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Theorem clwwisshclwwslem 29267
Description: Lemma for clwwisshclwws 29268. (Contributed by AV, 24-Mar-2018.) (Revised by AV, 28-Apr-2021.)
Assertion
Ref Expression
clwwisshclwwslem ((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) β†’ ((βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ 𝐸 ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ 𝐸) β†’ βˆ€π‘— ∈ (0..^((β™―β€˜(π‘Š cyclShift 𝑁)) βˆ’ 1)){((π‘Š cyclShift 𝑁)β€˜π‘—), ((π‘Š cyclShift 𝑁)β€˜(𝑗 + 1))} ∈ 𝐸))
Distinct variable groups:   𝑖,𝐸,𝑗   𝑖,𝑁,𝑗   𝑖,𝑉,𝑗   𝑖,π‘Š,𝑗

Proof of Theorem clwwisshclwwslem
StepHypRef Expression
1 elfzoelz 13632 . . . . . . . . 9 (𝑁 ∈ (1..^(β™―β€˜π‘Š)) β†’ 𝑁 ∈ β„€)
2 cshwlen 14749 . . . . . . . . 9 ((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ β„€) β†’ (β™―β€˜(π‘Š cyclShift 𝑁)) = (β™―β€˜π‘Š))
31, 2sylan2 594 . . . . . . . 8 ((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) β†’ (β™―β€˜(π‘Š cyclShift 𝑁)) = (β™―β€˜π‘Š))
43oveq1d 7424 . . . . . . 7 ((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) β†’ ((β™―β€˜(π‘Š cyclShift 𝑁)) βˆ’ 1) = ((β™―β€˜π‘Š) βˆ’ 1))
54oveq2d 7425 . . . . . 6 ((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) β†’ (0..^((β™―β€˜(π‘Š cyclShift 𝑁)) βˆ’ 1)) = (0..^((β™―β€˜π‘Š) βˆ’ 1)))
65eleq2d 2820 . . . . 5 ((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) β†’ (𝑗 ∈ (0..^((β™―β€˜(π‘Š cyclShift 𝑁)) βˆ’ 1)) ↔ 𝑗 ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1))))
76adantr 482 . . . 4 (((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) ∧ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ 𝐸 ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ 𝐸)) β†’ (𝑗 ∈ (0..^((β™―β€˜(π‘Š cyclShift 𝑁)) βˆ’ 1)) ↔ 𝑗 ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1))))
8 simpll 766 . . . . . . . . 9 (((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) ∧ 𝑗 ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1))) β†’ π‘Š ∈ Word 𝑉)
91ad2antlr 726 . . . . . . . . 9 (((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) ∧ 𝑗 ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1))) β†’ 𝑁 ∈ β„€)
10 lencl 14483 . . . . . . . . . . . . 13 (π‘Š ∈ Word 𝑉 β†’ (β™―β€˜π‘Š) ∈ β„•0)
11 nn0z 12583 . . . . . . . . . . . . . . 15 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (β™―β€˜π‘Š) ∈ β„€)
12 peano2zm 12605 . . . . . . . . . . . . . . 15 ((β™―β€˜π‘Š) ∈ β„€ β†’ ((β™―β€˜π‘Š) βˆ’ 1) ∈ β„€)
1311, 12syl 17 . . . . . . . . . . . . . 14 ((β™―β€˜π‘Š) ∈ β„•0 β†’ ((β™―β€˜π‘Š) βˆ’ 1) ∈ β„€)
14 nn0re 12481 . . . . . . . . . . . . . . 15 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (β™―β€˜π‘Š) ∈ ℝ)
1514lem1d 12147 . . . . . . . . . . . . . 14 ((β™―β€˜π‘Š) ∈ β„•0 β†’ ((β™―β€˜π‘Š) βˆ’ 1) ≀ (β™―β€˜π‘Š))
16 eluz2 12828 . . . . . . . . . . . . . 14 ((β™―β€˜π‘Š) ∈ (β„€β‰₯β€˜((β™―β€˜π‘Š) βˆ’ 1)) ↔ (((β™―β€˜π‘Š) βˆ’ 1) ∈ β„€ ∧ (β™―β€˜π‘Š) ∈ β„€ ∧ ((β™―β€˜π‘Š) βˆ’ 1) ≀ (β™―β€˜π‘Š)))
1713, 11, 15, 16syl3anbrc 1344 . . . . . . . . . . . . 13 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (β™―β€˜π‘Š) ∈ (β„€β‰₯β€˜((β™―β€˜π‘Š) βˆ’ 1)))
1810, 17syl 17 . . . . . . . . . . . 12 (π‘Š ∈ Word 𝑉 β†’ (β™―β€˜π‘Š) ∈ (β„€β‰₯β€˜((β™―β€˜π‘Š) βˆ’ 1)))
1918adantr 482 . . . . . . . . . . 11 ((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) β†’ (β™―β€˜π‘Š) ∈ (β„€β‰₯β€˜((β™―β€˜π‘Š) βˆ’ 1)))
20 fzoss2 13660 . . . . . . . . . . 11 ((β™―β€˜π‘Š) ∈ (β„€β‰₯β€˜((β™―β€˜π‘Š) βˆ’ 1)) β†’ (0..^((β™―β€˜π‘Š) βˆ’ 1)) βŠ† (0..^(β™―β€˜π‘Š)))
2119, 20syl 17 . . . . . . . . . 10 ((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) β†’ (0..^((β™―β€˜π‘Š) βˆ’ 1)) βŠ† (0..^(β™―β€˜π‘Š)))
2221sselda 3983 . . . . . . . . 9 (((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) ∧ 𝑗 ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1))) β†’ 𝑗 ∈ (0..^(β™―β€˜π‘Š)))
23 cshwidxmod 14753 . . . . . . . . 9 ((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ β„€ ∧ 𝑗 ∈ (0..^(β™―β€˜π‘Š))) β†’ ((π‘Š cyclShift 𝑁)β€˜π‘—) = (π‘Šβ€˜((𝑗 + 𝑁) mod (β™―β€˜π‘Š))))
248, 9, 22, 23syl3anc 1372 . . . . . . . 8 (((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) ∧ 𝑗 ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1))) β†’ ((π‘Š cyclShift 𝑁)β€˜π‘—) = (π‘Šβ€˜((𝑗 + 𝑁) mod (β™―β€˜π‘Š))))
25 elfzo1 13682 . . . . . . . . . . . 12 (𝑁 ∈ (1..^(β™―β€˜π‘Š)) ↔ (𝑁 ∈ β„• ∧ (β™―β€˜π‘Š) ∈ β„• ∧ 𝑁 < (β™―β€˜π‘Š)))
2625simp2bi 1147 . . . . . . . . . . 11 (𝑁 ∈ (1..^(β™―β€˜π‘Š)) β†’ (β™―β€˜π‘Š) ∈ β„•)
2726adantl 483 . . . . . . . . . 10 ((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) β†’ (β™―β€˜π‘Š) ∈ β„•)
28 elfzom1p1elfzo 13712 . . . . . . . . . 10 (((β™―β€˜π‘Š) ∈ β„• ∧ 𝑗 ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1))) β†’ (𝑗 + 1) ∈ (0..^(β™―β€˜π‘Š)))
2927, 28sylan 581 . . . . . . . . 9 (((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) ∧ 𝑗 ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1))) β†’ (𝑗 + 1) ∈ (0..^(β™―β€˜π‘Š)))
30 cshwidxmod 14753 . . . . . . . . 9 ((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ β„€ ∧ (𝑗 + 1) ∈ (0..^(β™―β€˜π‘Š))) β†’ ((π‘Š cyclShift 𝑁)β€˜(𝑗 + 1)) = (π‘Šβ€˜(((𝑗 + 1) + 𝑁) mod (β™―β€˜π‘Š))))
318, 9, 29, 30syl3anc 1372 . . . . . . . 8 (((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) ∧ 𝑗 ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1))) β†’ ((π‘Š cyclShift 𝑁)β€˜(𝑗 + 1)) = (π‘Šβ€˜(((𝑗 + 1) + 𝑁) mod (β™―β€˜π‘Š))))
3224, 31preq12d 4746 . . . . . . 7 (((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) ∧ 𝑗 ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1))) β†’ {((π‘Š cyclShift 𝑁)β€˜π‘—), ((π‘Š cyclShift 𝑁)β€˜(𝑗 + 1))} = {(π‘Šβ€˜((𝑗 + 𝑁) mod (β™―β€˜π‘Š))), (π‘Šβ€˜(((𝑗 + 1) + 𝑁) mod (β™―β€˜π‘Š)))})
3332adantlr 714 . . . . . 6 ((((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) ∧ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ 𝐸 ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ 𝐸)) ∧ 𝑗 ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1))) β†’ {((π‘Š cyclShift 𝑁)β€˜π‘—), ((π‘Š cyclShift 𝑁)β€˜(𝑗 + 1))} = {(π‘Šβ€˜((𝑗 + 𝑁) mod (β™―β€˜π‘Š))), (π‘Šβ€˜(((𝑗 + 1) + 𝑁) mod (β™―β€˜π‘Š)))})
34 2z 12594 . . . . . . . . . . 11 2 ∈ β„€
3534a1i 11 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ (β™―β€˜π‘Š) ∈ β„• ∧ 𝑁 < (β™―β€˜π‘Š)) β†’ 2 ∈ β„€)
36 nnz 12579 . . . . . . . . . . 11 ((β™―β€˜π‘Š) ∈ β„• β†’ (β™―β€˜π‘Š) ∈ β„€)
37363ad2ant2 1135 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ (β™―β€˜π‘Š) ∈ β„• ∧ 𝑁 < (β™―β€˜π‘Š)) β†’ (β™―β€˜π‘Š) ∈ β„€)
38 nnnn0 12479 . . . . . . . . . . . 12 ((β™―β€˜π‘Š) ∈ β„• β†’ (β™―β€˜π‘Š) ∈ β„•0)
39383ad2ant2 1135 . . . . . . . . . . 11 ((𝑁 ∈ β„• ∧ (β™―β€˜π‘Š) ∈ β„• ∧ 𝑁 < (β™―β€˜π‘Š)) β†’ (β™―β€˜π‘Š) ∈ β„•0)
40 nnne0 12246 . . . . . . . . . . . 12 ((β™―β€˜π‘Š) ∈ β„• β†’ (β™―β€˜π‘Š) β‰  0)
41403ad2ant2 1135 . . . . . . . . . . 11 ((𝑁 ∈ β„• ∧ (β™―β€˜π‘Š) ∈ β„• ∧ 𝑁 < (β™―β€˜π‘Š)) β†’ (β™―β€˜π‘Š) β‰  0)
42 1red 11215 . . . . . . . . . . . 12 ((𝑁 ∈ β„• ∧ (β™―β€˜π‘Š) ∈ β„• ∧ 𝑁 < (β™―β€˜π‘Š)) β†’ 1 ∈ ℝ)
43 nnre 12219 . . . . . . . . . . . . . 14 (𝑁 ∈ β„• β†’ 𝑁 ∈ ℝ)
44433ad2ant1 1134 . . . . . . . . . . . . 13 ((𝑁 ∈ β„• ∧ (β™―β€˜π‘Š) ∈ β„• ∧ 𝑁 < (β™―β€˜π‘Š)) β†’ 𝑁 ∈ ℝ)
45 nnre 12219 . . . . . . . . . . . . . 14 ((β™―β€˜π‘Š) ∈ β„• β†’ (β™―β€˜π‘Š) ∈ ℝ)
46453ad2ant2 1135 . . . . . . . . . . . . 13 ((𝑁 ∈ β„• ∧ (β™―β€˜π‘Š) ∈ β„• ∧ 𝑁 < (β™―β€˜π‘Š)) β†’ (β™―β€˜π‘Š) ∈ ℝ)
47 nnge1 12240 . . . . . . . . . . . . . 14 (𝑁 ∈ β„• β†’ 1 ≀ 𝑁)
48473ad2ant1 1134 . . . . . . . . . . . . 13 ((𝑁 ∈ β„• ∧ (β™―β€˜π‘Š) ∈ β„• ∧ 𝑁 < (β™―β€˜π‘Š)) β†’ 1 ≀ 𝑁)
49 simp3 1139 . . . . . . . . . . . . 13 ((𝑁 ∈ β„• ∧ (β™―β€˜π‘Š) ∈ β„• ∧ 𝑁 < (β™―β€˜π‘Š)) β†’ 𝑁 < (β™―β€˜π‘Š))
5042, 44, 46, 48, 49lelttrd 11372 . . . . . . . . . . . 12 ((𝑁 ∈ β„• ∧ (β™―β€˜π‘Š) ∈ β„• ∧ 𝑁 < (β™―β€˜π‘Š)) β†’ 1 < (β™―β€˜π‘Š))
5142, 50gtned 11349 . . . . . . . . . . 11 ((𝑁 ∈ β„• ∧ (β™―β€˜π‘Š) ∈ β„• ∧ 𝑁 < (β™―β€˜π‘Š)) β†’ (β™―β€˜π‘Š) β‰  1)
52 nn0n0n1ge2 12539 . . . . . . . . . . 11 (((β™―β€˜π‘Š) ∈ β„•0 ∧ (β™―β€˜π‘Š) β‰  0 ∧ (β™―β€˜π‘Š) β‰  1) β†’ 2 ≀ (β™―β€˜π‘Š))
5339, 41, 51, 52syl3anc 1372 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ (β™―β€˜π‘Š) ∈ β„• ∧ 𝑁 < (β™―β€˜π‘Š)) β†’ 2 ≀ (β™―β€˜π‘Š))
54 eluz2 12828 . . . . . . . . . 10 ((β™―β€˜π‘Š) ∈ (β„€β‰₯β€˜2) ↔ (2 ∈ β„€ ∧ (β™―β€˜π‘Š) ∈ β„€ ∧ 2 ≀ (β™―β€˜π‘Š)))
5535, 37, 53, 54syl3anbrc 1344 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ (β™―β€˜π‘Š) ∈ β„• ∧ 𝑁 < (β™―β€˜π‘Š)) β†’ (β™―β€˜π‘Š) ∈ (β„€β‰₯β€˜2))
5625, 55sylbi 216 . . . . . . . 8 (𝑁 ∈ (1..^(β™―β€˜π‘Š)) β†’ (β™―β€˜π‘Š) ∈ (β„€β‰₯β€˜2))
5756ad3antlr 730 . . . . . . 7 ((((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) ∧ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ 𝐸 ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ 𝐸)) ∧ 𝑗 ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1))) β†’ (β™―β€˜π‘Š) ∈ (β„€β‰₯β€˜2))
58 elfzoelz 13632 . . . . . . . 8 (𝑗 ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)) β†’ 𝑗 ∈ β„€)
5958adantl 483 . . . . . . 7 ((((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) ∧ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ 𝐸 ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ 𝐸)) ∧ 𝑗 ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1))) β†’ 𝑗 ∈ β„€)
601ad3antlr 730 . . . . . . 7 ((((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) ∧ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ 𝐸 ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ 𝐸)) ∧ 𝑗 ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1))) β†’ 𝑁 ∈ β„€)
61 simplrl 776 . . . . . . 7 ((((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) ∧ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ 𝐸 ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ 𝐸)) ∧ 𝑗 ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1))) β†’ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ 𝐸)
62 lsw 14514 . . . . . . . . . . . . . 14 (π‘Š ∈ Word 𝑉 β†’ (lastSβ€˜π‘Š) = (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1)))
6362adantr 482 . . . . . . . . . . . . 13 ((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) β†’ (lastSβ€˜π‘Š) = (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1)))
6463preq1d 4744 . . . . . . . . . . . 12 ((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) β†’ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} = {(π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1)), (π‘Šβ€˜0)})
6564eleq1d 2819 . . . . . . . . . . 11 ((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) β†’ ({(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ 𝐸 ↔ {(π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1)), (π‘Šβ€˜0)} ∈ 𝐸))
6665biimpcd 248 . . . . . . . . . 10 ({(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ 𝐸 β†’ ((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) β†’ {(π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1)), (π‘Šβ€˜0)} ∈ 𝐸))
6766adantl 483 . . . . . . . . 9 ((βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ 𝐸 ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ 𝐸) β†’ ((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) β†’ {(π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1)), (π‘Šβ€˜0)} ∈ 𝐸))
6867impcom 409 . . . . . . . 8 (((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) ∧ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ 𝐸 ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ 𝐸)) β†’ {(π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1)), (π‘Šβ€˜0)} ∈ 𝐸)
6968adantr 482 . . . . . . 7 ((((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) ∧ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ 𝐸 ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ 𝐸)) ∧ 𝑗 ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1))) β†’ {(π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1)), (π‘Šβ€˜0)} ∈ 𝐸)
70 clwwisshclwwslemlem 29266 . . . . . . 7 ((((β™―β€˜π‘Š) ∈ (β„€β‰₯β€˜2) ∧ 𝑗 ∈ β„€ ∧ 𝑁 ∈ β„€) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ 𝐸 ∧ {(π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1)), (π‘Šβ€˜0)} ∈ 𝐸) β†’ {(π‘Šβ€˜((𝑗 + 𝑁) mod (β™―β€˜π‘Š))), (π‘Šβ€˜(((𝑗 + 1) + 𝑁) mod (β™―β€˜π‘Š)))} ∈ 𝐸)
7157, 59, 60, 61, 69, 70syl311anc 1385 . . . . . 6 ((((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) ∧ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ 𝐸 ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ 𝐸)) ∧ 𝑗 ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1))) β†’ {(π‘Šβ€˜((𝑗 + 𝑁) mod (β™―β€˜π‘Š))), (π‘Šβ€˜(((𝑗 + 1) + 𝑁) mod (β™―β€˜π‘Š)))} ∈ 𝐸)
7233, 71eqeltrd 2834 . . . . 5 ((((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) ∧ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ 𝐸 ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ 𝐸)) ∧ 𝑗 ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1))) β†’ {((π‘Š cyclShift 𝑁)β€˜π‘—), ((π‘Š cyclShift 𝑁)β€˜(𝑗 + 1))} ∈ 𝐸)
7372ex 414 . . . 4 (((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) ∧ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ 𝐸 ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ 𝐸)) β†’ (𝑗 ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)) β†’ {((π‘Š cyclShift 𝑁)β€˜π‘—), ((π‘Š cyclShift 𝑁)β€˜(𝑗 + 1))} ∈ 𝐸))
747, 73sylbid 239 . . 3 (((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) ∧ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ 𝐸 ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ 𝐸)) β†’ (𝑗 ∈ (0..^((β™―β€˜(π‘Š cyclShift 𝑁)) βˆ’ 1)) β†’ {((π‘Š cyclShift 𝑁)β€˜π‘—), ((π‘Š cyclShift 𝑁)β€˜(𝑗 + 1))} ∈ 𝐸))
7574ralrimiv 3146 . 2 (((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) ∧ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ 𝐸 ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ 𝐸)) β†’ βˆ€π‘— ∈ (0..^((β™―β€˜(π‘Š cyclShift 𝑁)) βˆ’ 1)){((π‘Š cyclShift 𝑁)β€˜π‘—), ((π‘Š cyclShift 𝑁)β€˜(𝑗 + 1))} ∈ 𝐸)
7675ex 414 1 ((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) β†’ ((βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ 𝐸 ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ 𝐸) β†’ βˆ€π‘— ∈ (0..^((β™―β€˜(π‘Š cyclShift 𝑁)) βˆ’ 1)){((π‘Š cyclShift 𝑁)β€˜π‘—), ((π‘Š cyclShift 𝑁)β€˜(𝑗 + 1))} ∈ 𝐸))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062   βŠ† wss 3949  {cpr 4631   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409  β„cr 11109  0cc0 11110  1c1 11111   + caddc 11113   < clt 11248   ≀ cle 11249   βˆ’ cmin 11444  β„•cn 12212  2c2 12267  β„•0cn0 12472  β„€cz 12558  β„€β‰₯cuz 12822  ..^cfzo 13627   mod cmo 13834  β™―chash 14290  Word cword 14464  lastSclsw 14512   cyclShift ccsh 14738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9437  df-inf 9438  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-n0 12473  df-z 12559  df-uz 12823  df-rp 12975  df-fz 13485  df-fzo 13628  df-fl 13757  df-mod 13835  df-hash 14291  df-word 14465  df-lsw 14513  df-concat 14521  df-substr 14591  df-pfx 14621  df-csh 14739
This theorem is referenced by:  clwwisshclwws  29268
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