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Theorem wwlksnextbi 29412
Description: Extension of a walk (as word) by adding an edge/vertex. (Contributed by Alexander van der Vekens, 5-Aug-2018.) (Revised by AV, 16-Apr-2021.) (Proof shortened by AV, 27-Oct-2022.)
Hypotheses
Ref Expression
wwlksnext.v 𝑉 = (Vtxβ€˜πΊ)
wwlksnext.e 𝐸 = (Edgβ€˜πΊ)
Assertion
Ref Expression
wwlksnextbi (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ π‘Š = (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸)) β†’ (π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ 𝑇 ∈ (𝑁 WWalksN 𝐺)))

Proof of Theorem wwlksnextbi
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 wwlksnext.v . . . . 5 𝑉 = (Vtxβ€˜πΊ)
2 wwlksnext.e . . . . 5 𝐸 = (Edgβ€˜πΊ)
31, 2wwlknp 29361 . . . 4 (π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺) β†’ (π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = ((𝑁 + 1) + 1) ∧ βˆ€π‘– ∈ (0..^(𝑁 + 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ 𝐸))
4 wwlksnred 29410 . . . . . . . 8 (𝑁 ∈ β„•0 β†’ (π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺) β†’ (π‘Š prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺)))
54ad2antrr 723 . . . . . . 7 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ π‘Š = (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸)) β†’ (π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺) β†’ (π‘Š prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺)))
6 fveqeq2 6901 . . . . . . . . . . . . . . . 16 (π‘Š = (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) β†’ ((β™―β€˜π‘Š) = ((𝑁 + 1) + 1) ↔ (β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) = ((𝑁 + 1) + 1)))
763ad2ant2 1133 . . . . . . . . . . . . . . 15 ((𝑇 ∈ Word 𝑉 ∧ π‘Š = (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸) β†’ ((β™―β€˜π‘Š) = ((𝑁 + 1) + 1) ↔ (β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) = ((𝑁 + 1) + 1)))
87adantl 481 . . . . . . . . . . . . . 14 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ π‘Š = (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸)) β†’ ((β™―β€˜π‘Š) = ((𝑁 + 1) + 1) ↔ (β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) = ((𝑁 + 1) + 1)))
9 s1cl 14557 . . . . . . . . . . . . . . . . . . . . 21 (𝑆 ∈ 𝑉 β†’ βŸ¨β€œπ‘†β€βŸ© ∈ Word 𝑉)
109adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) β†’ βŸ¨β€œπ‘†β€βŸ© ∈ Word 𝑉)
1110anim1ci 615 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) β†’ (𝑇 ∈ Word 𝑉 ∧ βŸ¨β€œπ‘†β€βŸ© ∈ Word 𝑉))
12 ccatlen 14530 . . . . . . . . . . . . . . . . . . 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©)))
1311, 12syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) β†’ (β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) = ((β™―β€˜π‘‡) + (β™―β€˜βŸ¨β€œπ‘†β€βŸ©)))
1413eqeq1d 2733 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) β†’ ((β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) = ((𝑁 + 1) + 1) ↔ ((β™―β€˜π‘‡) + (β™―β€˜βŸ¨β€œπ‘†β€βŸ©)) = ((𝑁 + 1) + 1)))
15 s1len 14561 . . . . . . . . . . . . . . . . . . . 20 (β™―β€˜βŸ¨β€œπ‘†β€βŸ©) = 1
1615a1i 11 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) β†’ (β™―β€˜βŸ¨β€œπ‘†β€βŸ©) = 1)
1716oveq2d 7428 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) β†’ ((β™―β€˜π‘‡) + (β™―β€˜βŸ¨β€œπ‘†β€βŸ©)) = ((β™―β€˜π‘‡) + 1))
1817eqeq1d 2733 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) β†’ (((β™―β€˜π‘‡) + (β™―β€˜βŸ¨β€œπ‘†β€βŸ©)) = ((𝑁 + 1) + 1) ↔ ((β™―β€˜π‘‡) + 1) = ((𝑁 + 1) + 1)))
19 lencl 14488 . . . . . . . . . . . . . . . . . . . 20 (𝑇 ∈ Word 𝑉 β†’ (β™―β€˜π‘‡) ∈ β„•0)
2019nn0cnd 12539 . . . . . . . . . . . . . . . . . . 19 (𝑇 ∈ Word 𝑉 β†’ (β™―β€˜π‘‡) ∈ β„‚)
2120adantl 481 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) β†’ (β™―β€˜π‘‡) ∈ β„‚)
22 peano2nn0 12517 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„•0)
2322nn0cnd 12539 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„‚)
2423ad2antrr 723 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) β†’ (𝑁 + 1) ∈ β„‚)
25 1cnd 11214 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) β†’ 1 ∈ β„‚)
2621, 24, 25addcan2d 11423 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) β†’ (((β™―β€˜π‘‡) + 1) = ((𝑁 + 1) + 1) ↔ (β™―β€˜π‘‡) = (𝑁 + 1)))
2714, 18, 263bitrd 304 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) β†’ ((β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) = ((𝑁 + 1) + 1) ↔ (β™―β€˜π‘‡) = (𝑁 + 1)))
28 oveq2 7420 . . . . . . . . . . . . . . . . . . 19 ((𝑁 + 1) = (β™―β€˜π‘‡) β†’ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) prefix (𝑁 + 1)) = ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) prefix (β™―β€˜π‘‡)))
2928eqcoms 2739 . . . . . . . . . . . . . . . . . 18 ((β™―β€˜π‘‡) = (𝑁 + 1) β†’ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) prefix (𝑁 + 1)) = ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) prefix (β™―β€˜π‘‡)))
30 pfxccat1 14657 . . . . . . . . . . . . . . . . . . 19 ((𝑇 ∈ Word 𝑉 ∧ βŸ¨β€œπ‘†β€βŸ© ∈ Word 𝑉) β†’ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) prefix (β™―β€˜π‘‡)) = 𝑇)
3111, 30syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) β†’ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) prefix (β™―β€˜π‘‡)) = 𝑇)
3229, 31sylan9eqr 2793 . . . . . . . . . . . . . . . . 17 ((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) ∧ (β™―β€˜π‘‡) = (𝑁 + 1)) β†’ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) prefix (𝑁 + 1)) = 𝑇)
3332ex 412 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) β†’ ((β™―β€˜π‘‡) = (𝑁 + 1) β†’ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) prefix (𝑁 + 1)) = 𝑇))
3427, 33sylbid 239 . . . . . . . . . . . . . . 15 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) β†’ ((β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) = ((𝑁 + 1) + 1) β†’ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) prefix (𝑁 + 1)) = 𝑇))
35343ad2antr1 1187 . . . . . . . . . . . . . 14 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ π‘Š = (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸)) β†’ ((β™―β€˜(𝑇 ++ βŸ¨β€œπ‘†β€βŸ©)) = ((𝑁 + 1) + 1) β†’ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) prefix (𝑁 + 1)) = 𝑇))
368, 35sylbid 239 . . . . . . . . . . . . 13 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ π‘Š = (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸)) β†’ ((β™―β€˜π‘Š) = ((𝑁 + 1) + 1) β†’ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) prefix (𝑁 + 1)) = 𝑇))
3736imp 406 . . . . . . . . . . . 12 ((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ π‘Š = (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸)) ∧ (β™―β€˜π‘Š) = ((𝑁 + 1) + 1)) β†’ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) prefix (𝑁 + 1)) = 𝑇)
38 oveq1 7419 . . . . . . . . . . . . . . 15 (π‘Š = (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) β†’ (π‘Š prefix (𝑁 + 1)) = ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) prefix (𝑁 + 1)))
3938eqeq1d 2733 . . . . . . . . . . . . . 14 (π‘Š = (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) β†’ ((π‘Š prefix (𝑁 + 1)) = 𝑇 ↔ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) prefix (𝑁 + 1)) = 𝑇))
40393ad2ant2 1133 . . . . . . . . . . . . 13 ((𝑇 ∈ Word 𝑉 ∧ π‘Š = (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸) β†’ ((π‘Š prefix (𝑁 + 1)) = 𝑇 ↔ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) prefix (𝑁 + 1)) = 𝑇))
4140ad2antlr 724 . . . . . . . . . . . 12 ((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ π‘Š = (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸)) ∧ (β™―β€˜π‘Š) = ((𝑁 + 1) + 1)) β†’ ((π‘Š prefix (𝑁 + 1)) = 𝑇 ↔ ((𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) prefix (𝑁 + 1)) = 𝑇))
4237, 41mpbird 256 . . . . . . . . . . 11 ((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ π‘Š = (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸)) ∧ (β™―β€˜π‘Š) = ((𝑁 + 1) + 1)) β†’ (π‘Š prefix (𝑁 + 1)) = 𝑇)
4342eleq1d 2817 . . . . . . . . . 10 ((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ π‘Š = (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸)) ∧ (β™―β€˜π‘Š) = ((𝑁 + 1) + 1)) β†’ ((π‘Š prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺) ↔ 𝑇 ∈ (𝑁 WWalksN 𝐺)))
4443biimpd 228 . . . . . . . . 9 ((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ π‘Š = (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸)) ∧ (β™―β€˜π‘Š) = ((𝑁 + 1) + 1)) β†’ ((π‘Š prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺) β†’ 𝑇 ∈ (𝑁 WWalksN 𝐺)))
4544ex 412 . . . . . . . 8 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ π‘Š = (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸)) β†’ ((β™―β€˜π‘Š) = ((𝑁 + 1) + 1) β†’ ((π‘Š prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺) β†’ 𝑇 ∈ (𝑁 WWalksN 𝐺))))
4645com23 86 . . . . . . 7 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ π‘Š = (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸)) β†’ ((π‘Š prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺) β†’ ((β™―β€˜π‘Š) = ((𝑁 + 1) + 1) β†’ 𝑇 ∈ (𝑁 WWalksN 𝐺))))
475, 46syld 47 . . . . . 6 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ π‘Š = (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸)) β†’ (π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺) β†’ ((β™―β€˜π‘Š) = ((𝑁 + 1) + 1) β†’ 𝑇 ∈ (𝑁 WWalksN 𝐺))))
4847com13 88 . . . . 5 ((β™―β€˜π‘Š) = ((𝑁 + 1) + 1) β†’ (π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺) β†’ (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ π‘Š = (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸)) β†’ 𝑇 ∈ (𝑁 WWalksN 𝐺))))
49483ad2ant2 1133 . . . 4 ((π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = ((𝑁 + 1) + 1) ∧ βˆ€π‘– ∈ (0..^(𝑁 + 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ 𝐸) β†’ (π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺) β†’ (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ π‘Š = (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸)) β†’ 𝑇 ∈ (𝑁 WWalksN 𝐺))))
503, 49mpcom 38 . . 3 (π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺) β†’ (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ π‘Š = (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸)) β†’ 𝑇 ∈ (𝑁 WWalksN 𝐺)))
5150com12 32 . 2 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ π‘Š = (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸)) β†’ (π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺) β†’ 𝑇 ∈ (𝑁 WWalksN 𝐺)))
521, 2wwlksnext 29411 . . . . . . . . . . 11 ((𝑇 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸) β†’ (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∈ ((𝑁 + 1) WWalksN 𝐺))
53 eleq1 2820 . . . . . . . . . . 11 (π‘Š = (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) β†’ (π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∈ ((𝑁 + 1) WWalksN 𝐺)))
5452, 53syl5ibrcom 246 . . . . . . . . . 10 ((𝑇 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸) β†’ (π‘Š = (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) β†’ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺)))
55543exp 1118 . . . . . . . . 9 (𝑇 ∈ (𝑁 WWalksN 𝐺) β†’ (𝑆 ∈ 𝑉 β†’ ({(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸 β†’ (π‘Š = (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) β†’ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺)))))
5655com23 86 . . . . . . . 8 (𝑇 ∈ (𝑁 WWalksN 𝐺) β†’ ({(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸 β†’ (𝑆 ∈ 𝑉 β†’ (π‘Š = (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) β†’ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺)))))
5756com14 96 . . . . . . 7 (π‘Š = (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) β†’ ({(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸 β†’ (𝑆 ∈ 𝑉 β†’ (𝑇 ∈ (𝑁 WWalksN 𝐺) β†’ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺)))))
5857imp 406 . . . . . 6 ((π‘Š = (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸) β†’ (𝑆 ∈ 𝑉 β†’ (𝑇 ∈ (𝑁 WWalksN 𝐺) β†’ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺))))
59583adant1 1129 . . . . 5 ((𝑇 ∈ Word 𝑉 ∧ π‘Š = (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸) β†’ (𝑆 ∈ 𝑉 β†’ (𝑇 ∈ (𝑁 WWalksN 𝐺) β†’ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺))))
6059com12 32 . . . 4 (𝑆 ∈ 𝑉 β†’ ((𝑇 ∈ Word 𝑉 ∧ π‘Š = (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸) β†’ (𝑇 ∈ (𝑁 WWalksN 𝐺) β†’ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺))))
6160adantl 481 . . 3 ((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) β†’ ((𝑇 ∈ Word 𝑉 ∧ π‘Š = (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸) β†’ (𝑇 ∈ (𝑁 WWalksN 𝐺) β†’ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺))))
6261imp 406 . 2 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ π‘Š = (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸)) β†’ (𝑇 ∈ (𝑁 WWalksN 𝐺) β†’ π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺)))
6351, 62impbid 211 1 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ π‘Š = (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸)) β†’ (π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ 𝑇 ∈ (𝑁 WWalksN 𝐺)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  βˆ€wral 3060  {cpr 4631  β€˜cfv 6544  (class class class)co 7412  β„‚cc 11111  0cc0 11113  1c1 11114   + caddc 11116  β„•0cn0 12477  ..^cfzo 13632  β™―chash 14295  Word cword 14469  lastSclsw 14517   ++ cconcat 14525  βŸ¨β€œcs1 14550   prefix cpfx 14625  Vtxcvtx 28520  Edgcedg 28571   WWalksN cwwlksn 29344
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-1st 7978  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-1o 8469  df-er 8706  df-map 8825  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-card 9937  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-n0 12478  df-z 12564  df-uz 12828  df-fz 13490  df-fzo 13633  df-hash 14296  df-word 14470  df-lsw 14518  df-concat 14526  df-s1 14551  df-substr 14596  df-pfx 14626  df-wwlks 29348  df-wwlksn 29349
This theorem is referenced by:  wwlksnextwrd  29415
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