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Theorem wwlksnextbi 29415
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 29364 . . . 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 29413 . . . . . . . 8 (𝑁 ∈ β„•0 β†’ (π‘Š ∈ ((𝑁 + 1) WWalksN 𝐺) β†’ (π‘Š prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺)))
54ad2antrr 722 . . . . . . 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 6899 . . . . . . . . . . . . . . . 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 1132 . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . 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 14556 . . . . . . . . . . . . . . . . . . . . 21 (𝑆 ∈ 𝑉 β†’ βŸ¨β€œπ‘†β€βŸ© ∈ Word 𝑉)
109adantl 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) β†’ βŸ¨β€œπ‘†β€βŸ© ∈ Word 𝑉)
1110anim1ci 614 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) β†’ (𝑇 ∈ Word 𝑉 ∧ βŸ¨β€œπ‘†β€βŸ© ∈ Word 𝑉))
12 ccatlen 14529 . . . . . . . . . . . . . . . . . . 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 2732 . . . . . . . . . . . . . . . . 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 14560 . . . . . . . . . . . . . . . . . . . 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 7427 . . . . . . . . . . . . . . . . . 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 2732 . . . . . . . . . . . . . . . . 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 14487 . . . . . . . . . . . . . . . . . . . 20 (𝑇 ∈ Word 𝑉 β†’ (β™―β€˜π‘‡) ∈ β„•0)
2019nn0cnd 12538 . . . . . . . . . . . . . . . . . . 19 (𝑇 ∈ Word 𝑉 β†’ (β™―β€˜π‘‡) ∈ β„‚)
2120adantl 480 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) β†’ (β™―β€˜π‘‡) ∈ β„‚)
22 peano2nn0 12516 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„•0)
2322nn0cnd 12538 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„‚)
2423ad2antrr 722 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) β†’ (𝑁 + 1) ∈ β„‚)
25 1cnd 11213 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) β†’ 1 ∈ β„‚)
2621, 24, 25addcan2d 11422 . . . . . . . . . . . . . . . . 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 7419 . . . . . . . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . . . . . . 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 14656 . . . . . . . . . . . . . . . . . . 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 2792 . . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . . . 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 1186 . . . . . . . . . . . . . 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 405 . . . . . . . . . . . 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 7418 . . . . . . . . . . . . . . 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 2732 . . . . . . . . . . . . . 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 1132 . . . . . . . . . . . . 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 723 . . . . . . . . . . . 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 2816 . . . . . . . . . 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 411 . . . . . . . 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 1132 . . . 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 29414 . . . . . . . . . . 11 ((𝑇 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑆 ∈ 𝑉 ∧ {(lastSβ€˜π‘‡), 𝑆} ∈ 𝐸) β†’ (𝑇 ++ βŸ¨β€œπ‘†β€βŸ©) ∈ ((𝑁 + 1) WWalksN 𝐺))
53 eleq1 2819 . . . . . . . . . . 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 1117 . . . . . . . . 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 405 . . . . . 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 1128 . . . . 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 480 . . 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 405 . 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 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  {cpr 4629  β€˜cfv 6542  (class class class)co 7411  β„‚cc 11110  0cc0 11112  1c1 11113   + caddc 11115  β„•0cn0 12476  ..^cfzo 13631  β™―chash 14294  Word cword 14468  lastSclsw 14516   ++ cconcat 14524  βŸ¨β€œcs1 14549   prefix cpfx 14624  Vtxcvtx 28523  Edgcedg 28574   WWalksN cwwlksn 29347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13489  df-fzo 13632  df-hash 14295  df-word 14469  df-lsw 14517  df-concat 14525  df-s1 14550  df-substr 14595  df-pfx 14625  df-wwlks 29351  df-wwlksn 29352
This theorem is referenced by:  wwlksnextwrd  29418
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