64syl5bi 231 . . . . . . . . . . . . . . . . . . . . . . 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))))
6665exp31 628 . . . . . . . . . . . . . . . . . . . . . 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))))))
6766com15 99 . . . . . . . . . . . . . . . . . . . . 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))))))
6867impcom 445 . . . . . . . . . . . . . . . . . . . 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)))))
69683adant1 1072 . . . . . . . . . . . . . . . . . . 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)))))
7069impcom 445 . . . . . . . . . . . . . . . . . 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))))
7170com13 86 . . . . . . . . . . . . . . . . 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))))
72713impia 1253 . . . . . . . . . . . . . . . 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)))
7372impcom 445 . . . . . . . . . . . . . . 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))
7414, 15, 733jca 1235 . . . . . . . . . . . . . 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)))
754, 5erclwwlkneq 26351 . . . . . . . . . . . . . . 15 ((𝑥 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑧 ↔ (𝑥𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))
76753adant2 1073 . . . . . . . . . . . . . 14 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑧 ↔ (𝑥𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))
7774, 76syl5ibrcom 236 . . . . . . . . . . . . 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))
7877exp31 628 . . . . . . . . . . . 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))))
7978com24 93 . . . . . . . . . . 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))))
8079ex 449 . . . . . . . . . 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)))))
8180com4t 91 . . . . . . . . 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)))))
8213, 81sylbid 229 . . . . . . . 8 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑦 → ((#‘𝑦) = (#‘𝑧) → ((#‘𝑥) = (#‘𝑦) → ((𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛)) → 𝑥 𝑧)))))
8382com25 97 . . . . . . 7 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛)) → ((#‘𝑦) = (#‘𝑧) → ((#‘𝑥) = (#‘𝑦) → (𝑥 𝑦𝑥 𝑧)))))
8411, 83sylbid 229 . . . . . 6 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 𝑧 → ((#‘𝑦) = (#‘𝑧) → ((#‘𝑥) = (#‘𝑦) → (𝑥 𝑦𝑥 𝑧)))))
859, 84mpdd 42 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 𝑧 → ((#‘𝑥) = (#‘𝑦) → (𝑥 𝑦𝑥 𝑧))))
8685com24 93 . . . 4 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑦 → ((#‘𝑥) = (#‘𝑦) → (𝑦 𝑧𝑥 𝑧))))
877, 86mpdd 42 . . 3 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑦 → (𝑦 𝑧𝑥 𝑧)))
8887impd 446 . 2 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))
891, 2, 3, 88mp3an 1416 1 ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173   class class class wbr 4583  {copab 4642  ‘cfv 5804  (class class class)co 6549  0cc0 9815  ℕ0cn0 11169  ...cfz 12197  #chash 12979  Word cword 13146   cyclShift ccsh 13385   ClWWalksN cclwwlkn 26277 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-hash 12980  df-word 13154  df-concat 13156  df-substr 13158  df-csh 13386  df-clwwlk 26279  df-clwwlkn 26280 This theorem is referenced by:  erclwwlkn  26356
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