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Theorem wwlksnextwrd 28884
Description: Lemma for wwlksnextbij 28889. (Contributed by Alexander van der Vekens, 5-Aug-2018.) (Revised by AV, 18-Apr-2021.) (Revised by AV, 27-Oct-2022.)
Hypotheses
Ref Expression
wwlksnextbij0.v 𝑉 = (Vtxβ€˜πΊ)
wwlksnextbij0.e 𝐸 = (Edgβ€˜πΊ)
wwlksnextbij0.d 𝐷 = {𝑀 ∈ Word 𝑉 ∣ ((β™―β€˜π‘€) = (𝑁 + 2) ∧ (𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)}
Assertion
Ref Expression
wwlksnextwrd (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ 𝐷 = {𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)})
Distinct variable groups:   𝑀,𝐺   𝑀,𝑁   𝑀,π‘Š
Allowed substitution hints:   𝐷(𝑀)   𝐸(𝑀)   𝑉(𝑀)

Proof of Theorem wwlksnextwrd
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 wwlksnextbij0.d . 2 𝐷 = {𝑀 ∈ Word 𝑉 ∣ ((β™―β€˜π‘€) = (𝑁 + 2) ∧ (𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)}
2 3anass 1096 . . . . 5 (((β™―β€˜π‘€) = (𝑁 + 2) ∧ (𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸) ↔ ((β™―β€˜π‘€) = (𝑁 + 2) ∧ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)))
32bianass 641 . . . 4 ((𝑀 ∈ Word 𝑉 ∧ ((β™―β€˜π‘€) = (𝑁 + 2) ∧ (𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)) ↔ ((𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2)) ∧ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)))
4 wwlksnextbij0.v . . . . . . . . . . 11 𝑉 = (Vtxβ€˜πΊ)
54wwlknbp 28829 . . . . . . . . . 10 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ (𝐺 ∈ V ∧ 𝑁 ∈ β„•0 ∧ π‘Š ∈ Word 𝑉))
6 simpl 484 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„•0 ∧ ((𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2)) ∧ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸))) β†’ 𝑁 ∈ β„•0)
7 simpl 484 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2)) β†’ 𝑀 ∈ Word 𝑉)
8 nn0re 12429 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ ℝ)
9 2re 12234 . . . . . . . . . . . . . . . . . . . . 21 2 ∈ ℝ
109a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ β„•0 β†’ 2 ∈ ℝ)
11 nn0ge0 12445 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ β„•0 β†’ 0 ≀ 𝑁)
12 2pos 12263 . . . . . . . . . . . . . . . . . . . . 21 0 < 2
1312a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ β„•0 β†’ 0 < 2)
148, 10, 11, 13addgegt0d 11735 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ β„•0 β†’ 0 < (𝑁 + 2))
1514adantr 482 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ β„•0 ∧ (𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2))) β†’ 0 < (𝑁 + 2))
16 breq2 5114 . . . . . . . . . . . . . . . . . . 19 ((β™―β€˜π‘€) = (𝑁 + 2) β†’ (0 < (β™―β€˜π‘€) ↔ 0 < (𝑁 + 2)))
1716ad2antll 728 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ β„•0 ∧ (𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2))) β†’ (0 < (β™―β€˜π‘€) ↔ 0 < (𝑁 + 2)))
1815, 17mpbird 257 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ β„•0 ∧ (𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2))) β†’ 0 < (β™―β€˜π‘€))
19 hashgt0n0 14272 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘€)) β†’ 𝑀 β‰  βˆ…)
207, 18, 19syl2an2 685 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ β„•0 ∧ (𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2))) β†’ 𝑀 β‰  βˆ…)
21 lswcl 14463 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ Word 𝑉 ∧ 𝑀 β‰  βˆ…) β†’ (lastSβ€˜π‘€) ∈ 𝑉)
227, 20, 21syl2an2 685 . . . . . . . . . . . . . . 15 ((𝑁 ∈ β„•0 ∧ (𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2))) β†’ (lastSβ€˜π‘€) ∈ 𝑉)
2322adantrr 716 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„•0 ∧ ((𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2)) ∧ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸))) β†’ (lastSβ€˜π‘€) ∈ 𝑉)
24 pfxcl 14572 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 ∈ Word 𝑉 β†’ (𝑀 prefix (𝑁 + 1)) ∈ Word 𝑉)
25 eleq1 2826 . . . . . . . . . . . . . . . . . . . . 21 (π‘Š = (𝑀 prefix (𝑁 + 1)) β†’ (π‘Š ∈ Word 𝑉 ↔ (𝑀 prefix (𝑁 + 1)) ∈ Word 𝑉))
2624, 25syl5ibr 246 . . . . . . . . . . . . . . . . . . . 20 (π‘Š = (𝑀 prefix (𝑁 + 1)) β†’ (𝑀 ∈ Word 𝑉 β†’ π‘Š ∈ Word 𝑉))
2726eqcoms 2745 . . . . . . . . . . . . . . . . . . 19 ((𝑀 prefix (𝑁 + 1)) = π‘Š β†’ (𝑀 ∈ Word 𝑉 β†’ π‘Š ∈ Word 𝑉))
2827adantr 482 . . . . . . . . . . . . . . . . . 18 (((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸) β†’ (𝑀 ∈ Word 𝑉 β†’ π‘Š ∈ Word 𝑉))
2928com12 32 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ Word 𝑉 β†’ (((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸) β†’ π‘Š ∈ Word 𝑉))
3029adantr 482 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2)) β†’ (((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸) β†’ π‘Š ∈ Word 𝑉))
3130imp 408 . . . . . . . . . . . . . . 15 (((𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2)) ∧ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)) β†’ π‘Š ∈ Word 𝑉)
3231adantl 483 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„•0 ∧ ((𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2)) ∧ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸))) β†’ π‘Š ∈ Word 𝑉)
33 oveq1 7369 . . . . . . . . . . . . . . . . . 18 (π‘Š = (𝑀 prefix (𝑁 + 1)) β†’ (π‘Š ++ βŸ¨β€œ(lastSβ€˜π‘€)β€βŸ©) = ((𝑀 prefix (𝑁 + 1)) ++ βŸ¨β€œ(lastSβ€˜π‘€)β€βŸ©))
3433eqcoms 2745 . . . . . . . . . . . . . . . . 17 ((𝑀 prefix (𝑁 + 1)) = π‘Š β†’ (π‘Š ++ βŸ¨β€œ(lastSβ€˜π‘€)β€βŸ©) = ((𝑀 prefix (𝑁 + 1)) ++ βŸ¨β€œ(lastSβ€˜π‘€)β€βŸ©))
3534adantr 482 . . . . . . . . . . . . . . . 16 (((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸) β†’ (π‘Š ++ βŸ¨β€œ(lastSβ€˜π‘€)β€βŸ©) = ((𝑀 prefix (𝑁 + 1)) ++ βŸ¨β€œ(lastSβ€˜π‘€)β€βŸ©))
3635ad2antll 728 . . . . . . . . . . . . . . 15 ((𝑁 ∈ β„•0 ∧ ((𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2)) ∧ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸))) β†’ (π‘Š ++ βŸ¨β€œ(lastSβ€˜π‘€)β€βŸ©) = ((𝑀 prefix (𝑁 + 1)) ++ βŸ¨β€œ(lastSβ€˜π‘€)β€βŸ©))
37 oveq1 7369 . . . . . . . . . . . . . . . . . . . . 21 ((β™―β€˜π‘€) = (𝑁 + 2) β†’ ((β™―β€˜π‘€) βˆ’ 1) = ((𝑁 + 2) βˆ’ 1))
3837adantl 483 . . . . . . . . . . . . . . . . . . . 20 ((𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2)) β†’ ((β™―β€˜π‘€) βˆ’ 1) = ((𝑁 + 2) βˆ’ 1))
39 nn0cn 12430 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ β„‚)
40 2cnd 12238 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ β„•0 β†’ 2 ∈ β„‚)
41 1cnd 11157 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ β„•0 β†’ 1 ∈ β„‚)
4239, 40, 41addsubassd 11539 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ β„•0 β†’ ((𝑁 + 2) βˆ’ 1) = (𝑁 + (2 βˆ’ 1)))
43 2m1e1 12286 . . . . . . . . . . . . . . . . . . . . . . 23 (2 βˆ’ 1) = 1
4443a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ β„•0 β†’ (2 βˆ’ 1) = 1)
4544oveq2d 7378 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ β„•0 β†’ (𝑁 + (2 βˆ’ 1)) = (𝑁 + 1))
4642, 45eqtrd 2777 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ β„•0 β†’ ((𝑁 + 2) βˆ’ 1) = (𝑁 + 1))
4738, 46sylan9eqr 2799 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ β„•0 ∧ (𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2))) β†’ ((β™―β€˜π‘€) βˆ’ 1) = (𝑁 + 1))
4847oveq2d 7378 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ β„•0 ∧ (𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2))) β†’ (𝑀 prefix ((β™―β€˜π‘€) βˆ’ 1)) = (𝑀 prefix (𝑁 + 1)))
4948oveq1d 7377 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ β„•0 ∧ (𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2))) β†’ ((𝑀 prefix ((β™―β€˜π‘€) βˆ’ 1)) ++ βŸ¨β€œ(lastSβ€˜π‘€)β€βŸ©) = ((𝑀 prefix (𝑁 + 1)) ++ βŸ¨β€œ(lastSβ€˜π‘€)β€βŸ©))
50 pfxlswccat 14608 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ Word 𝑉 ∧ 𝑀 β‰  βˆ…) β†’ ((𝑀 prefix ((β™―β€˜π‘€) βˆ’ 1)) ++ βŸ¨β€œ(lastSβ€˜π‘€)β€βŸ©) = 𝑀)
517, 20, 50syl2an2 685 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ β„•0 ∧ (𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2))) β†’ ((𝑀 prefix ((β™―β€˜π‘€) βˆ’ 1)) ++ βŸ¨β€œ(lastSβ€˜π‘€)β€βŸ©) = 𝑀)
5249, 51eqtr3d 2779 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ β„•0 ∧ (𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2))) β†’ ((𝑀 prefix (𝑁 + 1)) ++ βŸ¨β€œ(lastSβ€˜π‘€)β€βŸ©) = 𝑀)
5352adantrr 716 . . . . . . . . . . . . . . 15 ((𝑁 ∈ β„•0 ∧ ((𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2)) ∧ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸))) β†’ ((𝑀 prefix (𝑁 + 1)) ++ βŸ¨β€œ(lastSβ€˜π‘€)β€βŸ©) = 𝑀)
5436, 53eqtr2d 2778 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„•0 ∧ ((𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2)) ∧ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸))) β†’ 𝑀 = (π‘Š ++ βŸ¨β€œ(lastSβ€˜π‘€)β€βŸ©))
55 simprrr 781 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„•0 ∧ ((𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2)) ∧ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸))) β†’ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)
56 wwlksnextbij0.e . . . . . . . . . . . . . . 15 𝐸 = (Edgβ€˜πΊ)
574, 56wwlksnextbi 28881 . . . . . . . . . . . . . 14 (((𝑁 ∈ β„•0 ∧ (lastSβ€˜π‘€) ∈ 𝑉) ∧ (π‘Š ∈ Word 𝑉 ∧ 𝑀 = (π‘Š ++ βŸ¨β€œ(lastSβ€˜π‘€)β€βŸ©) ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)) β†’ (𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ π‘Š ∈ (𝑁 WWalksN 𝐺)))
586, 23, 32, 54, 55, 57syl23anc 1378 . . . . . . . . . . . . 13 ((𝑁 ∈ β„•0 ∧ ((𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2)) ∧ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸))) β†’ (𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ π‘Š ∈ (𝑁 WWalksN 𝐺)))
5958exbiri 810 . . . . . . . . . . . 12 (𝑁 ∈ β„•0 β†’ (((𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2)) ∧ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)) β†’ (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ 𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺))))
6059com23 86 . . . . . . . . . . 11 (𝑁 ∈ β„•0 β†’ (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ (((𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2)) ∧ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)) β†’ 𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺))))
61603ad2ant2 1135 . . . . . . . . . 10 ((𝐺 ∈ V ∧ 𝑁 ∈ β„•0 ∧ π‘Š ∈ Word 𝑉) β†’ (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ (((𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2)) ∧ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)) β†’ 𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺))))
625, 61mpcom 38 . . . . . . . . 9 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ (((𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2)) ∧ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)) β†’ 𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺)))
6362expcomd 418 . . . . . . . 8 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ (((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸) β†’ ((𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2)) β†’ 𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺))))
6463imp 408 . . . . . . 7 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)) β†’ ((𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2)) β†’ 𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺)))
654, 56wwlknp 28830 . . . . . . . . . . . 12 (𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) β†’ (𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = ((𝑁 + 1) + 1) ∧ βˆ€π‘– ∈ (0..^(𝑁 + 1)){(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} ∈ 𝐸))
6639, 41, 41addassd 11184 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ β„•0 β†’ ((𝑁 + 1) + 1) = (𝑁 + (1 + 1)))
67 1p1e2 12285 . . . . . . . . . . . . . . . . . . . . . 22 (1 + 1) = 2
6867a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ β„•0 β†’ (1 + 1) = 2)
6968oveq2d 7378 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ β„•0 β†’ (𝑁 + (1 + 1)) = (𝑁 + 2))
7066, 69eqtrd 2777 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ β„•0 β†’ ((𝑁 + 1) + 1) = (𝑁 + 2))
7170eqeq2d 2748 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ β„•0 β†’ ((β™―β€˜π‘€) = ((𝑁 + 1) + 1) ↔ (β™―β€˜π‘€) = (𝑁 + 2)))
7271biimpd 228 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ β„•0 β†’ ((β™―β€˜π‘€) = ((𝑁 + 1) + 1) β†’ (β™―β€˜π‘€) = (𝑁 + 2)))
7372adantr 482 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word 𝑉) β†’ ((β™―β€˜π‘€) = ((𝑁 + 1) + 1) β†’ (β™―β€˜π‘€) = (𝑁 + 2)))
7473com12 32 . . . . . . . . . . . . . . 15 ((β™―β€˜π‘€) = ((𝑁 + 1) + 1) β†’ ((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word 𝑉) β†’ (β™―β€˜π‘€) = (𝑁 + 2)))
7574adantl 483 . . . . . . . . . . . . . 14 ((𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = ((𝑁 + 1) + 1)) β†’ ((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word 𝑉) β†’ (β™―β€˜π‘€) = (𝑁 + 2)))
76 simpl 484 . . . . . . . . . . . . . 14 ((𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = ((𝑁 + 1) + 1)) β†’ 𝑀 ∈ Word 𝑉)
7775, 76jctild 527 . . . . . . . . . . . . 13 ((𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = ((𝑁 + 1) + 1)) β†’ ((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word 𝑉) β†’ (𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2))))
78773adant3 1133 . . . . . . . . . . . 12 ((𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = ((𝑁 + 1) + 1) ∧ βˆ€π‘– ∈ (0..^(𝑁 + 1)){(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} ∈ 𝐸) β†’ ((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word 𝑉) β†’ (𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2))))
7965, 78syl 17 . . . . . . . . . . 11 (𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) β†’ ((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word 𝑉) β†’ (𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2))))
8079com12 32 . . . . . . . . . 10 ((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word 𝑉) β†’ (𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) β†’ (𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2))))
81803adant1 1131 . . . . . . . . 9 ((𝐺 ∈ V ∧ 𝑁 ∈ β„•0 ∧ π‘Š ∈ Word 𝑉) β†’ (𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) β†’ (𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2))))
825, 81syl 17 . . . . . . . 8 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ (𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) β†’ (𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2))))
8382adantr 482 . . . . . . 7 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)) β†’ (𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) β†’ (𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2))))
8464, 83impbid 211 . . . . . 6 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)) β†’ ((𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2)) ↔ 𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺)))
8584ex 414 . . . . 5 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ (((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸) β†’ ((𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2)) ↔ 𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺))))
8685pm5.32rd 579 . . . 4 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ (((𝑀 ∈ Word 𝑉 ∧ (β™―β€˜π‘€) = (𝑁 + 2)) ∧ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)) ↔ (𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸))))
873, 86bitrid 283 . . 3 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ ((𝑀 ∈ Word 𝑉 ∧ ((β™―β€˜π‘€) = (𝑁 + 2) ∧ (𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)) ↔ (𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸))))
8887rabbidva2 3412 . 2 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ {𝑀 ∈ Word 𝑉 ∣ ((β™―β€˜π‘€) = (𝑁 + 2) ∧ (𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)} = {𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)})
891, 88eqtrid 2789 1 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ 𝐷 = {𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆ€wral 3065  {crab 3410  Vcvv 3448  βˆ…c0 4287  {cpr 4593   class class class wbr 5110  β€˜cfv 6501  (class class class)co 7362  β„cr 11057  0cc0 11058  1c1 11059   + caddc 11061   < clt 11196   βˆ’ cmin 11392  2c2 12215  β„•0cn0 12420  ..^cfzo 13574  β™―chash 14237  Word cword 14409  lastSclsw 14457   ++ cconcat 14465  βŸ¨β€œcs1 14490   prefix cpfx 14565  Vtxcvtx 27989  Edgcedg 28040   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-2 12223  df-n0 12421  df-xnn0 12493  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-substr 14536  df-pfx 14566  df-wwlks 28817  df-wwlksn 28818
This theorem is referenced by:  wwlksnextsurj  28887  wwlksnextbij  28889
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