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Theorem clwlkclwwlk 28995
Description: A closed walk as word of length at least 2 corresponds to a closed walk in a simple pseudograph. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 24-Apr-2021.) (Revised by AV, 30-Oct-2022.)
Hypotheses
Ref Expression
clwlkclwwlk.v 𝑉 = (Vtxβ€˜πΊ)
clwlkclwwlk.e 𝐸 = (iEdgβ€˜πΊ)
Assertion
Ref Expression
clwlkclwwlk ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (βˆƒπ‘“ 𝑓(ClWalksβ€˜πΊ)𝑃 ↔ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ (𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1)) ∈ (ClWWalksβ€˜πΊ))))
Distinct variable groups:   𝑓,𝐸   𝑃,𝑓   𝑓,𝑉   𝑓,𝐺

Proof of Theorem clwlkclwwlk
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 clwlkclwwlk.e . . . . . 6 𝐸 = (iEdgβ€˜πΊ)
21uspgrf1oedg 28173 . . . . 5 (𝐺 ∈ USPGraph β†’ 𝐸:dom 𝐸–1-1-ontoβ†’(Edgβ€˜πΊ))
3 f1of1 6787 . . . . 5 (𝐸:dom 𝐸–1-1-ontoβ†’(Edgβ€˜πΊ) β†’ 𝐸:dom 𝐸–1-1β†’(Edgβ€˜πΊ))
42, 3syl 17 . . . 4 (𝐺 ∈ USPGraph β†’ 𝐸:dom 𝐸–1-1β†’(Edgβ€˜πΊ))
5 clwlkclwwlklem3 28994 . . . 4 ((𝐸:dom 𝐸–1-1β†’(Edgβ€˜πΊ) ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (βˆƒπ‘“((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(β™―β€˜π‘“))βŸΆπ‘‰ ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))(πΈβ€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜π‘“))) ↔ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ (βˆ€π‘– ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜0)} ∈ ran 𝐸))))
64, 5syl3an1 1164 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (βˆƒπ‘“((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(β™―β€˜π‘“))βŸΆπ‘‰ ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))(πΈβ€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜π‘“))) ↔ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ (βˆ€π‘– ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜0)} ∈ ran 𝐸))))
7 lencl 14430 . . . . . . . . . . . . . 14 (𝑃 ∈ Word 𝑉 β†’ (β™―β€˜π‘ƒ) ∈ β„•0)
8 ige2m1fz 13540 . . . . . . . . . . . . . 14 (((β™―β€˜π‘ƒ) ∈ β„•0 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ (0...(β™―β€˜π‘ƒ)))
97, 8sylan 581 . . . . . . . . . . . . 13 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ (0...(β™―β€˜π‘ƒ)))
10 pfxlen 14580 . . . . . . . . . . . . 13 ((𝑃 ∈ Word 𝑉 ∧ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ (0...(β™―β€˜π‘ƒ))) β†’ (β™―β€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))) = ((β™―β€˜π‘ƒ) βˆ’ 1))
119, 10syldan 592 . . . . . . . . . . . 12 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (β™―β€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))) = ((β™―β€˜π‘ƒ) βˆ’ 1))
127nn0cnd 12483 . . . . . . . . . . . . . . . 16 (𝑃 ∈ Word 𝑉 β†’ (β™―β€˜π‘ƒ) ∈ β„‚)
13 1cnd 11158 . . . . . . . . . . . . . . . 16 (𝑃 ∈ Word 𝑉 β†’ 1 ∈ β„‚)
1412, 13subcld 11520 . . . . . . . . . . . . . . 15 (𝑃 ∈ Word 𝑉 β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ β„‚)
1514subid1d 11509 . . . . . . . . . . . . . 14 (𝑃 ∈ Word 𝑉 β†’ (((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) = ((β™―β€˜π‘ƒ) βˆ’ 1))
1615eqcomd 2739 . . . . . . . . . . . . 13 (𝑃 ∈ Word 𝑉 β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) = (((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0))
1716adantr 482 . . . . . . . . . . . 12 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) = (((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0))
1811, 17eqtrd 2773 . . . . . . . . . . 11 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (β™―β€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))) = (((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0))
1918oveq1d 7376 . . . . . . . . . 10 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((β™―β€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))) βˆ’ 1) = ((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1))
2019oveq2d 7377 . . . . . . . . 9 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (0..^((β™―β€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))) βˆ’ 1)) = (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)))
2111oveq1d 7376 . . . . . . . . . . . . 13 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((β™―β€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))) βˆ’ 1) = (((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 1))
2221oveq2d 7377 . . . . . . . . . . . 12 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (0..^((β™―β€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))) βˆ’ 1)) = (0..^(((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 1)))
2322eleq2d 2820 . . . . . . . . . . 11 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (𝑖 ∈ (0..^((β™―β€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))) βˆ’ 1)) ↔ 𝑖 ∈ (0..^(((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 1))))
24 simpll 766 . . . . . . . . . . . . . . 15 (((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝑖 ∈ (0..^(((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 1))) β†’ 𝑃 ∈ Word 𝑉)
25 wrdlenge2n0 14449 . . . . . . . . . . . . . . . 16 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ 𝑃 β‰  βˆ…)
2625adantr 482 . . . . . . . . . . . . . . 15 (((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝑖 ∈ (0..^(((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 1))) β†’ 𝑃 β‰  βˆ…)
27 nn0z 12532 . . . . . . . . . . . . . . . . . . 19 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (β™―β€˜π‘ƒ) ∈ β„€)
28 peano2zm 12554 . . . . . . . . . . . . . . . . . . 19 ((β™―β€˜π‘ƒ) ∈ β„€ β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ β„€)
2927, 28syl 17 . . . . . . . . . . . . . . . . . 18 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ β„€)
307, 29syl 17 . . . . . . . . . . . . . . . . 17 (𝑃 ∈ Word 𝑉 β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ β„€)
3130adantr 482 . . . . . . . . . . . . . . . 16 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ β„€)
32 elfzom1elfzo 13649 . . . . . . . . . . . . . . . 16 ((((β™―β€˜π‘ƒ) βˆ’ 1) ∈ β„€ ∧ 𝑖 ∈ (0..^(((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 1))) β†’ 𝑖 ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)))
3331, 32sylan 581 . . . . . . . . . . . . . . 15 (((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝑖 ∈ (0..^(((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 1))) β†’ 𝑖 ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)))
34 pfxtrcfv 14590 . . . . . . . . . . . . . . 15 ((𝑃 ∈ Word 𝑉 ∧ 𝑃 β‰  βˆ… ∧ 𝑖 ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1))) β†’ ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜π‘–) = (π‘ƒβ€˜π‘–))
3524, 26, 33, 34syl3anc 1372 . . . . . . . . . . . . . 14 (((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝑖 ∈ (0..^(((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 1))) β†’ ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜π‘–) = (π‘ƒβ€˜π‘–))
367adantr 482 . . . . . . . . . . . . . . . 16 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (β™―β€˜π‘ƒ) ∈ β„•0)
37 elfzom1elp1fzo 13648 . . . . . . . . . . . . . . . . 17 ((((β™―β€˜π‘ƒ) βˆ’ 1) ∈ β„€ ∧ 𝑖 ∈ (0..^(((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 1))) β†’ (𝑖 + 1) ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)))
3829, 37sylan 581 . . . . . . . . . . . . . . . 16 (((β™―β€˜π‘ƒ) ∈ β„•0 ∧ 𝑖 ∈ (0..^(((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 1))) β†’ (𝑖 + 1) ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)))
3936, 38sylan 581 . . . . . . . . . . . . . . 15 (((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝑖 ∈ (0..^(((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 1))) β†’ (𝑖 + 1) ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)))
40 pfxtrcfv 14590 . . . . . . . . . . . . . . 15 ((𝑃 ∈ Word 𝑉 ∧ 𝑃 β‰  βˆ… ∧ (𝑖 + 1) ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1))) β†’ ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜(𝑖 + 1)) = (π‘ƒβ€˜(𝑖 + 1)))
4124, 26, 39, 40syl3anc 1372 . . . . . . . . . . . . . 14 (((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝑖 ∈ (0..^(((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 1))) β†’ ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜(𝑖 + 1)) = (π‘ƒβ€˜(𝑖 + 1)))
4235, 41preq12d 4706 . . . . . . . . . . . . 13 (((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝑖 ∈ (0..^(((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 1))) β†’ {((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜π‘–), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜(𝑖 + 1))} = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})
4342eleq1d 2819 . . . . . . . . . . . 12 (((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝑖 ∈ (0..^(((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 1))) β†’ ({((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜π‘–), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜(𝑖 + 1))} ∈ ran 𝐸 ↔ {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸))
4443ex 414 . . . . . . . . . . 11 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (𝑖 ∈ (0..^(((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 1)) β†’ ({((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜π‘–), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜(𝑖 + 1))} ∈ ran 𝐸 ↔ {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸)))
4523, 44sylbid 239 . . . . . . . . . 10 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (𝑖 ∈ (0..^((β™―β€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))) βˆ’ 1)) β†’ ({((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜π‘–), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜(𝑖 + 1))} ∈ ran 𝐸 ↔ {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸)))
4645imp 408 . . . . . . . . 9 (((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝑖 ∈ (0..^((β™―β€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))) βˆ’ 1))) β†’ ({((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜π‘–), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜(𝑖 + 1))} ∈ ran 𝐸 ↔ {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸))
4720, 46raleqbidva 3320 . . . . . . . 8 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))) βˆ’ 1)){((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜π‘–), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜(𝑖 + 1))} ∈ ran 𝐸 ↔ βˆ€π‘– ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸))
48 pfxtrcfvl 14594 . . . . . . . . . 10 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (lastSβ€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))) = (π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)))
49 pfxtrcfv0 14591 . . . . . . . . . 10 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜0) = (π‘ƒβ€˜0))
5048, 49preq12d 4706 . . . . . . . . 9 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ {(lastSβ€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜0)} = {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜0)})
5150eleq1d 2819 . . . . . . . 8 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ({(lastSβ€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜0)} ∈ ran 𝐸 ↔ {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜0)} ∈ ran 𝐸))
5247, 51anbi12d 632 . . . . . . 7 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((βˆ€π‘– ∈ (0..^((β™―β€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))) βˆ’ 1)){((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜π‘–), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(lastSβ€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜0)} ∈ ran 𝐸) ↔ (βˆ€π‘– ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜0)} ∈ ran 𝐸)))
5352bicomd 222 . . . . . 6 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((βˆ€π‘– ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜0)} ∈ ran 𝐸) ↔ (βˆ€π‘– ∈ (0..^((β™―β€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))) βˆ’ 1)){((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜π‘–), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(lastSβ€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜0)} ∈ ran 𝐸)))
54533adant1 1131 . . . . 5 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((βˆ€π‘– ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜0)} ∈ ran 𝐸) ↔ (βˆ€π‘– ∈ (0..^((β™―β€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))) βˆ’ 1)){((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜π‘–), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(lastSβ€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜0)} ∈ ran 𝐸)))
55 pfxcl 14574 . . . . . . 7 (𝑃 ∈ Word 𝑉 β†’ (𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1)) ∈ Word 𝑉)
56553ad2ant2 1135 . . . . . 6 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1)) ∈ Word 𝑉)
57563biant1d 1479 . . . . 5 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((βˆ€π‘– ∈ (0..^((β™―β€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))) βˆ’ 1)){((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜π‘–), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(lastSβ€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜0)} ∈ ran 𝐸) ↔ ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1)) ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))) βˆ’ 1)){((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜π‘–), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(lastSβ€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜0)} ∈ ran 𝐸)))
5854, 57bitrd 279 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((βˆ€π‘– ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜0)} ∈ ran 𝐸) ↔ ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1)) ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))) βˆ’ 1)){((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜π‘–), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(lastSβ€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜0)} ∈ ran 𝐸)))
5958anbi2d 630 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ (βˆ€π‘– ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜0)} ∈ ran 𝐸)) ↔ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1)) ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))) βˆ’ 1)){((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜π‘–), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(lastSβ€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜0)} ∈ ran 𝐸))))
606, 59bitrd 279 . 2 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (βˆƒπ‘“((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(β™―β€˜π‘“))βŸΆπ‘‰ ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))(πΈβ€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜π‘“))) ↔ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1)) ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))) βˆ’ 1)){((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜π‘–), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(lastSβ€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜0)} ∈ ran 𝐸))))
61 uspgrupgr 28176 . . . . . 6 (𝐺 ∈ USPGraph β†’ 𝐺 ∈ UPGraph)
62 clwlkclwwlk.v . . . . . . . 8 𝑉 = (Vtxβ€˜πΊ)
6362, 1isclwlkupgr 28775 . . . . . . 7 (𝐺 ∈ UPGraph β†’ (𝑓(ClWalksβ€˜πΊ)𝑃 ↔ ((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(β™―β€˜π‘“))βŸΆπ‘‰) ∧ (βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))(πΈβ€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜π‘“))))))
64 3an4anass 1106 . . . . . . 7 (((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(β™―β€˜π‘“))βŸΆπ‘‰ ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))(πΈβ€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜π‘“))) ↔ ((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(β™―β€˜π‘“))βŸΆπ‘‰) ∧ (βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))(πΈβ€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜π‘“)))))
6563, 64bitr4di 289 . . . . . 6 (𝐺 ∈ UPGraph β†’ (𝑓(ClWalksβ€˜πΊ)𝑃 ↔ ((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(β™―β€˜π‘“))βŸΆπ‘‰ ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))(πΈβ€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜π‘“)))))
6661, 65syl 17 . . . . 5 (𝐺 ∈ USPGraph β†’ (𝑓(ClWalksβ€˜πΊ)𝑃 ↔ ((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(β™―β€˜π‘“))βŸΆπ‘‰ ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))(πΈβ€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜π‘“)))))
6766adantr 482 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉) β†’ (𝑓(ClWalksβ€˜πΊ)𝑃 ↔ ((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(β™―β€˜π‘“))βŸΆπ‘‰ ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))(πΈβ€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜π‘“)))))
6867exbidv 1925 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉) β†’ (βˆƒπ‘“ 𝑓(ClWalksβ€˜πΊ)𝑃 ↔ βˆƒπ‘“((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(β™―β€˜π‘“))βŸΆπ‘‰ ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))(πΈβ€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜π‘“)))))
69683adant3 1133 . 2 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (βˆƒπ‘“ 𝑓(ClWalksβ€˜πΊ)𝑃 ↔ βˆƒπ‘“((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(β™―β€˜π‘“))βŸΆπ‘‰ ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))(πΈβ€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜π‘“)))))
70 eqid 2733 . . . . . 6 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
7162, 70isclwwlk 28977 . . . . 5 ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1)) ∈ (ClWWalksβ€˜πΊ) ↔ (((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1)) ∈ Word 𝑉 ∧ (𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1)) β‰  βˆ…) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))) βˆ’ 1)){((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜π‘–), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜0)} ∈ (Edgβ€˜πΊ)))
72 simpl 484 . . . . . . . . . . 11 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ 𝑃 ∈ Word 𝑉)
73 nn0ge2m1nn 12490 . . . . . . . . . . . 12 (((β™―β€˜π‘ƒ) ∈ β„•0 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ β„•)
747, 73sylan 581 . . . . . . . . . . 11 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ β„•)
75 nn0re 12430 . . . . . . . . . . . . . . 15 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (β™―β€˜π‘ƒ) ∈ ℝ)
7675lem1d 12096 . . . . . . . . . . . . . 14 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) ≀ (β™―β€˜π‘ƒ))
7776a1d 25 . . . . . . . . . . . . 13 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (2 ≀ (β™―β€˜π‘ƒ) β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) ≀ (β™―β€˜π‘ƒ)))
787, 77syl 17 . . . . . . . . . . . 12 (𝑃 ∈ Word 𝑉 β†’ (2 ≀ (β™―β€˜π‘ƒ) β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) ≀ (β™―β€˜π‘ƒ)))
7978imp 408 . . . . . . . . . . 11 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) ≀ (β™―β€˜π‘ƒ))
8072, 74, 793jca 1129 . . . . . . . . . 10 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (𝑃 ∈ Word 𝑉 ∧ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ β„• ∧ ((β™―β€˜π‘ƒ) βˆ’ 1) ≀ (β™―β€˜π‘ƒ)))
81803adant1 1131 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (𝑃 ∈ Word 𝑉 ∧ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ β„• ∧ ((β™―β€˜π‘ƒ) βˆ’ 1) ≀ (β™―β€˜π‘ƒ)))
82 pfxn0 14583 . . . . . . . . 9 ((𝑃 ∈ Word 𝑉 ∧ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ β„• ∧ ((β™―β€˜π‘ƒ) βˆ’ 1) ≀ (β™―β€˜π‘ƒ)) β†’ (𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1)) β‰  βˆ…)
8381, 82syl 17 . . . . . . . 8 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1)) β‰  βˆ…)
8483biantrud 533 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1)) ∈ Word 𝑉 ↔ ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1)) ∈ Word 𝑉 ∧ (𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1)) β‰  βˆ…)))
8584bicomd 222 . . . . . 6 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1)) ∈ Word 𝑉 ∧ (𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1)) β‰  βˆ…) ↔ (𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1)) ∈ Word 𝑉))
86853anbi1d 1441 . . . . 5 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1)) ∈ Word 𝑉 ∧ (𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1)) β‰  βˆ…) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))) βˆ’ 1)){((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜π‘–), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜0)} ∈ (Edgβ€˜πΊ)) ↔ ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1)) ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))) βˆ’ 1)){((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜π‘–), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜0)} ∈ (Edgβ€˜πΊ))))
8771, 86bitrid 283 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1)) ∈ (ClWWalksβ€˜πΊ) ↔ ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1)) ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))) βˆ’ 1)){((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜π‘–), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜0)} ∈ (Edgβ€˜πΊ))))
88 biid 261 . . . . 5 ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1)) ∈ Word 𝑉 ↔ (𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1)) ∈ Word 𝑉)
89 edgval 28049 . . . . . . . 8 (Edgβ€˜πΊ) = ran (iEdgβ€˜πΊ)
901eqcomi 2742 . . . . . . . . 9 (iEdgβ€˜πΊ) = 𝐸
9190rneqi 5896 . . . . . . . 8 ran (iEdgβ€˜πΊ) = ran 𝐸
9289, 91eqtri 2761 . . . . . . 7 (Edgβ€˜πΊ) = ran 𝐸
9392eleq2i 2826 . . . . . 6 ({((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜π‘–), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ↔ {((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜π‘–), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜(𝑖 + 1))} ∈ ran 𝐸)
9493ralbii 3093 . . . . 5 (βˆ€π‘– ∈ (0..^((β™―β€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))) βˆ’ 1)){((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜π‘–), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ↔ βˆ€π‘– ∈ (0..^((β™―β€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))) βˆ’ 1)){((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜π‘–), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜(𝑖 + 1))} ∈ ran 𝐸)
9592eleq2i 2826 . . . . 5 ({(lastSβ€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜0)} ∈ (Edgβ€˜πΊ) ↔ {(lastSβ€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜0)} ∈ ran 𝐸)
9688, 94, 953anbi123i 1156 . . . 4 (((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1)) ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))) βˆ’ 1)){((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜π‘–), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜0)} ∈ (Edgβ€˜πΊ)) ↔ ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1)) ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))) βˆ’ 1)){((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜π‘–), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(lastSβ€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜0)} ∈ ran 𝐸))
9787, 96bitrdi 287 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1)) ∈ (ClWWalksβ€˜πΊ) ↔ ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1)) ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))) βˆ’ 1)){((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜π‘–), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(lastSβ€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜0)} ∈ ran 𝐸)))
9897anbi2d 630 . 2 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ (𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1)) ∈ (ClWWalksβ€˜πΊ)) ↔ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1)) ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))) βˆ’ 1)){((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜π‘–), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(lastSβ€˜(𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))), ((𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1))β€˜0)} ∈ ran 𝐸))))
9960, 69, 983bitr4d 311 1 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (βˆƒπ‘“ 𝑓(ClWalksβ€˜πΊ)𝑃 ↔ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ (𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1)) ∈ (ClWWalksβ€˜πΊ))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061  βˆ…c0 4286  {cpr 4592   class class class wbr 5109  dom cdm 5637  ran crn 5638  βŸΆwf 6496  β€“1-1β†’wf1 6497  β€“1-1-ontoβ†’wf1o 6499  β€˜cfv 6500  (class class class)co 7361  0cc0 11059  1c1 11060   + caddc 11062   ≀ cle 11198   βˆ’ cmin 11393  β„•cn 12161  2c2 12216  β„•0cn0 12421  β„€cz 12507  ...cfz 13433  ..^cfzo 13576  β™―chash 14239  Word cword 14411  lastSclsw 14459   prefix cpfx 14567  Vtxcvtx 27996  iEdgciedg 27997  Edgcedg 28047  UPGraphcupgr 28080  USPGraphcuspgr 28148  ClWalkscclwlks 28767  ClWWalkscclwwlk 28974
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ifp 1063  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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-2o 8417  df-oadd 8420  df-er 8654  df-map 8773  df-pm 8774  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-dju 9845  df-card 9883  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-2 12224  df-n0 12422  df-xnn0 12494  df-z 12508  df-uz 12772  df-fz 13434  df-fzo 13577  df-hash 14240  df-word 14412  df-lsw 14460  df-substr 14538  df-pfx 14568  df-edg 28048  df-uhgr 28058  df-upgr 28082  df-uspgr 28150  df-wlks 28596  df-clwlks 28768  df-clwwlk 28975
This theorem is referenced by:  clwlkclwwlk2  28996  clwlkclwwlkf  29001
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