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Theorem wwlksnext 29748
Description: Extension of a walk (as word) by adding an edge/vertex. (Contributed by Alexander van der Vekens, 4-Aug-2018.) (Revised by AV, 16-Apr-2021.)
Hypotheses
Ref Expression
wwlksnext.v 𝑉 = (Vtxβ€˜πΊ)
wwlksnext.e 𝐸 = (Edgβ€˜πΊ)
Assertion
Ref Expression
wwlksnext ((𝑇 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸) β†’ (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∈ ((𝑁 + 1) WWalksN 𝐺))

Proof of Theorem wwlksnext
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 wwlksnext.v . . . 4 𝑉 = (Vtxβ€˜πΊ)
21wwlknbp 29697 . . 3 (𝑇 ∈ (𝑁 WWalksN 𝐺) β†’ (𝐺 ∈ V ∧ 𝑁 ∈ β„•0 ∧ 𝑇 ∈ Word 𝑉))
3 wwlksnext.e . . . . . . . . . . 11 𝐸 = (Edgβ€˜πΊ)
41, 3wwlknp 29698 . . . . . . . . . 10 (𝑇 ∈ (𝑁 WWalksN 𝐺) β†’ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))} ∈ 𝐸))
5 simp1 1133 . . . . . . . . . . . . . . 15 ((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))} ∈ 𝐸) β†’ 𝑇 ∈ Word 𝑉)
6 simprl 769 . . . . . . . . . . . . . . 15 ((𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸)) β†’ 𝑆 ∈ 𝑉)
7 cats1un 14703 . . . . . . . . . . . . . . 15 ((𝑇 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉) β†’ (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) = (𝑇 βˆͺ {⟨(β™―β€˜π‘‡), π‘†βŸ©}))
85, 6, 7syl2an 594 . . . . . . . . . . . . . 14 (((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸))) β†’ (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) = (𝑇 βˆͺ {⟨(β™―β€˜π‘‡), π‘†βŸ©}))
9 opex 5460 . . . . . . . . . . . . . . . . . . 19 ⟨(β™―β€˜π‘‡), π‘†βŸ© ∈ V
109snnz 4776 . . . . . . . . . . . . . . . . . 18 {⟨(β™―β€˜π‘‡), π‘†βŸ©} β‰  βˆ…
1110neii 2932 . . . . . . . . . . . . . . . . 17 Β¬ {⟨(β™―β€˜π‘‡), π‘†βŸ©} = βˆ…
1211intnan 485 . . . . . . . . . . . . . . . 16 Β¬ (𝑇 = βˆ… ∧ {⟨(β™―β€˜π‘‡), π‘†βŸ©} = βˆ…)
13 df-ne 2931 . . . . . . . . . . . . . . . . 17 ((𝑇 βˆͺ {⟨(β™―β€˜π‘‡), π‘†βŸ©}) β‰  βˆ… ↔ Β¬ (𝑇 βˆͺ {⟨(β™―β€˜π‘‡), π‘†βŸ©}) = βˆ…)
14 un00 4438 . . . . . . . . . . . . . . . . 17 ((𝑇 = βˆ… ∧ {⟨(β™―β€˜π‘‡), π‘†βŸ©} = βˆ…) ↔ (𝑇 βˆͺ {⟨(β™―β€˜π‘‡), π‘†βŸ©}) = βˆ…)
1513, 14xchbinxr 334 . . . . . . . . . . . . . . . 16 ((𝑇 βˆͺ {⟨(β™―β€˜π‘‡), π‘†βŸ©}) β‰  βˆ… ↔ Β¬ (𝑇 = βˆ… ∧ {⟨(β™―β€˜π‘‡), π‘†βŸ©} = βˆ…))
1612, 15mpbir 230 . . . . . . . . . . . . . . 15 (𝑇 βˆͺ {⟨(β™―β€˜π‘‡), π‘†βŸ©}) β‰  βˆ…
1716a1i 11 . . . . . . . . . . . . . 14 (((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸))) β†’ (𝑇 βˆͺ {⟨(β™―β€˜π‘‡), π‘†βŸ©}) β‰  βˆ…)
188, 17eqnetrd 2998 . . . . . . . . . . . . 13 (((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸))) β†’ (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) β‰  βˆ…)
19 s1cl 14584 . . . . . . . . . . . . . . 15 (𝑆 ∈ 𝑉 β†’ βŸ¨β€œπ‘†β€βŸ© ∈ Word 𝑉)
2019ad2antrl 726 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸)) β†’ βŸ¨β€œπ‘†β€βŸ© ∈ Word 𝑉)
21 ccatcl 14556 . . . . . . . . . . . . . 14 ((𝑇 ∈ Word 𝑉 ∧ βŸ¨β€œπ‘†β€βŸ© ∈ Word 𝑉) β†’ (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∈ Word 𝑉)
225, 20, 21syl2an 594 . . . . . . . . . . . . 13 (((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸))) β†’ (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∈ Word 𝑉)
23 simplrl 775 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) β†’ 𝑇 ∈ Word 𝑉)
24 fzossfzop1 13742 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑁 ∈ β„•0 β†’ (0..^𝑁) βŠ† (0..^(𝑁 + 1)))
2524ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) β†’ (0..^𝑁) βŠ† (0..^(𝑁 + 1)))
2625sselda 3972 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) β†’ 𝑖 ∈ (0..^(𝑁 + 1)))
27 oveq2 7424 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((β™―β€˜π‘‡) = (𝑁 + 1) β†’ (0..^(β™―β€˜π‘‡)) = (0..^(𝑁 + 1)))
2827eleq2d 2811 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((β™―β€˜π‘‡) = (𝑁 + 1) β†’ (𝑖 ∈ (0..^(β™―β€˜π‘‡)) ↔ 𝑖 ∈ (0..^(𝑁 + 1))))
2928adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1)) β†’ (𝑖 ∈ (0..^(β™―β€˜π‘‡)) ↔ 𝑖 ∈ (0..^(𝑁 + 1))))
3029ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) β†’ (𝑖 ∈ (0..^(β™―β€˜π‘‡)) ↔ 𝑖 ∈ (0..^(𝑁 + 1))))
3126, 30mpbird 256 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) β†’ 𝑖 ∈ (0..^(β™―β€˜π‘‡)))
32 ccats1val1 14608 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑇 ∈ Word 𝑉 ∧ 𝑖 ∈ (0..^(β™―β€˜π‘‡))) β†’ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–) = (π‘‡β€˜π‘–))
3323, 31, 32syl2anc 582 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) β†’ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–) = (π‘‡β€˜π‘–))
34 fzonn0p1p1 13743 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑖 ∈ (0..^𝑁) β†’ (𝑖 + 1) ∈ (0..^(𝑁 + 1)))
3534adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) β†’ (𝑖 + 1) ∈ (0..^(𝑁 + 1)))
3627adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1)) β†’ (0..^(β™―β€˜π‘‡)) = (0..^(𝑁 + 1)))
3736ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) β†’ (0..^(β™―β€˜π‘‡)) = (0..^(𝑁 + 1)))
3835, 37eleqtrrd 2828 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) β†’ (𝑖 + 1) ∈ (0..^(β™―β€˜π‘‡)))
39 ccats1val1 14608 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑇 ∈ Word 𝑉 ∧ (𝑖 + 1) ∈ (0..^(β™―β€˜π‘‡))) β†’ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1)) = (π‘‡β€˜(𝑖 + 1)))
4023, 38, 39syl2anc 582 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) β†’ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1)) = (π‘‡β€˜(𝑖 + 1)))
4133, 40preq12d 4741 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) β†’ {((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} = {(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))})
4241exp31 418 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) β†’ ((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1)) β†’ (𝑖 ∈ (0..^𝑁) β†’ {((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} = {(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))})))
4342adantrr 715 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸)) β†’ ((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1)) β†’ (𝑖 ∈ (0..^𝑁) β†’ {((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} = {(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))})))
4443impcom 406 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1)) ∧ (𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸))) β†’ (𝑖 ∈ (0..^𝑁) β†’ {((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} = {(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))}))
4544imp 405 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1)) ∧ (𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸))) ∧ 𝑖 ∈ (0..^𝑁)) β†’ {((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} = {(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))})
4645eleq1d 2810 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1)) ∧ (𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸))) ∧ 𝑖 ∈ (0..^𝑁)) β†’ ({((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} ∈ 𝐸 ↔ {(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))} ∈ 𝐸))
4746ralbidva 3166 . . . . . . . . . . . . . . . . . . . 20 (((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1)) ∧ (𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸))) β†’ (βˆ€π‘– ∈ (0..^𝑁){((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} ∈ 𝐸 ↔ βˆ€π‘– ∈ (0..^𝑁){(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))} ∈ 𝐸))
4847exbiri 809 . . . . . . . . . . . . . . . . . . 19 ((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1)) β†’ ((𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸)) β†’ (βˆ€π‘– ∈ (0..^𝑁){(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))} ∈ 𝐸 β†’ βˆ€π‘– ∈ (0..^𝑁){((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} ∈ 𝐸)))
4948com23 86 . . . . . . . . . . . . . . . . . 18 ((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1)) β†’ (βˆ€π‘– ∈ (0..^𝑁){(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))} ∈ 𝐸 β†’ ((𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸)) β†’ βˆ€π‘– ∈ (0..^𝑁){((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} ∈ 𝐸)))
50493impia 1114 . . . . . . . . . . . . . . . . 17 ((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))} ∈ 𝐸) β†’ ((𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸)) β†’ βˆ€π‘– ∈ (0..^𝑁){((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} ∈ 𝐸))
5150imp 405 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸))) β†’ βˆ€π‘– ∈ (0..^𝑁){((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} ∈ 𝐸)
52 oveq1 7423 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((β™―β€˜π‘‡) = (𝑁 + 1) β†’ ((β™―β€˜π‘‡) βˆ’ 1) = ((𝑁 + 1) βˆ’ 1))
5352adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1)) β†’ ((β™―β€˜π‘‡) βˆ’ 1) = ((𝑁 + 1) βˆ’ 1))
54 nn0cn 12512 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ β„‚)
55 1cnd 11239 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑁 ∈ β„•0 β†’ 1 ∈ β„‚)
5654, 55pncand 11602 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑁 ∈ β„•0 β†’ ((𝑁 + 1) βˆ’ 1) = 𝑁)
5756adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ ((𝑁 + 1) βˆ’ 1) = 𝑁)
5853, 57sylan9eqr 2787 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) β†’ ((β™―β€˜π‘‡) βˆ’ 1) = 𝑁)
5958fveq2d 6896 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) β†’ (π‘‡β€˜((β™―β€˜π‘‡) βˆ’ 1)) = (π‘‡β€˜π‘))
60 lsw 14546 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑇 ∈ Word 𝑉 β†’ (lastSβ€˜π‘‡) = (π‘‡β€˜((β™―β€˜π‘‡) βˆ’ 1)))
6160ad2antrl 726 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) β†’ (lastSβ€˜π‘‡) = (π‘‡β€˜((β™―β€˜π‘‡) βˆ’ 1)))
62 simprl 769 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) β†’ 𝑇 ∈ Word 𝑉)
63 fzonn0p1 13741 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ (0..^(𝑁 + 1)))
6463ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) β†’ 𝑁 ∈ (0..^(𝑁 + 1)))
6527eleq2d 2811 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((β™―β€˜π‘‡) = (𝑁 + 1) β†’ (𝑁 ∈ (0..^(β™―β€˜π‘‡)) ↔ 𝑁 ∈ (0..^(𝑁 + 1))))
6665ad2antll 727 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) β†’ (𝑁 ∈ (0..^(β™―β€˜π‘‡)) ↔ 𝑁 ∈ (0..^(𝑁 + 1))))
6764, 66mpbird 256 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) β†’ 𝑁 ∈ (0..^(β™―β€˜π‘‡)))
68 ccats1val1 14608 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑇 ∈ Word 𝑉 ∧ 𝑁 ∈ (0..^(β™―β€˜π‘‡))) β†’ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘) = (π‘‡β€˜π‘))
6962, 67, 68syl2anc 582 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) β†’ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘) = (π‘‡β€˜π‘))
7059, 61, 693eqtr4d 2775 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) β†’ (lastSβ€˜π‘‡) = ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘))
71 simpll 765 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) β†’ 𝑆 ∈ 𝑉)
72 simprr 771 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) β†’ (β™―β€˜π‘‡) = (𝑁 + 1))
7372eqcomd 2731 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) β†’ (𝑁 + 1) = (β™―β€˜π‘‡))
74 ccats1val2 14609 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑇 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ (𝑁 + 1) = (β™―β€˜π‘‡)) β†’ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑁 + 1)) = 𝑆)
7574eqcomd 2731 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑇 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ (𝑁 + 1) = (β™―β€˜π‘‡)) β†’ 𝑆 = ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑁 + 1)))
7662, 71, 73, 75syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) β†’ 𝑆 = ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑁 + 1)))
7770, 76preq12d 4741 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) β†’ {(lastSβ€˜π‘‡), 𝑆} = {((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑁 + 1))})
7877eleq1d 2810 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) β†’ ({(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸 ↔ {((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑁 + 1))} ∈ 𝐸))
7978biimpcd 248 . . . . . . . . . . . . . . . . . . . . . 22 ({(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸 β†’ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) β†’ {((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑁 + 1))} ∈ 𝐸))
8079exp4c 431 . . . . . . . . . . . . . . . . . . . . 21 ({(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸 β†’ (𝑆 ∈ 𝑉 β†’ (𝑁 ∈ β„•0 β†’ ((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1)) β†’ {((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑁 + 1))} ∈ 𝐸))))
8180impcom 406 . . . . . . . . . . . . . . . . . . . 20 ((𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸) β†’ (𝑁 ∈ β„•0 β†’ ((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1)) β†’ {((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑁 + 1))} ∈ 𝐸)))
8281impcom 406 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸)) β†’ ((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1)) β†’ {((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑁 + 1))} ∈ 𝐸))
8382impcom 406 . . . . . . . . . . . . . . . . . 18 (((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1)) ∧ (𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸))) β†’ {((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑁 + 1))} ∈ 𝐸)
84833adantl3 1165 . . . . . . . . . . . . . . . . 17 (((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸))) β†’ {((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑁 + 1))} ∈ 𝐸)
85 fveq2 6892 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = 𝑁 β†’ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–) = ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘))
86 fvoveq1 7439 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = 𝑁 β†’ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1)) = ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑁 + 1)))
8785, 86preq12d 4741 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 𝑁 β†’ {((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} = {((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑁 + 1))})
8887eleq1d 2810 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 𝑁 β†’ ({((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} ∈ 𝐸 ↔ {((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑁 + 1))} ∈ 𝐸))
8988ralsng 4673 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ β„•0 β†’ (βˆ€π‘– ∈ {𝑁} {((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} ∈ 𝐸 ↔ {((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑁 + 1))} ∈ 𝐸))
9089ad2antrl 726 . . . . . . . . . . . . . . . . 17 (((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸))) β†’ (βˆ€π‘– ∈ {𝑁} {((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} ∈ 𝐸 ↔ {((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑁 + 1))} ∈ 𝐸))
9184, 90mpbird 256 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸))) β†’ βˆ€π‘– ∈ {𝑁} {((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} ∈ 𝐸)
92 ralunb 4185 . . . . . . . . . . . . . . . 16 (βˆ€π‘– ∈ ((0..^𝑁) βˆͺ {𝑁}){((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} ∈ 𝐸 ↔ (βˆ€π‘– ∈ (0..^𝑁){((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} ∈ 𝐸 ∧ βˆ€π‘– ∈ {𝑁} {((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} ∈ 𝐸))
9351, 91, 92sylanbrc 581 . . . . . . . . . . . . . . 15 (((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸))) β†’ βˆ€π‘– ∈ ((0..^𝑁) βˆͺ {𝑁}){((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} ∈ 𝐸)
94 elnn0uz 12897 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ β„•0 ↔ 𝑁 ∈ (β„€β‰₯β€˜0))
95 eluzfz2 13541 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ (β„€β‰₯β€˜0) β†’ 𝑁 ∈ (0...𝑁))
9694, 95sylbi 216 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ (0...𝑁))
97 fzelp1 13585 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ (0...𝑁) β†’ 𝑁 ∈ (0...(𝑁 + 1)))
98 fzosplit 13697 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ (0...(𝑁 + 1)) β†’ (0..^(𝑁 + 1)) = ((0..^𝑁) βˆͺ (𝑁..^(𝑁 + 1))))
9996, 97, 983syl 18 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ β„•0 β†’ (0..^(𝑁 + 1)) = ((0..^𝑁) βˆͺ (𝑁..^(𝑁 + 1))))
100 nn0z 12613 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ β„€)
101 fzosn 13735 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ β„€ β†’ (𝑁..^(𝑁 + 1)) = {𝑁})
102100, 101syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ β„•0 β†’ (𝑁..^(𝑁 + 1)) = {𝑁})
103102uneq2d 4156 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ β„•0 β†’ ((0..^𝑁) βˆͺ (𝑁..^(𝑁 + 1))) = ((0..^𝑁) βˆͺ {𝑁}))
10499, 103eqtrd 2765 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ β„•0 β†’ (0..^(𝑁 + 1)) = ((0..^𝑁) βˆͺ {𝑁}))
105104ad2antrl 726 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸))) β†’ (0..^(𝑁 + 1)) = ((0..^𝑁) βˆͺ {𝑁}))
106105raleqdv 3315 . . . . . . . . . . . . . . 15 (((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸))) β†’ (βˆ€π‘– ∈ (0..^(𝑁 + 1)){((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} ∈ 𝐸 ↔ βˆ€π‘– ∈ ((0..^𝑁) βˆͺ {𝑁}){((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} ∈ 𝐸))
10793, 106mpbird 256 . . . . . . . . . . . . . 14 (((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸))) β†’ βˆ€π‘– ∈ (0..^(𝑁 + 1)){((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} ∈ 𝐸)
108 ccatlen 14557 . . . . . . . . . . . . . . . . . . 19 ((𝑇 ∈ Word 𝑉 ∧ βŸ¨β€œπ‘†β€βŸ© ∈ Word 𝑉) β†’ (β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) = ((β™―β€˜π‘‡) + (β™―β€˜βŸ¨β€œπ‘†β€βŸ©)))
1095, 20, 108syl2an 594 . . . . . . . . . . . . . . . . . 18 (((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸))) β†’ (β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) = ((β™―β€˜π‘‡) + (β™―β€˜βŸ¨β€œπ‘†β€βŸ©)))
110109oveq1d 7431 . . . . . . . . . . . . . . . . 17 (((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸))) β†’ ((β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) βˆ’ 1) = (((β™―β€˜π‘‡) + (β™―β€˜βŸ¨β€œπ‘†β€βŸ©)) βˆ’ 1))
111 simpl2 1189 . . . . . . . . . . . . . . . . . . 19 (((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸))) β†’ (β™―β€˜π‘‡) = (𝑁 + 1))
112 s1len 14588 . . . . . . . . . . . . . . . . . . . 20 (β™―β€˜βŸ¨β€œπ‘†β€βŸ©) = 1
113112a1i 11 . . . . . . . . . . . . . . . . . . 19 (((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸))) β†’ (β™―β€˜βŸ¨β€œπ‘†β€βŸ©) = 1)
114111, 113oveq12d 7434 . . . . . . . . . . . . . . . . . 18 (((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸))) β†’ ((β™―β€˜π‘‡) + (β™―β€˜βŸ¨β€œπ‘†β€βŸ©)) = ((𝑁 + 1) + 1))
115114oveq1d 7431 . . . . . . . . . . . . . . . . 17 (((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸))) β†’ (((β™―β€˜π‘‡) + (β™―β€˜βŸ¨β€œπ‘†β€βŸ©)) βˆ’ 1) = (((𝑁 + 1) + 1) βˆ’ 1))
116 peano2nn0 12542 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„•0)
117116nn0cnd 12564 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„‚)
118117, 55pncand 11602 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ β„•0 β†’ (((𝑁 + 1) + 1) βˆ’ 1) = (𝑁 + 1))
119118ad2antrl 726 . . . . . . . . . . . . . . . . 17 (((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸))) β†’ (((𝑁 + 1) + 1) βˆ’ 1) = (𝑁 + 1))
120110, 115, 1193eqtrd 2769 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸))) β†’ ((β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) βˆ’ 1) = (𝑁 + 1))
121120oveq2d 7432 . . . . . . . . . . . . . . 15 (((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸))) β†’ (0..^((β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) βˆ’ 1)) = (0..^(𝑁 + 1)))
122121raleqdv 3315 . . . . . . . . . . . . . 14 (((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸))) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) βˆ’ 1)){((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} ∈ 𝐸 ↔ βˆ€π‘– ∈ (0..^(𝑁 + 1)){((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} ∈ 𝐸))
123107, 122mpbird 256 . . . . . . . . . . . . 13 (((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸))) β†’ βˆ€π‘– ∈ (0..^((β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) βˆ’ 1)){((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} ∈ 𝐸)
12418, 22, 1233jca 1125 . . . . . . . . . . . 12 (((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸))) β†’ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) β‰  βˆ… ∧ (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) βˆ’ 1)){((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} ∈ 𝐸))
125109, 114eqtrd 2765 . . . . . . . . . . . 12 (((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸))) β†’ (β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) = ((𝑁 + 1) + 1))
126124, 125jca 510 . . . . . . . . . . 11 (((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸))) β†’ (((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) β‰  βˆ… ∧ (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) βˆ’ 1)){((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) = ((𝑁 + 1) + 1)))
127126ex 411 . . . . . . . . . 10 ((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘‡β€˜π‘–), (π‘‡β€˜(𝑖 + 1))} ∈ 𝐸) β†’ ((𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸)) β†’ (((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) β‰  βˆ… ∧ (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) βˆ’ 1)){((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) = ((𝑁 + 1) + 1))))
1284, 127syl 17 . . . . . . . . 9 (𝑇 ∈ (𝑁 WWalksN 𝐺) β†’ ((𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸)) β†’ (((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) β‰  βˆ… ∧ (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) βˆ’ 1)){((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) = ((𝑁 + 1) + 1))))
129128expd 414 . . . . . . . 8 (𝑇 ∈ (𝑁 WWalksN 𝐺) β†’ (𝑁 ∈ β„•0 β†’ ((𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸) β†’ (((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) β‰  βˆ… ∧ (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) βˆ’ 1)){((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) = ((𝑁 + 1) + 1)))))
130129impcom 406 . . . . . . 7 ((𝑁 ∈ β„•0 ∧ 𝑇 ∈ (𝑁 WWalksN 𝐺)) β†’ ((𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸) β†’ (((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) β‰  βˆ… ∧ (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) βˆ’ 1)){((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) = ((𝑁 + 1) + 1))))
131130adantll 712 . . . . . 6 (((𝐺 ∈ V ∧ 𝑁 ∈ β„•0) ∧ 𝑇 ∈ (𝑁 WWalksN 𝐺)) β†’ ((𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸) β†’ (((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) β‰  βˆ… ∧ (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) βˆ’ 1)){((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) = ((𝑁 + 1) + 1))))
132 iswwlksn 29693 . . . . . . . . . 10 ((𝑁 + 1) ∈ β„•0 β†’ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∈ (WWalksβ€˜πΊ) ∧ (β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) = ((𝑁 + 1) + 1))))
133116, 132syl 17 . . . . . . . . 9 (𝑁 ∈ β„•0 β†’ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∈ (WWalksβ€˜πΊ) ∧ (β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) = ((𝑁 + 1) + 1))))
134133adantl 480 . . . . . . . 8 ((𝐺 ∈ V ∧ 𝑁 ∈ β„•0) β†’ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∈ (WWalksβ€˜πΊ) ∧ (β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) = ((𝑁 + 1) + 1))))
1351, 3iswwlks 29691 . . . . . . . . 9 ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∈ (WWalksβ€˜πΊ) ↔ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) β‰  βˆ… ∧ (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) βˆ’ 1)){((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} ∈ 𝐸))
136135anbi1i 622 . . . . . . . 8 (((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∈ (WWalksβ€˜πΊ) ∧ (β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) = ((𝑁 + 1) + 1)) ↔ (((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) β‰  βˆ… ∧ (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) βˆ’ 1)){((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) = ((𝑁 + 1) + 1)))
137134, 136bitrdi 286 . . . . . . 7 ((𝐺 ∈ V ∧ 𝑁 ∈ β„•0) β†’ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ (((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) β‰  βˆ… ∧ (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) βˆ’ 1)){((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) = ((𝑁 + 1) + 1))))
138137adantr 479 . . . . . 6 (((𝐺 ∈ V ∧ 𝑁 ∈ β„•0) ∧ 𝑇 ∈ (𝑁 WWalksN 𝐺)) β†’ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ (((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) β‰  βˆ… ∧ (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) βˆ’ 1)){((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–), ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1))} ∈ 𝐸) ∧ (β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) = ((𝑁 + 1) + 1))))
139131, 138sylibrd 258 . . . . 5 (((𝐺 ∈ V ∧ 𝑁 ∈ β„•0) ∧ 𝑇 ∈ (𝑁 WWalksN 𝐺)) β†’ ((𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸) β†’ (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∈ ((𝑁 + 1) WWalksN 𝐺)))
140139ex 411 . . . 4 ((𝐺 ∈ V ∧ 𝑁 ∈ β„•0) β†’ (𝑇 ∈ (𝑁 WWalksN 𝐺) β†’ ((𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸) β†’ (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∈ ((𝑁 + 1) WWalksN 𝐺))))
1411403adant3 1129 . . 3 ((𝐺 ∈ V ∧ 𝑁 ∈ β„•0 ∧ 𝑇 ∈ Word 𝑉) β†’ (𝑇 ∈ (𝑁 WWalksN 𝐺) β†’ ((𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸) β†’ (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∈ ((𝑁 + 1) WWalksN 𝐺))))
1422, 141mpcom 38 . 2 (𝑇 ∈ (𝑁 WWalksN 𝐺) β†’ ((𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸) β†’ (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∈ ((𝑁 + 1) WWalksN 𝐺)))
1431423impib 1113 1 ((𝑇 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸) β†’ (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∈ ((𝑁 + 1) WWalksN 𝐺))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2930  βˆ€wral 3051  Vcvv 3463   βˆͺ cun 3937   βŠ† wss 3939  βˆ…c0 4318  {csn 4624  {cpr 4626  βŸ¨cop 4630  β€˜cfv 6543  (class class class)co 7416  0cc0 11138  1c1 11139   + caddc 11141   βˆ’ cmin 11474  β„•0cn0 12502  β„€cz 12588  β„€β‰₯cuz 12852  ...cfz 13516  ..^cfzo 13659  β™―chash 14321  Word cword 14496  lastSclsw 14544   ++ cconcat 14552  βŸ¨β€œcs1 14577  Vtxcvtx 28853  Edgcedg 28904  WWalkscwwlks 29680   WWalksN cwwlksn 29681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  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 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7991  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8723  df-map 8845  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-card 9962  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-nn 12243  df-n0 12503  df-z 12589  df-uz 12853  df-fz 13517  df-fzo 13660  df-hash 14322  df-word 14497  df-lsw 14545  df-concat 14553  df-s1 14578  df-wwlks 29685  df-wwlksn 29686
This theorem is referenced by:  wwlksnextbi  29749  wwlksnextsurj  29755
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