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Theorem wwlksnext 28880
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 28829 . . 3 (𝑇 ∈ (𝑁 WWalksN 𝐺) β†’ (𝐺 ∈ V ∧ 𝑁 ∈ β„•0 ∧ 𝑇 ∈ Word 𝑉))
3 wwlksnext.e . . . . . . . . . . 11 𝐸 = (Edgβ€˜πΊ)
41, 3wwlknp 28830 . . . . . . . . . 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 1137 . . . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . . 15 ((𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸)) β†’ 𝑆 ∈ 𝑉)
7 cats1un 14616 . . . . . . . . . . . . . . 15 ((𝑇 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉) β†’ (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) = (𝑇 βˆͺ {⟨(β™―β€˜π‘‡), π‘†βŸ©}))
85, 6, 7syl2an 597 . . . . . . . . . . . . . 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 5426 . . . . . . . . . . . . . . . . . . 19 ⟨(β™―β€˜π‘‡), π‘†βŸ© ∈ V
109snnz 4742 . . . . . . . . . . . . . . . . . 18 {⟨(β™―β€˜π‘‡), π‘†βŸ©} β‰  βˆ…
1110neii 2946 . . . . . . . . . . . . . . . . 17 Β¬ {⟨(β™―β€˜π‘‡), π‘†βŸ©} = βˆ…
1211intnan 488 . . . . . . . . . . . . . . . 16 Β¬ (𝑇 = βˆ… ∧ {⟨(β™―β€˜π‘‡), π‘†βŸ©} = βˆ…)
13 df-ne 2945 . . . . . . . . . . . . . . . . 17 ((𝑇 βˆͺ {⟨(β™―β€˜π‘‡), π‘†βŸ©}) β‰  βˆ… ↔ Β¬ (𝑇 βˆͺ {⟨(β™―β€˜π‘‡), π‘†βŸ©}) = βˆ…)
14 un00 4407 . . . . . . . . . . . . . . . . 17 ((𝑇 = βˆ… ∧ {⟨(β™―β€˜π‘‡), π‘†βŸ©} = βˆ…) ↔ (𝑇 βˆͺ {⟨(β™―β€˜π‘‡), π‘†βŸ©}) = βˆ…)
1513, 14xchbinxr 335 . . . . . . . . . . . . . . . 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 3012 . . . . . . . . . . . . 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 14497 . . . . . . . . . . . . . . 15 (𝑆 ∈ 𝑉 β†’ βŸ¨β€œπ‘†β€βŸ© ∈ Word 𝑉)
2019ad2antrl 727 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„•0 ∧ (𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸)) β†’ βŸ¨β€œπ‘†β€βŸ© ∈ Word 𝑉)
21 ccatcl 14469 . . . . . . . . . . . . . 14 ((𝑇 ∈ Word 𝑉 ∧ βŸ¨β€œπ‘†β€βŸ© ∈ Word 𝑉) β†’ (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∈ Word 𝑉)
225, 20, 21syl2an 597 . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) β†’ 𝑇 ∈ Word 𝑉)
24 fzossfzop1 13657 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑁 ∈ β„•0 β†’ (0..^𝑁) βŠ† (0..^(𝑁 + 1)))
2524ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) β†’ (0..^𝑁) βŠ† (0..^(𝑁 + 1)))
2625sselda 3949 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) β†’ 𝑖 ∈ (0..^(𝑁 + 1)))
27 oveq2 7370 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((β™―β€˜π‘‡) = (𝑁 + 1) β†’ (0..^(β™―β€˜π‘‡)) = (0..^(𝑁 + 1)))
2827eleq2d 2824 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((β™―β€˜π‘‡) = (𝑁 + 1) β†’ (𝑖 ∈ (0..^(β™―β€˜π‘‡)) ↔ 𝑖 ∈ (0..^(𝑁 + 1))))
2928adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1)) β†’ (𝑖 ∈ (0..^(β™―β€˜π‘‡)) ↔ 𝑖 ∈ (0..^(𝑁 + 1))))
3029ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 14521 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑇 ∈ Word 𝑉 ∧ 𝑖 ∈ (0..^(β™―β€˜π‘‡))) β†’ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–) = (π‘‡β€˜π‘–))
3323, 31, 32syl2anc 585 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 13658 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑖 ∈ (0..^𝑁) β†’ (𝑖 + 1) ∈ (0..^(𝑁 + 1)))
3534adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) β†’ (𝑖 + 1) ∈ (0..^(𝑁 + 1)))
3627adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1)) β†’ (0..^(β™―β€˜π‘‡)) = (0..^(𝑁 + 1)))
3736ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2841 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 14521 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 585 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4707 . . . . . . . . . . . . . . . . . . . . . . . . . 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 421 . . . . . . . . . . . . . . . . . . . . . . . . 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 716 . . . . . . . . . . . . . . . . . . . . . . . 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 409 . . . . . . . . . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . . . . . . . . . . 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 2823 . . . . . . . . . . . . . . . . . . . . 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 3173 . . . . . . . . . . . . . . . . . . . 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 810 . . . . . . . . . . . . . . . . . . 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 1118 . . . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . . . . 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 7369 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((β™―β€˜π‘‡) = (𝑁 + 1) β†’ ((β™―β€˜π‘‡) βˆ’ 1) = ((𝑁 + 1) βˆ’ 1))
5352adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1)) β†’ ((β™―β€˜π‘‡) βˆ’ 1) = ((𝑁 + 1) βˆ’ 1))
54 nn0cn 12430 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ β„‚)
55 1cnd 11157 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑁 ∈ β„•0 β†’ 1 ∈ β„‚)
5654, 55pncand 11520 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑁 ∈ β„•0 β†’ ((𝑁 + 1) βˆ’ 1) = 𝑁)
5756adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ ((𝑁 + 1) βˆ’ 1) = 𝑁)
5853, 57sylan9eqr 2799 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) β†’ ((β™―β€˜π‘‡) βˆ’ 1) = 𝑁)
5958fveq2d 6851 . . . . . . . . . . . . . . . . . . . . . . . . . 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 14459 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑇 ∈ Word 𝑉 β†’ (lastSβ€˜π‘‡) = (π‘‡β€˜((β™―β€˜π‘‡) βˆ’ 1)))
6160ad2antrl 727 . . . . . . . . . . . . . . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) β†’ 𝑇 ∈ Word 𝑉)
63 fzonn0p1 13656 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ (0..^(𝑁 + 1)))
6463ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) β†’ 𝑁 ∈ (0..^(𝑁 + 1)))
6527eleq2d 2824 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((β™―β€˜π‘‡) = (𝑁 + 1) β†’ (𝑁 ∈ (0..^(β™―β€˜π‘‡)) ↔ 𝑁 ∈ (0..^(𝑁 + 1))))
6665ad2antll 728 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) β†’ 𝑁 ∈ (0..^(β™―β€˜π‘‡)))
68 ccats1val1 14521 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑇 ∈ Word 𝑉 ∧ 𝑁 ∈ (0..^(β™―β€˜π‘‡))) β†’ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘) = (π‘‡β€˜π‘))
6962, 67, 68syl2anc 585 . . . . . . . . . . . . . . . . . . . . . . . . . 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 2787 . . . . . . . . . . . . . . . . . . . . . . . . 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 766 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) β†’ 𝑆 ∈ 𝑉)
72 simprr 772 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) β†’ (β™―β€˜π‘‡) = (𝑁 + 1))
7372eqcomd 2743 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) ∧ (𝑇 ∈ Word 𝑉 ∧ (β™―β€˜π‘‡) = (𝑁 + 1))) β†’ (𝑁 + 1) = (β™―β€˜π‘‡))
74 ccats1val2 14522 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑇 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ (𝑁 + 1) = (β™―β€˜π‘‡)) β†’ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑁 + 1)) = 𝑆)
7574eqcomd 2743 . . . . . . . . . . . . . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . . . . . . . . . . . . . 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 4707 . . . . . . . . . . . . . . . . . . . . . . . 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 2823 . . . . . . . . . . . . . . . . . . . . . . 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 249 . . . . . . . . . . . . . . . . . . . . . 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 434 . . . . . . . . . . . . . . . . . . . . 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 409 . . . . . . . . . . . . . . . . . . . 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 409 . . . . . . . . . . . . . . . . . . 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 409 . . . . . . . . . . . . . . . . . 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 1169 . . . . . . . . . . . . . . . . 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 6847 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = 𝑁 β†’ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘–) = ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜π‘))
86 fvoveq1 7385 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = 𝑁 β†’ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑖 + 1)) = ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)β€˜(𝑁 + 1)))
8785, 86preq12d 4707 . . . . . . . . . . . . . . . . . . . 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 2823 . . . . . . . . . . . . . . . . . . 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 4639 . . . . . . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . . . . 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 4156 . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . 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 12815 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ β„•0 ↔ 𝑁 ∈ (β„€β‰₯β€˜0))
95 eluzfz2 13456 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ (β„€β‰₯β€˜0) β†’ 𝑁 ∈ (0...𝑁))
9694, 95sylbi 216 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ (0...𝑁))
97 fzelp1 13500 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ (0...𝑁) β†’ 𝑁 ∈ (0...(𝑁 + 1)))
98 fzosplit 13612 . . . . . . . . . . . . . . . . . . 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 12531 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ β„€)
101 fzosn 13650 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ β„€ β†’ (𝑁..^(𝑁 + 1)) = {𝑁})
102100, 101syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ β„•0 β†’ (𝑁..^(𝑁 + 1)) = {𝑁})
103102uneq2d 4128 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ β„•0 β†’ ((0..^𝑁) βˆͺ (𝑁..^(𝑁 + 1))) = ((0..^𝑁) βˆͺ {𝑁}))
10499, 103eqtrd 2777 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ β„•0 β†’ (0..^(𝑁 + 1)) = ((0..^𝑁) βˆͺ {𝑁}))
105104ad2antrl 727 . . . . . . . . . . . . . . . 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 3316 . . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . . 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 14470 . . . . . . . . . . . . . . . . . . 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 597 . . . . . . . . . . . . . . . . . 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 7377 . . . . . . . . . . . . . . . . 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 1193 . . . . . . . . . . . . . . . . . . 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 14501 . . . . . . . . . . . . . . . . . . . 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 7380 . . . . . . . . . . . . . . . . . 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 7377 . . . . . . . . . . . . . . . . 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 12460 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„•0)
117116nn0cnd 12482 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„‚)
118117, 55pncand 11520 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ β„•0 β†’ (((𝑁 + 1) + 1) βˆ’ 1) = (𝑁 + 1))
119118ad2antrl 727 . . . . . . . . . . . . . . . . 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 2781 . . . . . . . . . . . . . . . 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 7378 . . . . . . . . . . . . . . 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 3316 . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . 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 1129 . . . . . . . . . . . 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 2777 . . . . . . . . . . . 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 513 . . . . . . . . . . 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 414 . . . . . . . . . 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 417 . . . . . . . 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 409 . . . . . . 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 713 . . . . . 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 28825 . . . . . . . . . 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 483 . . . . . . . 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 28823 . . . . . . . . 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 625 . . . . . . . 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 287 . . . . . . 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 482 . . . . . 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 259 . . . . 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 414 . . . 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 1133 . . 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 1117 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆ€wral 3065  Vcvv 3448   βˆͺ cun 3913   βŠ† wss 3915  βˆ…c0 4287  {csn 4591  {cpr 4593  βŸ¨cop 4597  β€˜cfv 6501  (class class class)co 7362  0cc0 11058  1c1 11059   + caddc 11061   βˆ’ cmin 11392  β„•0cn0 12420  β„€cz 12506  β„€β‰₯cuz 12770  ...cfz 13431  ..^cfzo 13574  β™―chash 14237  Word cword 14409  lastSclsw 14457   ++ cconcat 14465  βŸ¨β€œcs1 14490  Vtxcvtx 27989  Edgcedg 28040  WWalkscwwlks 28812   WWalksN cwwlksn 28813
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-n0 12421  df-z 12507  df-uz 12771  df-fz 13432  df-fzo 13575  df-hash 14238  df-word 14410  df-lsw 14458  df-concat 14466  df-s1 14491  df-wwlks 28817  df-wwlksn 28818
This theorem is referenced by:  wwlksnextbi  28881  wwlksnextsurj  28887
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