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Theorem clwwlkinwwlk 28453
Description: If the initial vertex of a walk occurs another time in the walk, the walk starts with a closed walk. Since the walk is expressed as a word over vertices, the closed walk can be expressed as a subword of this word. (Contributed by Alexander van der Vekens, 15-Sep-2018.) (Revised by AV, 23-Jan-2022.) (Revised by AV, 30-Oct-2022.)
Assertion
Ref Expression
clwwlkinwwlk (((𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘)) ∧ π‘Š ∈ (𝑀 WWalksN 𝐺) ∧ (π‘Šβ€˜π‘) = (π‘Šβ€˜0)) β†’ (π‘Š prefix 𝑁) ∈ (𝑁 ClWWalksN 𝐺))

Proof of Theorem clwwlkinwwlk
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 eqid 2736 . . 3 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
2 eqid 2736 . . 3 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
31, 2wwlknp 28257 . 2 (π‘Š ∈ (𝑀 WWalksN 𝐺) β†’ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1) ∧ βˆ€π‘– ∈ (0..^𝑀){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))
4 pfxcl 14439 . . . . . . . . . . . 12 (π‘Š ∈ Word (Vtxβ€˜πΊ) β†’ (π‘Š prefix 𝑁) ∈ Word (Vtxβ€˜πΊ))
54adantr 482 . . . . . . . . . . 11 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) β†’ (π‘Š prefix 𝑁) ∈ Word (Vtxβ€˜πΊ))
65adantr 482 . . . . . . . . . 10 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ (π‘Š prefix 𝑁) ∈ Word (Vtxβ€˜πΊ))
7 simpll 765 . . . . . . . . . . 11 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ π‘Š ∈ Word (Vtxβ€˜πΊ))
8 simprl 769 . . . . . . . . . . 11 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ 𝑁 ∈ β„•)
9 eluz2 12638 . . . . . . . . . . . . . . . 16 (𝑀 ∈ (β„€β‰₯β€˜π‘) ↔ (𝑁 ∈ β„€ ∧ 𝑀 ∈ β„€ ∧ 𝑁 ≀ 𝑀))
10 zre 12373 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ β„€ β†’ 𝑁 ∈ ℝ)
11 zre 12373 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ β„€ β†’ 𝑀 ∈ ℝ)
12 id 22 . . . . . . . . . . . . . . . . 17 (𝑁 ≀ 𝑀 β†’ 𝑁 ≀ 𝑀)
1310, 11, 123anim123i 1151 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ β„€ ∧ 𝑀 ∈ β„€ ∧ 𝑁 ≀ 𝑀) β†’ (𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝑁 ≀ 𝑀))
149, 13sylbi 216 . . . . . . . . . . . . . . 15 (𝑀 ∈ (β„€β‰₯β€˜π‘) β†’ (𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝑁 ≀ 𝑀))
15 letrp1 11869 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝑁 ≀ 𝑀) β†’ 𝑁 ≀ (𝑀 + 1))
1614, 15syl 17 . . . . . . . . . . . . . 14 (𝑀 ∈ (β„€β‰₯β€˜π‘) β†’ 𝑁 ≀ (𝑀 + 1))
1716adantl 483 . . . . . . . . . . . . 13 ((𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘)) β†’ 𝑁 ≀ (𝑀 + 1))
1817adantl 483 . . . . . . . . . . . 12 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ 𝑁 ≀ (𝑀 + 1))
19 breq2 5085 . . . . . . . . . . . . 13 ((β™―β€˜π‘Š) = (𝑀 + 1) β†’ (𝑁 ≀ (β™―β€˜π‘Š) ↔ 𝑁 ≀ (𝑀 + 1)))
2019ad2antlr 725 . . . . . . . . . . . 12 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ (𝑁 ≀ (β™―β€˜π‘Š) ↔ 𝑁 ≀ (𝑀 + 1)))
2118, 20mpbird 257 . . . . . . . . . . 11 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ 𝑁 ≀ (β™―β€˜π‘Š))
22 pfxn0 14448 . . . . . . . . . . 11 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ 𝑁 ∈ β„• ∧ 𝑁 ≀ (β™―β€˜π‘Š)) β†’ (π‘Š prefix 𝑁) β‰  βˆ…)
237, 8, 21, 22syl3anc 1371 . . . . . . . . . 10 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ (π‘Š prefix 𝑁) β‰  βˆ…)
246, 23jca 513 . . . . . . . . 9 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ ((π‘Š prefix 𝑁) ∈ Word (Vtxβ€˜πΊ) ∧ (π‘Š prefix 𝑁) β‰  βˆ…))
25243adantl3 1168 . . . . . . . 8 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1) ∧ βˆ€π‘– ∈ (0..^𝑀){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ ((π‘Š prefix 𝑁) ∈ Word (Vtxβ€˜πΊ) ∧ (π‘Š prefix 𝑁) β‰  βˆ…))
2625adantr 482 . . . . . . 7 ((((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1) ∧ βˆ€π‘– ∈ (0..^𝑀){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) ∧ (π‘Šβ€˜π‘) = (π‘Šβ€˜0)) β†’ ((π‘Š prefix 𝑁) ∈ Word (Vtxβ€˜πΊ) ∧ (π‘Š prefix 𝑁) β‰  βˆ…))
27 nnz 12392 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ β„• β†’ 𝑁 ∈ β„€)
28 1nn0 12299 . . . . . . . . . . . . . . . . . 18 1 ∈ β„•0
29 eluzmn 12639 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ β„€ ∧ 1 ∈ β„•0) β†’ 𝑁 ∈ (β„€β‰₯β€˜(𝑁 βˆ’ 1)))
3027, 28, 29sylancl 587 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ β„• β†’ 𝑁 ∈ (β„€β‰₯β€˜(𝑁 βˆ’ 1)))
31 uzss 12655 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ (β„€β‰₯β€˜(𝑁 βˆ’ 1)) β†’ (β„€β‰₯β€˜π‘) βŠ† (β„€β‰₯β€˜(𝑁 βˆ’ 1)))
3230, 31syl 17 . . . . . . . . . . . . . . . 16 (𝑁 ∈ β„• β†’ (β„€β‰₯β€˜π‘) βŠ† (β„€β‰₯β€˜(𝑁 βˆ’ 1)))
3332sselda 3926 . . . . . . . . . . . . . . 15 ((𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘)) β†’ 𝑀 ∈ (β„€β‰₯β€˜(𝑁 βˆ’ 1)))
34 fzoss2 13465 . . . . . . . . . . . . . . 15 (𝑀 ∈ (β„€β‰₯β€˜(𝑁 βˆ’ 1)) β†’ (0..^(𝑁 βˆ’ 1)) βŠ† (0..^𝑀))
3533, 34syl 17 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘)) β†’ (0..^(𝑁 βˆ’ 1)) βŠ† (0..^𝑀))
36353ad2ant3 1135 . . . . . . . . . . . . 13 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ (0..^(𝑁 βˆ’ 1)) βŠ† (0..^𝑀))
37 ssralv 3992 . . . . . . . . . . . . 13 ((0..^(𝑁 βˆ’ 1)) βŠ† (0..^𝑀) β†’ (βˆ€π‘– ∈ (0..^𝑀){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) β†’ βˆ€π‘– ∈ (0..^(𝑁 βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))
3836, 37syl 17 . . . . . . . . . . . 12 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ (βˆ€π‘– ∈ (0..^𝑀){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) β†’ βˆ€π‘– ∈ (0..^(𝑁 βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))
39383exp 1119 . . . . . . . . . . 11 (π‘Š ∈ Word (Vtxβ€˜πΊ) β†’ ((β™―β€˜π‘Š) = (𝑀 + 1) β†’ ((𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘)) β†’ (βˆ€π‘– ∈ (0..^𝑀){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) β†’ βˆ€π‘– ∈ (0..^(𝑁 βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))))
4039com34 91 . . . . . . . . . 10 (π‘Š ∈ Word (Vtxβ€˜πΊ) β†’ ((β™―β€˜π‘Š) = (𝑀 + 1) β†’ (βˆ€π‘– ∈ (0..^𝑀){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) β†’ ((𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘)) β†’ βˆ€π‘– ∈ (0..^(𝑁 βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))))
41403imp1 1347 . . . . . . . . 9 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1) ∧ βˆ€π‘– ∈ (0..^𝑀){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ βˆ€π‘– ∈ (0..^(𝑁 βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ))
4241adantr 482 . . . . . . . 8 ((((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1) ∧ βˆ€π‘– ∈ (0..^𝑀){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) ∧ (π‘Šβ€˜π‘) = (π‘Šβ€˜0)) β†’ βˆ€π‘– ∈ (0..^(𝑁 βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ))
43 nnnn0 12290 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ β„• β†’ 𝑁 ∈ β„•0)
44 elnn0uz 12673 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ β„•0 ↔ 𝑁 ∈ (β„€β‰₯β€˜0))
4543, 44sylib 217 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ β„• β†’ 𝑁 ∈ (β„€β‰₯β€˜0))
46 eluzfz 13301 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ (β„€β‰₯β€˜0) ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘)) β†’ 𝑁 ∈ (0...𝑀))
4745, 46sylan 581 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘)) β†’ 𝑁 ∈ (0...𝑀))
48 fzelp1 13358 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ (0...𝑀) β†’ 𝑁 ∈ (0...(𝑀 + 1)))
4947, 48syl 17 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘)) β†’ 𝑁 ∈ (0...(𝑀 + 1)))
5049adantl 483 . . . . . . . . . . . . . . . 16 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ 𝑁 ∈ (0...(𝑀 + 1)))
51 oveq2 7315 . . . . . . . . . . . . . . . . . 18 ((β™―β€˜π‘Š) = (𝑀 + 1) β†’ (0...(β™―β€˜π‘Š)) = (0...(𝑀 + 1)))
5251eleq2d 2822 . . . . . . . . . . . . . . . . 17 ((β™―β€˜π‘Š) = (𝑀 + 1) β†’ (𝑁 ∈ (0...(β™―β€˜π‘Š)) ↔ 𝑁 ∈ (0...(𝑀 + 1))))
5352ad2antlr 725 . . . . . . . . . . . . . . . 16 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ (𝑁 ∈ (0...(β™―β€˜π‘Š)) ↔ 𝑁 ∈ (0...(𝑀 + 1))))
5450, 53mpbird 257 . . . . . . . . . . . . . . 15 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ 𝑁 ∈ (0...(β™―β€˜π‘Š)))
55 pfxlen 14445 . . . . . . . . . . . . . . 15 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ 𝑁 ∈ (0...(β™―β€˜π‘Š))) β†’ (β™―β€˜(π‘Š prefix 𝑁)) = 𝑁)
567, 54, 55syl2anc 585 . . . . . . . . . . . . . 14 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ (β™―β€˜(π‘Š prefix 𝑁)) = 𝑁)
5756oveq1d 7322 . . . . . . . . . . . . 13 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ ((β™―β€˜(π‘Š prefix 𝑁)) βˆ’ 1) = (𝑁 βˆ’ 1))
5857oveq2d 7323 . . . . . . . . . . . 12 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ (0..^((β™―β€˜(π‘Š prefix 𝑁)) βˆ’ 1)) = (0..^(𝑁 βˆ’ 1)))
5958raleqdv 3367 . . . . . . . . . . 11 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜(π‘Š prefix 𝑁)) βˆ’ 1)){((π‘Š prefix 𝑁)β€˜π‘–), ((π‘Š prefix 𝑁)β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ↔ βˆ€π‘– ∈ (0..^(𝑁 βˆ’ 1)){((π‘Š prefix 𝑁)β€˜π‘–), ((π‘Š prefix 𝑁)β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))
607adantr 482 . . . . . . . . . . . . . . 15 ((((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) ∧ 𝑖 ∈ (0..^(𝑁 βˆ’ 1))) β†’ π‘Š ∈ Word (Vtxβ€˜πΊ))
6154adantr 482 . . . . . . . . . . . . . . 15 ((((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) ∧ 𝑖 ∈ (0..^(𝑁 βˆ’ 1))) β†’ 𝑁 ∈ (0...(β™―β€˜π‘Š)))
6230ad2antrl 726 . . . . . . . . . . . . . . . . 17 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ 𝑁 ∈ (β„€β‰₯β€˜(𝑁 βˆ’ 1)))
63 fzoss2 13465 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ (β„€β‰₯β€˜(𝑁 βˆ’ 1)) β†’ (0..^(𝑁 βˆ’ 1)) βŠ† (0..^𝑁))
6462, 63syl 17 . . . . . . . . . . . . . . . 16 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ (0..^(𝑁 βˆ’ 1)) βŠ† (0..^𝑁))
6564sselda 3926 . . . . . . . . . . . . . . 15 ((((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) ∧ 𝑖 ∈ (0..^(𝑁 βˆ’ 1))) β†’ 𝑖 ∈ (0..^𝑁))
66 pfxfv 14444 . . . . . . . . . . . . . . 15 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ 𝑁 ∈ (0...(β™―β€˜π‘Š)) ∧ 𝑖 ∈ (0..^𝑁)) β†’ ((π‘Š prefix 𝑁)β€˜π‘–) = (π‘Šβ€˜π‘–))
6760, 61, 65, 66syl3anc 1371 . . . . . . . . . . . . . 14 ((((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) ∧ 𝑖 ∈ (0..^(𝑁 βˆ’ 1))) β†’ ((π‘Š prefix 𝑁)β€˜π‘–) = (π‘Šβ€˜π‘–))
6827ad2antrl 726 . . . . . . . . . . . . . . . 16 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ 𝑁 ∈ β„€)
69 elfzom1elp1fzo 13504 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ β„€ ∧ 𝑖 ∈ (0..^(𝑁 βˆ’ 1))) β†’ (𝑖 + 1) ∈ (0..^𝑁))
7068, 69sylan 581 . . . . . . . . . . . . . . 15 ((((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) ∧ 𝑖 ∈ (0..^(𝑁 βˆ’ 1))) β†’ (𝑖 + 1) ∈ (0..^𝑁))
71 pfxfv 14444 . . . . . . . . . . . . . . 15 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ 𝑁 ∈ (0...(β™―β€˜π‘Š)) ∧ (𝑖 + 1) ∈ (0..^𝑁)) β†’ ((π‘Š prefix 𝑁)β€˜(𝑖 + 1)) = (π‘Šβ€˜(𝑖 + 1)))
7260, 61, 70, 71syl3anc 1371 . . . . . . . . . . . . . 14 ((((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) ∧ 𝑖 ∈ (0..^(𝑁 βˆ’ 1))) β†’ ((π‘Š prefix 𝑁)β€˜(𝑖 + 1)) = (π‘Šβ€˜(𝑖 + 1)))
7367, 72preq12d 4681 . . . . . . . . . . . . 13 ((((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) ∧ 𝑖 ∈ (0..^(𝑁 βˆ’ 1))) β†’ {((π‘Š prefix 𝑁)β€˜π‘–), ((π‘Š prefix 𝑁)β€˜(𝑖 + 1))} = {(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))})
7473eleq1d 2821 . . . . . . . . . . . 12 ((((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) ∧ 𝑖 ∈ (0..^(𝑁 βˆ’ 1))) β†’ ({((π‘Š prefix 𝑁)β€˜π‘–), ((π‘Š prefix 𝑁)β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ↔ {(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))
7574ralbidva 3169 . . . . . . . . . . 11 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ (βˆ€π‘– ∈ (0..^(𝑁 βˆ’ 1)){((π‘Š prefix 𝑁)β€˜π‘–), ((π‘Š prefix 𝑁)β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ↔ βˆ€π‘– ∈ (0..^(𝑁 βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))
7659, 75bitrd 279 . . . . . . . . . 10 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜(π‘Š prefix 𝑁)) βˆ’ 1)){((π‘Š prefix 𝑁)β€˜π‘–), ((π‘Š prefix 𝑁)β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ↔ βˆ€π‘– ∈ (0..^(𝑁 βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))
77763adantl3 1168 . . . . . . . . 9 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1) ∧ βˆ€π‘– ∈ (0..^𝑀){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜(π‘Š prefix 𝑁)) βˆ’ 1)){((π‘Š prefix 𝑁)β€˜π‘–), ((π‘Š prefix 𝑁)β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ↔ βˆ€π‘– ∈ (0..^(𝑁 βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))
7877adantr 482 . . . . . . . 8 ((((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1) ∧ βˆ€π‘– ∈ (0..^𝑀){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) ∧ (π‘Šβ€˜π‘) = (π‘Šβ€˜0)) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜(π‘Š prefix 𝑁)) βˆ’ 1)){((π‘Š prefix 𝑁)β€˜π‘–), ((π‘Š prefix 𝑁)β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ↔ βˆ€π‘– ∈ (0..^(𝑁 βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))
7942, 78mpbird 257 . . . . . . 7 ((((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1) ∧ βˆ€π‘– ∈ (0..^𝑀){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) ∧ (π‘Šβ€˜π‘) = (π‘Šβ€˜0)) β†’ βˆ€π‘– ∈ (0..^((β™―β€˜(π‘Š prefix 𝑁)) βˆ’ 1)){((π‘Š prefix 𝑁)β€˜π‘–), ((π‘Š prefix 𝑁)β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ))
80 elfz1uz 13376 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘)) β†’ 𝑁 ∈ (1...𝑀))
81 fzelp1 13358 . . . . . . . . . . . . . 14 (𝑁 ∈ (1...𝑀) β†’ 𝑁 ∈ (1...(𝑀 + 1)))
8280, 81syl 17 . . . . . . . . . . . . 13 ((𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘)) β†’ 𝑁 ∈ (1...(𝑀 + 1)))
8382adantl 483 . . . . . . . . . . . 12 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ 𝑁 ∈ (1...(𝑀 + 1)))
84 oveq2 7315 . . . . . . . . . . . . . 14 ((β™―β€˜π‘Š) = (𝑀 + 1) β†’ (1...(β™―β€˜π‘Š)) = (1...(𝑀 + 1)))
8584eleq2d 2822 . . . . . . . . . . . . 13 ((β™―β€˜π‘Š) = (𝑀 + 1) β†’ (𝑁 ∈ (1...(β™―β€˜π‘Š)) ↔ 𝑁 ∈ (1...(𝑀 + 1))))
8685ad2antlr 725 . . . . . . . . . . . 12 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ (𝑁 ∈ (1...(β™―β€˜π‘Š)) ↔ 𝑁 ∈ (1...(𝑀 + 1))))
8783, 86mpbird 257 . . . . . . . . . . 11 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ 𝑁 ∈ (1...(β™―β€˜π‘Š)))
88 pfxfvlsw 14457 . . . . . . . . . . . 12 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ 𝑁 ∈ (1...(β™―β€˜π‘Š))) β†’ (lastSβ€˜(π‘Š prefix 𝑁)) = (π‘Šβ€˜(𝑁 βˆ’ 1)))
89 pfxfv0 14454 . . . . . . . . . . . 12 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ 𝑁 ∈ (1...(β™―β€˜π‘Š))) β†’ ((π‘Š prefix 𝑁)β€˜0) = (π‘Šβ€˜0))
9088, 89preq12d 4681 . . . . . . . . . . 11 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ 𝑁 ∈ (1...(β™―β€˜π‘Š))) β†’ {(lastSβ€˜(π‘Š prefix 𝑁)), ((π‘Š prefix 𝑁)β€˜0)} = {(π‘Šβ€˜(𝑁 βˆ’ 1)), (π‘Šβ€˜0)})
917, 87, 90syl2anc 585 . . . . . . . . . 10 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ {(lastSβ€˜(π‘Š prefix 𝑁)), ((π‘Š prefix 𝑁)β€˜0)} = {(π‘Šβ€˜(𝑁 βˆ’ 1)), (π‘Šβ€˜0)})
92913adantl3 1168 . . . . . . . . 9 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1) ∧ βˆ€π‘– ∈ (0..^𝑀){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ {(lastSβ€˜(π‘Š prefix 𝑁)), ((π‘Š prefix 𝑁)β€˜0)} = {(π‘Šβ€˜(𝑁 βˆ’ 1)), (π‘Šβ€˜0)})
9392adantr 482 . . . . . . . 8 ((((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1) ∧ βˆ€π‘– ∈ (0..^𝑀){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) ∧ (π‘Šβ€˜π‘) = (π‘Šβ€˜0)) β†’ {(lastSβ€˜(π‘Š prefix 𝑁)), ((π‘Š prefix 𝑁)β€˜0)} = {(π‘Šβ€˜(𝑁 βˆ’ 1)), (π‘Šβ€˜0)})
94 fz1fzo0m1 13485 . . . . . . . . . . . . . . . 16 (𝑁 ∈ (1...𝑀) β†’ (𝑁 βˆ’ 1) ∈ (0..^𝑀))
9580, 94syl 17 . . . . . . . . . . . . . . 15 ((𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘)) β†’ (𝑁 βˆ’ 1) ∈ (0..^𝑀))
96953ad2ant3 1135 . . . . . . . . . . . . . 14 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ (𝑁 βˆ’ 1) ∈ (0..^𝑀))
97 simpr 486 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ β„• ∧ 𝑖 = (𝑁 βˆ’ 1)) β†’ 𝑖 = (𝑁 βˆ’ 1))
9897fveq2d 6808 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ β„• ∧ 𝑖 = (𝑁 βˆ’ 1)) β†’ (π‘Šβ€˜π‘–) = (π‘Šβ€˜(𝑁 βˆ’ 1)))
99 oveq1 7314 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 = (𝑁 βˆ’ 1) β†’ (𝑖 + 1) = ((𝑁 βˆ’ 1) + 1))
100 nncn 12031 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ β„• β†’ 𝑁 ∈ β„‚)
101 npcan1 11450 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ β„‚ β†’ ((𝑁 βˆ’ 1) + 1) = 𝑁)
102100, 101syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ β„• β†’ ((𝑁 βˆ’ 1) + 1) = 𝑁)
10399, 102sylan9eqr 2798 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ β„• ∧ 𝑖 = (𝑁 βˆ’ 1)) β†’ (𝑖 + 1) = 𝑁)
104103fveq2d 6808 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ β„• ∧ 𝑖 = (𝑁 βˆ’ 1)) β†’ (π‘Šβ€˜(𝑖 + 1)) = (π‘Šβ€˜π‘))
10598, 104preq12d 4681 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ β„• ∧ 𝑖 = (𝑁 βˆ’ 1)) β†’ {(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} = {(π‘Šβ€˜(𝑁 βˆ’ 1)), (π‘Šβ€˜π‘)})
106105eleq1d 2821 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ β„• ∧ 𝑖 = (𝑁 βˆ’ 1)) β†’ ({(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ↔ {(π‘Šβ€˜(𝑁 βˆ’ 1)), (π‘Šβ€˜π‘)} ∈ (Edgβ€˜πΊ)))
107106ex 414 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ β„• β†’ (𝑖 = (𝑁 βˆ’ 1) β†’ ({(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ↔ {(π‘Šβ€˜(𝑁 βˆ’ 1)), (π‘Šβ€˜π‘)} ∈ (Edgβ€˜πΊ))))
108107adantr 482 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘)) β†’ (𝑖 = (𝑁 βˆ’ 1) β†’ ({(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ↔ {(π‘Šβ€˜(𝑁 βˆ’ 1)), (π‘Šβ€˜π‘)} ∈ (Edgβ€˜πΊ))))
1091083ad2ant3 1135 . . . . . . . . . . . . . . 15 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ (𝑖 = (𝑁 βˆ’ 1) β†’ ({(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ↔ {(π‘Šβ€˜(𝑁 βˆ’ 1)), (π‘Šβ€˜π‘)} ∈ (Edgβ€˜πΊ))))
110109imp 408 . . . . . . . . . . . . . 14 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) ∧ 𝑖 = (𝑁 βˆ’ 1)) β†’ ({(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ↔ {(π‘Šβ€˜(𝑁 βˆ’ 1)), (π‘Šβ€˜π‘)} ∈ (Edgβ€˜πΊ)))
11196, 110rspcdv 3558 . . . . . . . . . . . . 13 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ (βˆ€π‘– ∈ (0..^𝑀){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) β†’ {(π‘Šβ€˜(𝑁 βˆ’ 1)), (π‘Šβ€˜π‘)} ∈ (Edgβ€˜πΊ)))
1121113exp 1119 . . . . . . . . . . . 12 (π‘Š ∈ Word (Vtxβ€˜πΊ) β†’ ((β™―β€˜π‘Š) = (𝑀 + 1) β†’ ((𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘)) β†’ (βˆ€π‘– ∈ (0..^𝑀){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) β†’ {(π‘Šβ€˜(𝑁 βˆ’ 1)), (π‘Šβ€˜π‘)} ∈ (Edgβ€˜πΊ)))))
113112com34 91 . . . . . . . . . . 11 (π‘Š ∈ Word (Vtxβ€˜πΊ) β†’ ((β™―β€˜π‘Š) = (𝑀 + 1) β†’ (βˆ€π‘– ∈ (0..^𝑀){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) β†’ ((𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘)) β†’ {(π‘Šβ€˜(𝑁 βˆ’ 1)), (π‘Šβ€˜π‘)} ∈ (Edgβ€˜πΊ)))))
1141133imp1 1347 . . . . . . . . . 10 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1) ∧ βˆ€π‘– ∈ (0..^𝑀){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ {(π‘Šβ€˜(𝑁 βˆ’ 1)), (π‘Šβ€˜π‘)} ∈ (Edgβ€˜πΊ))
115114adantr 482 . . . . . . . . 9 ((((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1) ∧ βˆ€π‘– ∈ (0..^𝑀){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) ∧ (π‘Šβ€˜π‘) = (π‘Šβ€˜0)) β†’ {(π‘Šβ€˜(𝑁 βˆ’ 1)), (π‘Šβ€˜π‘)} ∈ (Edgβ€˜πΊ))
116 preq2 4674 . . . . . . . . . . 11 ((π‘Šβ€˜π‘) = (π‘Šβ€˜0) β†’ {(π‘Šβ€˜(𝑁 βˆ’ 1)), (π‘Šβ€˜π‘)} = {(π‘Šβ€˜(𝑁 βˆ’ 1)), (π‘Šβ€˜0)})
117116eleq1d 2821 . . . . . . . . . 10 ((π‘Šβ€˜π‘) = (π‘Šβ€˜0) β†’ ({(π‘Šβ€˜(𝑁 βˆ’ 1)), (π‘Šβ€˜π‘)} ∈ (Edgβ€˜πΊ) ↔ {(π‘Šβ€˜(𝑁 βˆ’ 1)), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)))
118117adantl 483 . . . . . . . . 9 ((((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1) ∧ βˆ€π‘– ∈ (0..^𝑀){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) ∧ (π‘Šβ€˜π‘) = (π‘Šβ€˜0)) β†’ ({(π‘Šβ€˜(𝑁 βˆ’ 1)), (π‘Šβ€˜π‘)} ∈ (Edgβ€˜πΊ) ↔ {(π‘Šβ€˜(𝑁 βˆ’ 1)), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)))
119115, 118mpbid 231 . . . . . . . 8 ((((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1) ∧ βˆ€π‘– ∈ (0..^𝑀){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) ∧ (π‘Šβ€˜π‘) = (π‘Šβ€˜0)) β†’ {(π‘Šβ€˜(𝑁 βˆ’ 1)), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ))
12093, 119eqeltrd 2837 . . . . . . 7 ((((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1) ∧ βˆ€π‘– ∈ (0..^𝑀){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) ∧ (π‘Šβ€˜π‘) = (π‘Šβ€˜0)) β†’ {(lastSβ€˜(π‘Š prefix 𝑁)), ((π‘Š prefix 𝑁)β€˜0)} ∈ (Edgβ€˜πΊ))
12126, 79, 1203jca 1128 . . . . . 6 ((((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1) ∧ βˆ€π‘– ∈ (0..^𝑀){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) ∧ (π‘Šβ€˜π‘) = (π‘Šβ€˜0)) β†’ (((π‘Š prefix 𝑁) ∈ Word (Vtxβ€˜πΊ) ∧ (π‘Š prefix 𝑁) β‰  βˆ…) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜(π‘Š prefix 𝑁)) βˆ’ 1)){((π‘Š prefix 𝑁)β€˜π‘–), ((π‘Š prefix 𝑁)β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜(π‘Š prefix 𝑁)), ((π‘Š prefix 𝑁)β€˜0)} ∈ (Edgβ€˜πΊ)))
122121exp31 421 . . . . 5 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1) ∧ βˆ€π‘– ∈ (0..^𝑀){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) β†’ ((𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘)) β†’ ((π‘Šβ€˜π‘) = (π‘Šβ€˜0) β†’ (((π‘Š prefix 𝑁) ∈ Word (Vtxβ€˜πΊ) ∧ (π‘Š prefix 𝑁) β‰  βˆ…) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜(π‘Š prefix 𝑁)) βˆ’ 1)){((π‘Š prefix 𝑁)β€˜π‘–), ((π‘Š prefix 𝑁)β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜(π‘Š prefix 𝑁)), ((π‘Š prefix 𝑁)β€˜0)} ∈ (Edgβ€˜πΊ)))))
1231223imp21 1114 . . . 4 (((𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘)) ∧ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1) ∧ βˆ€π‘– ∈ (0..^𝑀){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ∧ (π‘Šβ€˜π‘) = (π‘Šβ€˜0)) β†’ (((π‘Š prefix 𝑁) ∈ Word (Vtxβ€˜πΊ) ∧ (π‘Š prefix 𝑁) β‰  βˆ…) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜(π‘Š prefix 𝑁)) βˆ’ 1)){((π‘Š prefix 𝑁)β€˜π‘–), ((π‘Š prefix 𝑁)β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜(π‘Š prefix 𝑁)), ((π‘Š prefix 𝑁)β€˜0)} ∈ (Edgβ€˜πΊ)))
1241, 2isclwwlk 28397 . . . 4 ((π‘Š prefix 𝑁) ∈ (ClWWalksβ€˜πΊ) ↔ (((π‘Š prefix 𝑁) ∈ Word (Vtxβ€˜πΊ) ∧ (π‘Š prefix 𝑁) β‰  βˆ…) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜(π‘Š prefix 𝑁)) βˆ’ 1)){((π‘Š prefix 𝑁)β€˜π‘–), ((π‘Š prefix 𝑁)β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜(π‘Š prefix 𝑁)), ((π‘Š prefix 𝑁)β€˜0)} ∈ (Edgβ€˜πΊ)))
125123, 124sylibr 233 . . 3 (((𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘)) ∧ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1) ∧ βˆ€π‘– ∈ (0..^𝑀){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ∧ (π‘Šβ€˜π‘) = (π‘Šβ€˜0)) β†’ (π‘Š prefix 𝑁) ∈ (ClWWalksβ€˜πΊ))
12647adantl 483 . . . . . . . . . . 11 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ 𝑁 ∈ (0...𝑀))
127126, 48syl 17 . . . . . . . . . 10 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ 𝑁 ∈ (0...(𝑀 + 1)))
128127, 53mpbird 257 . . . . . . . . 9 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ 𝑁 ∈ (0...(β™―β€˜π‘Š)))
1297, 128jca 513 . . . . . . . 8 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) ∧ (𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘))) β†’ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ 𝑁 ∈ (0...(β™―β€˜π‘Š))))
130129ex 414 . . . . . . 7 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1)) β†’ ((𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘)) β†’ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ 𝑁 ∈ (0...(β™―β€˜π‘Š)))))
1311303adant3 1132 . . . . . 6 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1) ∧ βˆ€π‘– ∈ (0..^𝑀){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) β†’ ((𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘)) β†’ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ 𝑁 ∈ (0...(β™―β€˜π‘Š)))))
132131impcom 409 . . . . 5 (((𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘)) ∧ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1) ∧ βˆ€π‘– ∈ (0..^𝑀){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ))) β†’ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ 𝑁 ∈ (0...(β™―β€˜π‘Š))))
1331323adant3 1132 . . . 4 (((𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘)) ∧ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1) ∧ βˆ€π‘– ∈ (0..^𝑀){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ∧ (π‘Šβ€˜π‘) = (π‘Šβ€˜0)) β†’ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ 𝑁 ∈ (0...(β™―β€˜π‘Š))))
134133, 55syl 17 . . 3 (((𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘)) ∧ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1) ∧ βˆ€π‘– ∈ (0..^𝑀){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ∧ (π‘Šβ€˜π‘) = (π‘Šβ€˜0)) β†’ (β™―β€˜(π‘Š prefix 𝑁)) = 𝑁)
135 isclwwlkn 28440 . . 3 ((π‘Š prefix 𝑁) ∈ (𝑁 ClWWalksN 𝐺) ↔ ((π‘Š prefix 𝑁) ∈ (ClWWalksβ€˜πΊ) ∧ (β™―β€˜(π‘Š prefix 𝑁)) = 𝑁))
136125, 134, 135sylanbrc 584 . 2 (((𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘)) ∧ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑀 + 1) ∧ βˆ€π‘– ∈ (0..^𝑀){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ∧ (π‘Šβ€˜π‘) = (π‘Šβ€˜0)) β†’ (π‘Š prefix 𝑁) ∈ (𝑁 ClWWalksN 𝐺))
1373, 136syl3an2 1164 1 (((𝑁 ∈ β„• ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘)) ∧ π‘Š ∈ (𝑀 WWalksN 𝐺) ∧ (π‘Šβ€˜π‘) = (π‘Šβ€˜0)) β†’ (π‘Š prefix 𝑁) ∈ (𝑁 ClWWalksN 𝐺))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1087   = wceq 1539   ∈ wcel 2104   β‰  wne 2941  βˆ€wral 3062   βŠ† wss 3892  βˆ…c0 4262  {cpr 4567   class class class wbr 5081  β€˜cfv 6458  (class class class)co 7307  β„‚cc 10919  β„cr 10920  0cc0 10921  1c1 10922   + caddc 10924   ≀ cle 11060   βˆ’ cmin 11255  β„•cn 12023  β„•0cn0 12283  β„€cz 12369  β„€β‰₯cuz 12632  ...cfz 13289  ..^cfzo 13432  β™―chash 14094  Word cword 14266  lastSclsw 14314   prefix cpfx 14432  Vtxcvtx 27415  Edgcedg 27466   WWalksN cwwlksn 28240  ClWWalkscclwwlk 28394   ClWWalksN cclwwlkn 28437
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 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620  ax-cnex 10977  ax-resscn 10978  ax-1cn 10979  ax-icn 10980  ax-addcl 10981  ax-addrcl 10982  ax-mulcl 10983  ax-mulrcl 10984  ax-mulcom 10985  ax-addass 10986  ax-mulass 10987  ax-distr 10988  ax-i2m1 10989  ax-1ne0 10990  ax-1rid 10991  ax-rnegex 10992  ax-rrecex 10993  ax-cnre 10994  ax-pre-lttri 10995  ax-pre-lttrn 10996  ax-pre-ltadd 10997  ax-pre-mulgt0 10998
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3305  df-rab 3306  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-int 4887  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5500  df-eprel 5506  df-po 5514  df-so 5515  df-fr 5555  df-we 5557  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-pred 6217  df-ord 6284  df-on 6285  df-lim 6286  df-suc 6287  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-riota 7264  df-ov 7310  df-oprab 7311  df-mpo 7312  df-om 7745  df-1st 7863  df-2nd 7864  df-frecs 8128  df-wrecs 8159  df-recs 8233  df-rdg 8272  df-1o 8328  df-er 8529  df-map 8648  df-en 8765  df-dom 8766  df-sdom 8767  df-fin 8768  df-card 9745  df-pnf 11061  df-mnf 11062  df-xr 11063  df-ltxr 11064  df-le 11065  df-sub 11257  df-neg 11258  df-nn 12024  df-n0 12284  df-z 12370  df-uz 12633  df-fz 13290  df-fzo 13433  df-hash 14095  df-word 14267  df-lsw 14315  df-substr 14403  df-pfx 14433  df-wwlks 28244  df-wwlksn 28245  df-clwwlk 28395  df-clwwlkn 28438
This theorem is referenced by:  clwwnrepclwwn  28757
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