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Theorem clwlkclwwlk 29523
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 28701 . . . . 5 (𝐺 ∈ USPGraph β†’ 𝐸:dom 𝐸–1-1-ontoβ†’(Edgβ€˜πΊ))
3 f1of1 6832 . . . . 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 29522 . . . 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 1162 . . 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 14488 . . . . . . . . . . . . . 14 (𝑃 ∈ Word 𝑉 β†’ (β™―β€˜π‘ƒ) ∈ β„•0)
8 ige2m1fz 13596 . . . . . . . . . . . . . 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 579 . . . . . . . . . . . . 13 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ (0...(β™―β€˜π‘ƒ)))
10 pfxlen 14638 . . . . . . . . . . . . 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 590 . . . . . . . . . . . 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 12539 . . . . . . . . . . . . . . . 16 (𝑃 ∈ Word 𝑉 β†’ (β™―β€˜π‘ƒ) ∈ β„‚)
13 1cnd 11214 . . . . . . . . . . . . . . . 16 (𝑃 ∈ Word 𝑉 β†’ 1 ∈ β„‚)
1412, 13subcld 11576 . . . . . . . . . . . . . . 15 (𝑃 ∈ Word 𝑉 β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ β„‚)
1514subid1d 11565 . . . . . . . . . . . . . 14 (𝑃 ∈ Word 𝑉 β†’ (((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) = ((β™―β€˜π‘ƒ) βˆ’ 1))
1615eqcomd 2737 . . . . . . . . . . . . 13 (𝑃 ∈ Word 𝑉 β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) = (((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0))
1716adantr 480 . . . . . . . . . . . 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 2771 . . . . . . . . . . 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 7427 . . . . . . . . . 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 7428 . . . . . . . . 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 7427 . . . . . . . . . . . . 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 7428 . . . . . . . . . . . 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 2818 . . . . . . . . . . 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 764 . . . . . . . . . . . . . . 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 14507 . . . . . . . . . . . . . . . 16 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ 𝑃 β‰  βˆ…)
2625adantr 480 . . . . . . . . . . . . . . 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 12588 . . . . . . . . . . . . . . . . . . 19 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (β™―β€˜π‘ƒ) ∈ β„€)
28 peano2zm 12610 . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . 16 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ β„€)
32 elfzom1elfzo 13705 . . . . . . . . . . . . . . . 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 579 . . . . . . . . . . . . . . 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 14648 . . . . . . . . . . . . . . 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 1370 . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . 16 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (β™―β€˜π‘ƒ) ∈ β„•0)
37 elfzom1elp1fzo 13704 . . . . . . . . . . . . . . . . 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 579 . . . . . . . . . . . . . . . 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 579 . . . . . . . . . . . . . . 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 14648 . . . . . . . . . . . . . . 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 1370 . . . . . . . . . . . . . 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 4745 . . . . . . . . . . . . 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 2817 . . . . . . . . . . . 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 412 . . . . . . . . . . 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 406 . . . . . . . . 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 3326 . . . . . . . 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 14652 . . . . . . . . . 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 14649 . . . . . . . . . 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 4745 . . . . . . . . 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 2817 . . . . . . . 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 630 . . . . . . 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 1129 . . . . 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 14632 . . . . . . 7 (𝑃 ∈ Word 𝑉 β†’ (𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1)) ∈ Word 𝑉)
56553ad2ant2 1133 . . . . . 6 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (𝑃 prefix ((β™―β€˜π‘ƒ) βˆ’ 1)) ∈ Word 𝑉)
57563biant1d 1477 . . . . 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 628 . . 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 28704 . . . . . 6 (𝐺 ∈ USPGraph β†’ 𝐺 ∈ UPGraph)
62 clwlkclwwlk.v . . . . . . . 8 𝑉 = (Vtxβ€˜πΊ)
6362, 1isclwlkupgr 29303 . . . . . . 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 1104 . . . . . . 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 480 . . . 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 1923 . . 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 1131 . 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 2731 . . . . . 6 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
7162, 70isclwwlk 29505 . . . . 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 482 . . . . . . . . . . 11 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ 𝑃 ∈ Word 𝑉)
73 nn0ge2m1nn 12546 . . . . . . . . . . . 12 (((β™―β€˜π‘ƒ) ∈ β„•0 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ β„•)
747, 73sylan 579 . . . . . . . . . . 11 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ β„•)
75 nn0re 12486 . . . . . . . . . . . . . . 15 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (β™―β€˜π‘ƒ) ∈ ℝ)
7675lem1d 12152 . . . . . . . . . . . . . 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 406 . . . . . . . . . . 11 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) ≀ (β™―β€˜π‘ƒ))
8072, 74, 793jca 1127 . . . . . . . . . 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 1129 . . . . . . . . 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 14641 . . . . . . . . 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 531 . . . . . . 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 1439 . . . . 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 28577 . . . . . . . 8 (Edgβ€˜πΊ) = ran (iEdgβ€˜πΊ)
901eqcomi 2740 . . . . . . . . 9 (iEdgβ€˜πΊ) = 𝐸
9190rneqi 5936 . . . . . . . 8 ran (iEdgβ€˜πΊ) = ran 𝐸
9289, 91eqtri 2759 . . . . . . 7 (Edgβ€˜πΊ) = ran 𝐸
9392eleq2i 2824 . . . . . 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 3092 . . . . 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 2824 . . . . 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 1154 . . . 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 628 . 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 395   ∧ w3a 1086   = wceq 1540  βˆƒwex 1780   ∈ wcel 2105   β‰  wne 2939  βˆ€wral 3060  βˆ…c0 4322  {cpr 4630   class class class wbr 5148  dom cdm 5676  ran crn 5677  βŸΆwf 6539  β€“1-1β†’wf1 6540  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  (class class class)co 7412  0cc0 11114  1c1 11115   + caddc 11117   ≀ cle 11254   βˆ’ cmin 11449  β„•cn 12217  2c2 12272  β„•0cn0 12477  β„€cz 12563  ...cfz 13489  ..^cfzo 13632  β™―chash 14295  Word cword 14469  lastSclsw 14517   prefix cpfx 14625  Vtxcvtx 28524  iEdgciedg 28525  Edgcedg 28575  UPGraphcupgr 28608  USPGraphcuspgr 28676  ClWalkscclwlks 29295  ClWWalkscclwwlk 29502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  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
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ifp 1061  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  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-2o 8471  df-oadd 8474  df-er 8707  df-map 8826  df-pm 8827  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-dju 9900  df-card 9938  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-2 12280  df-n0 12478  df-xnn0 12550  df-z 12564  df-uz 12828  df-fz 13490  df-fzo 13633  df-hash 14296  df-word 14470  df-lsw 14518  df-substr 14596  df-pfx 14626  df-edg 28576  df-uhgr 28586  df-upgr 28610  df-uspgr 28678  df-wlks 29124  df-clwlks 29296  df-clwwlk 29503
This theorem is referenced by:  clwlkclwwlk2  29524  clwlkclwwlkf  29529
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