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Theorem clwwisshclwwslem 29532
Description: Lemma for clwwisshclwws 29533. (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 13638 . . . . . . . . 9 (𝑁 ∈ (1..^(β™―β€˜π‘Š)) β†’ 𝑁 ∈ β„€)
2 cshwlen 14755 . . . . . . . . 9 ((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ β„€) β†’ (β™―β€˜(π‘Š cyclShift 𝑁)) = (β™―β€˜π‘Š))
31, 2sylan2 591 . . . . . . . 8 ((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) β†’ (β™―β€˜(π‘Š cyclShift 𝑁)) = (β™―β€˜π‘Š))
43oveq1d 7428 . . . . . . 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 7429 . . . . . 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 2817 . . . . 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 479 . . . 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 763 . . . . . . . . 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 723 . . . . . . . . 9 (((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) ∧ 𝑗 ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1))) β†’ 𝑁 ∈ β„€)
10 lencl 14489 . . . . . . . . . . . . 13 (π‘Š ∈ Word 𝑉 β†’ (β™―β€˜π‘Š) ∈ β„•0)
11 nn0z 12589 . . . . . . . . . . . . . . 15 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (β™―β€˜π‘Š) ∈ β„€)
12 peano2zm 12611 . . . . . . . . . . . . . . 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 12487 . . . . . . . . . . . . . . 15 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (β™―β€˜π‘Š) ∈ ℝ)
1514lem1d 12153 . . . . . . . . . . . . . 14 ((β™―β€˜π‘Š) ∈ β„•0 β†’ ((β™―β€˜π‘Š) βˆ’ 1) ≀ (β™―β€˜π‘Š))
16 eluz2 12834 . . . . . . . . . . . . . 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 1341 . . . . . . . . . . . . 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 479 . . . . . . . . . . 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 13666 . . . . . . . . . . 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 14759 . . . . . . . . 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 1369 . . . . . . . 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 13688 . . . . . . . . . . . 12 (𝑁 ∈ (1..^(β™―β€˜π‘Š)) ↔ (𝑁 ∈ β„• ∧ (β™―β€˜π‘Š) ∈ β„• ∧ 𝑁 < (β™―β€˜π‘Š)))
2625simp2bi 1144 . . . . . . . . . . 11 (𝑁 ∈ (1..^(β™―β€˜π‘Š)) β†’ (β™―β€˜π‘Š) ∈ β„•)
2726adantl 480 . . . . . . . . . 10 ((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(β™―β€˜π‘Š))) β†’ (β™―β€˜π‘Š) ∈ β„•)
28 elfzom1p1elfzo 13718 . . . . . . . . . 10 (((β™―β€˜π‘Š) ∈ β„• ∧ 𝑗 ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1))) β†’ (𝑗 + 1) ∈ (0..^(β™―β€˜π‘Š)))
2927, 28sylan 578 . . . . . . . . 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 14759 . . . . . . . . 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 1369 . . . . . . . 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 711 . . . . . 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 12600 . . . . . . . . . . 11 2 ∈ β„€
3534a1i 11 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ (β™―β€˜π‘Š) ∈ β„• ∧ 𝑁 < (β™―β€˜π‘Š)) β†’ 2 ∈ β„€)
36 nnz 12585 . . . . . . . . . . 11 ((β™―β€˜π‘Š) ∈ β„• β†’ (β™―β€˜π‘Š) ∈ β„€)
37363ad2ant2 1132 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ (β™―β€˜π‘Š) ∈ β„• ∧ 𝑁 < (β™―β€˜π‘Š)) β†’ (β™―β€˜π‘Š) ∈ β„€)
38 nnnn0 12485 . . . . . . . . . . . 12 ((β™―β€˜π‘Š) ∈ β„• β†’ (β™―β€˜π‘Š) ∈ β„•0)
39383ad2ant2 1132 . . . . . . . . . . 11 ((𝑁 ∈ β„• ∧ (β™―β€˜π‘Š) ∈ β„• ∧ 𝑁 < (β™―β€˜π‘Š)) β†’ (β™―β€˜π‘Š) ∈ β„•0)
40 nnne0 12252 . . . . . . . . . . . 12 ((β™―β€˜π‘Š) ∈ β„• β†’ (β™―β€˜π‘Š) β‰  0)
41403ad2ant2 1132 . . . . . . . . . . 11 ((𝑁 ∈ β„• ∧ (β™―β€˜π‘Š) ∈ β„• ∧ 𝑁 < (β™―β€˜π‘Š)) β†’ (β™―β€˜π‘Š) β‰  0)
42 1red 11221 . . . . . . . . . . . 12 ((𝑁 ∈ β„• ∧ (β™―β€˜π‘Š) ∈ β„• ∧ 𝑁 < (β™―β€˜π‘Š)) β†’ 1 ∈ ℝ)
43 nnre 12225 . . . . . . . . . . . . . 14 (𝑁 ∈ β„• β†’ 𝑁 ∈ ℝ)
44433ad2ant1 1131 . . . . . . . . . . . . 13 ((𝑁 ∈ β„• ∧ (β™―β€˜π‘Š) ∈ β„• ∧ 𝑁 < (β™―β€˜π‘Š)) β†’ 𝑁 ∈ ℝ)
45 nnre 12225 . . . . . . . . . . . . . 14 ((β™―β€˜π‘Š) ∈ β„• β†’ (β™―β€˜π‘Š) ∈ ℝ)
46453ad2ant2 1132 . . . . . . . . . . . . 13 ((𝑁 ∈ β„• ∧ (β™―β€˜π‘Š) ∈ β„• ∧ 𝑁 < (β™―β€˜π‘Š)) β†’ (β™―β€˜π‘Š) ∈ ℝ)
47 nnge1 12246 . . . . . . . . . . . . . 14 (𝑁 ∈ β„• β†’ 1 ≀ 𝑁)
48473ad2ant1 1131 . . . . . . . . . . . . 13 ((𝑁 ∈ β„• ∧ (β™―β€˜π‘Š) ∈ β„• ∧ 𝑁 < (β™―β€˜π‘Š)) β†’ 1 ≀ 𝑁)
49 simp3 1136 . . . . . . . . . . . . 13 ((𝑁 ∈ β„• ∧ (β™―β€˜π‘Š) ∈ β„• ∧ 𝑁 < (β™―β€˜π‘Š)) β†’ 𝑁 < (β™―β€˜π‘Š))
5042, 44, 46, 48, 49lelttrd 11378 . . . . . . . . . . . 12 ((𝑁 ∈ β„• ∧ (β™―β€˜π‘Š) ∈ β„• ∧ 𝑁 < (β™―β€˜π‘Š)) β†’ 1 < (β™―β€˜π‘Š))
5142, 50gtned 11355 . . . . . . . . . . 11 ((𝑁 ∈ β„• ∧ (β™―β€˜π‘Š) ∈ β„• ∧ 𝑁 < (β™―β€˜π‘Š)) β†’ (β™―β€˜π‘Š) β‰  1)
52 nn0n0n1ge2 12545 . . . . . . . . . . 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 1369 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ (β™―β€˜π‘Š) ∈ β„• ∧ 𝑁 < (β™―β€˜π‘Š)) β†’ 2 ≀ (β™―β€˜π‘Š))
54 eluz2 12834 . . . . . . . . . 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 1341 . . . . . . . . 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 727 . . . . . . 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 13638 . . . . . . . 8 (𝑗 ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)) β†’ 𝑗 ∈ β„€)
5958adantl 480 . . . . . . 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 727 . . . . . . 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 773 . . . . . . 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 14520 . . . . . . . . . . . . . 14 (π‘Š ∈ Word 𝑉 β†’ (lastSβ€˜π‘Š) = (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1)))
6362adantr 479 . . . . . . . . . . . . 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 2816 . . . . . . . . . . 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 480 . . . . . . . . 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 406 . . . . . . . 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 479 . . . . . . 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 29531 . . . . . . 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 1382 . . . . . 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 2831 . . . . 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 411 . . . 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 3143 . 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 411 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 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  βˆ€wral 3059   βŠ† wss 3949  {cpr 4631   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7413  β„cr 11113  0cc0 11114  1c1 11115   + caddc 11117   < clt 11254   ≀ cle 11255   βˆ’ cmin 11450  β„•cn 12218  2c2 12273  β„•0cn0 12478  β„€cz 12564  β„€β‰₯cuz 12828  ..^cfzo 13633   mod cmo 13840  β™―chash 14296  Word cword 14470  lastSclsw 14518   cyclShift ccsh 14744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191  ax-pre-sup 11192
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  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 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-er 8707  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-sup 9441  df-inf 9442  df-card 9938  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11452  df-neg 11453  df-div 11878  df-nn 12219  df-2 12281  df-n0 12479  df-z 12565  df-uz 12829  df-rp 12981  df-fz 13491  df-fzo 13634  df-fl 13763  df-mod 13841  df-hash 14297  df-word 14471  df-lsw 14519  df-concat 14527  df-substr 14597  df-pfx 14627  df-csh 14745
This theorem is referenced by:  clwwisshclwws  29533
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