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Theorem wwlksnextinj 29420
Description: Lemma for wwlksnextbij 29423. (Contributed by Alexander van der Vekens, 7-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)}
wwlksnextbij0.r 𝑅 = {𝑛 ∈ 𝑉 ∣ {(lastSβ€˜π‘Š), 𝑛} ∈ 𝐸}
wwlksnextbij0.f 𝐹 = (𝑑 ∈ 𝐷 ↦ (lastSβ€˜π‘‘))
Assertion
Ref Expression
wwlksnextinj (𝑁 ∈ β„•0 β†’ 𝐹:𝐷–1-1→𝑅)
Distinct variable groups:   𝑀,𝐺   𝑀,𝑁   𝑀,π‘Š   𝑑,𝐷   𝑛,𝐸   𝑀,𝐸   𝑑,𝑁,𝑀   𝑑,𝑅   𝑛,𝑉   𝑀,𝑉   𝑛,π‘Š   𝑑,𝑛
Allowed substitution hints:   𝐷(𝑀,𝑛)   𝑅(𝑀,𝑛)   𝐸(𝑑)   𝐹(𝑀,𝑑,𝑛)   𝐺(𝑑,𝑛)   𝑁(𝑛)   𝑉(𝑑)   π‘Š(𝑑)

Proof of Theorem wwlksnextinj
Dummy variables 𝑑 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wwlksnextbij0.v . . 3 𝑉 = (Vtxβ€˜πΊ)
2 wwlksnextbij0.e . . 3 𝐸 = (Edgβ€˜πΊ)
3 wwlksnextbij0.d . . 3 𝐷 = {𝑀 ∈ Word 𝑉 ∣ ((β™―β€˜π‘€) = (𝑁 + 2) ∧ (𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)}
4 wwlksnextbij0.r . . 3 𝑅 = {𝑛 ∈ 𝑉 ∣ {(lastSβ€˜π‘Š), 𝑛} ∈ 𝐸}
5 wwlksnextbij0.f . . 3 𝐹 = (𝑑 ∈ 𝐷 ↦ (lastSβ€˜π‘‘))
61, 2, 3, 4, 5wwlksnextfun 29419 . 2 (𝑁 ∈ β„•0 β†’ 𝐹:π·βŸΆπ‘…)
7 fveq2 6890 . . . . . . 7 (𝑑 = 𝑑 β†’ (lastSβ€˜π‘‘) = (lastSβ€˜π‘‘))
8 fvex 6903 . . . . . . 7 (lastSβ€˜π‘‘) ∈ V
97, 5, 8fvmpt 6997 . . . . . 6 (𝑑 ∈ 𝐷 β†’ (πΉβ€˜π‘‘) = (lastSβ€˜π‘‘))
10 fveq2 6890 . . . . . . 7 (𝑑 = π‘₯ β†’ (lastSβ€˜π‘‘) = (lastSβ€˜π‘₯))
11 fvex 6903 . . . . . . 7 (lastSβ€˜π‘₯) ∈ V
1210, 5, 11fvmpt 6997 . . . . . 6 (π‘₯ ∈ 𝐷 β†’ (πΉβ€˜π‘₯) = (lastSβ€˜π‘₯))
139, 12eqeqan12d 2744 . . . . 5 ((𝑑 ∈ 𝐷 ∧ π‘₯ ∈ 𝐷) β†’ ((πΉβ€˜π‘‘) = (πΉβ€˜π‘₯) ↔ (lastSβ€˜π‘‘) = (lastSβ€˜π‘₯)))
1413adantl 480 . . . 4 ((𝑁 ∈ β„•0 ∧ (𝑑 ∈ 𝐷 ∧ π‘₯ ∈ 𝐷)) β†’ ((πΉβ€˜π‘‘) = (πΉβ€˜π‘₯) ↔ (lastSβ€˜π‘‘) = (lastSβ€˜π‘₯)))
15 fveqeq2 6899 . . . . . . . 8 (𝑀 = 𝑑 β†’ ((β™―β€˜π‘€) = (𝑁 + 2) ↔ (β™―β€˜π‘‘) = (𝑁 + 2)))
16 oveq1 7418 . . . . . . . . 9 (𝑀 = 𝑑 β†’ (𝑀 prefix (𝑁 + 1)) = (𝑑 prefix (𝑁 + 1)))
1716eqeq1d 2732 . . . . . . . 8 (𝑀 = 𝑑 β†’ ((𝑀 prefix (𝑁 + 1)) = π‘Š ↔ (𝑑 prefix (𝑁 + 1)) = π‘Š))
18 fveq2 6890 . . . . . . . . . 10 (𝑀 = 𝑑 β†’ (lastSβ€˜π‘€) = (lastSβ€˜π‘‘))
1918preq2d 4743 . . . . . . . . 9 (𝑀 = 𝑑 β†’ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} = {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)})
2019eleq1d 2816 . . . . . . . 8 (𝑀 = 𝑑 β†’ ({(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸 ↔ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸))
2115, 17, 203anbi123d 1434 . . . . . . 7 (𝑀 = 𝑑 β†’ (((β™―β€˜π‘€) = (𝑁 + 2) ∧ (𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸) ↔ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)))
2221, 3elrab2 3685 . . . . . 6 (𝑑 ∈ 𝐷 ↔ (𝑑 ∈ Word 𝑉 ∧ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)))
23 fveqeq2 6899 . . . . . . . 8 (𝑀 = π‘₯ β†’ ((β™―β€˜π‘€) = (𝑁 + 2) ↔ (β™―β€˜π‘₯) = (𝑁 + 2)))
24 oveq1 7418 . . . . . . . . 9 (𝑀 = π‘₯ β†’ (𝑀 prefix (𝑁 + 1)) = (π‘₯ prefix (𝑁 + 1)))
2524eqeq1d 2732 . . . . . . . 8 (𝑀 = π‘₯ β†’ ((𝑀 prefix (𝑁 + 1)) = π‘Š ↔ (π‘₯ prefix (𝑁 + 1)) = π‘Š))
26 fveq2 6890 . . . . . . . . . 10 (𝑀 = π‘₯ β†’ (lastSβ€˜π‘€) = (lastSβ€˜π‘₯))
2726preq2d 4743 . . . . . . . . 9 (𝑀 = π‘₯ β†’ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} = {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)})
2827eleq1d 2816 . . . . . . . 8 (𝑀 = π‘₯ β†’ ({(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸 ↔ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸))
2923, 25, 283anbi123d 1434 . . . . . . 7 (𝑀 = π‘₯ β†’ (((β™―β€˜π‘€) = (𝑁 + 2) ∧ (𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸) ↔ ((β™―β€˜π‘₯) = (𝑁 + 2) ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸)))
3029, 3elrab2 3685 . . . . . 6 (π‘₯ ∈ 𝐷 ↔ (π‘₯ ∈ Word 𝑉 ∧ ((β™―β€˜π‘₯) = (𝑁 + 2) ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸)))
31 eqtr3 2756 . . . . . . . . . . . . . . . . 17 (((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (β™―β€˜π‘₯) = (𝑁 + 2)) β†’ (β™―β€˜π‘‘) = (β™―β€˜π‘₯))
3231expcom 412 . . . . . . . . . . . . . . . 16 ((β™―β€˜π‘₯) = (𝑁 + 2) β†’ ((β™―β€˜π‘‘) = (𝑁 + 2) β†’ (β™―β€˜π‘‘) = (β™―β€˜π‘₯)))
33323ad2ant1 1131 . . . . . . . . . . . . . . 15 (((β™―β€˜π‘₯) = (𝑁 + 2) ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸) β†’ ((β™―β€˜π‘‘) = (𝑁 + 2) β†’ (β™―β€˜π‘‘) = (β™―β€˜π‘₯)))
3433adantl 480 . . . . . . . . . . . . . 14 ((π‘₯ ∈ Word 𝑉 ∧ ((β™―β€˜π‘₯) = (𝑁 + 2) ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸)) β†’ ((β™―β€˜π‘‘) = (𝑁 + 2) β†’ (β™―β€˜π‘‘) = (β™―β€˜π‘₯)))
3534com12 32 . . . . . . . . . . . . 13 ((β™―β€˜π‘‘) = (𝑁 + 2) β†’ ((π‘₯ ∈ Word 𝑉 ∧ ((β™―β€˜π‘₯) = (𝑁 + 2) ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸)) β†’ (β™―β€˜π‘‘) = (β™―β€˜π‘₯)))
36353ad2ant1 1131 . . . . . . . . . . . 12 (((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸) β†’ ((π‘₯ ∈ Word 𝑉 ∧ ((β™―β€˜π‘₯) = (𝑁 + 2) ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸)) β†’ (β™―β€˜π‘‘) = (β™―β€˜π‘₯)))
3736adantl 480 . . . . . . . . . . 11 ((𝑑 ∈ Word 𝑉 ∧ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)) β†’ ((π‘₯ ∈ Word 𝑉 ∧ ((β™―β€˜π‘₯) = (𝑁 + 2) ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸)) β†’ (β™―β€˜π‘‘) = (β™―β€˜π‘₯)))
3837imp 405 . . . . . . . . . 10 (((𝑑 ∈ Word 𝑉 ∧ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)) ∧ (π‘₯ ∈ Word 𝑉 ∧ ((β™―β€˜π‘₯) = (𝑁 + 2) ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸))) β†’ (β™―β€˜π‘‘) = (β™―β€˜π‘₯))
3938adantr 479 . . . . . . . . 9 ((((𝑑 ∈ Word 𝑉 ∧ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)) ∧ (π‘₯ ∈ Word 𝑉 ∧ ((β™―β€˜π‘₯) = (𝑁 + 2) ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸))) ∧ 𝑁 ∈ β„•0) β†’ (β™―β€˜π‘‘) = (β™―β€˜π‘₯))
4039adantr 479 . . . . . . . 8 (((((𝑑 ∈ Word 𝑉 ∧ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)) ∧ (π‘₯ ∈ Word 𝑉 ∧ ((β™―β€˜π‘₯) = (𝑁 + 2) ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸))) ∧ 𝑁 ∈ β„•0) ∧ (lastSβ€˜π‘‘) = (lastSβ€˜π‘₯)) β†’ (β™―β€˜π‘‘) = (β™―β€˜π‘₯))
41 simpr 483 . . . . . . . 8 (((((𝑑 ∈ Word 𝑉 ∧ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)) ∧ (π‘₯ ∈ Word 𝑉 ∧ ((β™―β€˜π‘₯) = (𝑁 + 2) ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸))) ∧ 𝑁 ∈ β„•0) ∧ (lastSβ€˜π‘‘) = (lastSβ€˜π‘₯)) β†’ (lastSβ€˜π‘‘) = (lastSβ€˜π‘₯))
42 eqtr3 2756 . . . . . . . . . . . . . . . . . . . 20 (((𝑑 prefix (𝑁 + 1)) = π‘Š ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š) β†’ (𝑑 prefix (𝑁 + 1)) = (π‘₯ prefix (𝑁 + 1)))
43 1e2m1 12343 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1 = (2 βˆ’ 1)
4443a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ β„•0 β†’ 1 = (2 βˆ’ 1))
4544oveq2d 7427 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) = (𝑁 + (2 βˆ’ 1)))
46 nn0cn 12486 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ β„‚)
47 2cnd 12294 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ β„•0 β†’ 2 ∈ β„‚)
48 1cnd 11213 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ β„•0 β†’ 1 ∈ β„‚)
4946, 47, 48addsubassd 11595 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ β„•0 β†’ ((𝑁 + 2) βˆ’ 1) = (𝑁 + (2 βˆ’ 1)))
5045, 49eqtr4d 2773 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) = ((𝑁 + 2) βˆ’ 1))
5150adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 ∈ β„•0 ∧ (β™―β€˜π‘‘) = (𝑁 + 2)) β†’ (𝑁 + 1) = ((𝑁 + 2) βˆ’ 1))
52 oveq1 7418 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((β™―β€˜π‘‘) = (𝑁 + 2) β†’ ((β™―β€˜π‘‘) βˆ’ 1) = ((𝑁 + 2) βˆ’ 1))
5352eqeq2d 2741 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((β™―β€˜π‘‘) = (𝑁 + 2) β†’ ((𝑁 + 1) = ((β™―β€˜π‘‘) βˆ’ 1) ↔ (𝑁 + 1) = ((𝑁 + 2) βˆ’ 1)))
5453adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 ∈ β„•0 ∧ (β™―β€˜π‘‘) = (𝑁 + 2)) β†’ ((𝑁 + 1) = ((β™―β€˜π‘‘) βˆ’ 1) ↔ (𝑁 + 1) = ((𝑁 + 2) βˆ’ 1)))
5551, 54mpbird 256 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 ∈ β„•0 ∧ (β™―β€˜π‘‘) = (𝑁 + 2)) β†’ (𝑁 + 1) = ((β™―β€˜π‘‘) βˆ’ 1))
56 oveq2 7419 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 + 1) = ((β™―β€˜π‘‘) βˆ’ 1) β†’ (𝑑 prefix (𝑁 + 1)) = (𝑑 prefix ((β™―β€˜π‘‘) βˆ’ 1)))
57 oveq2 7419 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 + 1) = ((β™―β€˜π‘‘) βˆ’ 1) β†’ (π‘₯ prefix (𝑁 + 1)) = (π‘₯ prefix ((β™―β€˜π‘‘) βˆ’ 1)))
5856, 57eqeq12d 2746 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 + 1) = ((β™―β€˜π‘‘) βˆ’ 1) β†’ ((𝑑 prefix (𝑁 + 1)) = (π‘₯ prefix (𝑁 + 1)) ↔ (𝑑 prefix ((β™―β€˜π‘‘) βˆ’ 1)) = (π‘₯ prefix ((β™―β€˜π‘‘) βˆ’ 1))))
5955, 58syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑁 ∈ β„•0 ∧ (β™―β€˜π‘‘) = (𝑁 + 2)) β†’ ((𝑑 prefix (𝑁 + 1)) = (π‘₯ prefix (𝑁 + 1)) ↔ (𝑑 prefix ((β™―β€˜π‘‘) βˆ’ 1)) = (π‘₯ prefix ((β™―β€˜π‘‘) βˆ’ 1))))
6059biimpd 228 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑁 ∈ β„•0 ∧ (β™―β€˜π‘‘) = (𝑁 + 2)) β†’ ((𝑑 prefix (𝑁 + 1)) = (π‘₯ prefix (𝑁 + 1)) β†’ (𝑑 prefix ((β™―β€˜π‘‘) βˆ’ 1)) = (π‘₯ prefix ((β™―β€˜π‘‘) βˆ’ 1))))
6160ex 411 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ β„•0 β†’ ((β™―β€˜π‘‘) = (𝑁 + 2) β†’ ((𝑑 prefix (𝑁 + 1)) = (π‘₯ prefix (𝑁 + 1)) β†’ (𝑑 prefix ((β™―β€˜π‘‘) βˆ’ 1)) = (π‘₯ prefix ((β™―β€˜π‘‘) βˆ’ 1)))))
6261com13 88 . . . . . . . . . . . . . . . . . . . 20 ((𝑑 prefix (𝑁 + 1)) = (π‘₯ prefix (𝑁 + 1)) β†’ ((β™―β€˜π‘‘) = (𝑁 + 2) β†’ (𝑁 ∈ β„•0 β†’ (𝑑 prefix ((β™―β€˜π‘‘) βˆ’ 1)) = (π‘₯ prefix ((β™―β€˜π‘‘) βˆ’ 1)))))
6342, 62syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝑑 prefix (𝑁 + 1)) = π‘Š ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š) β†’ ((β™―β€˜π‘‘) = (𝑁 + 2) β†’ (𝑁 ∈ β„•0 β†’ (𝑑 prefix ((β™―β€˜π‘‘) βˆ’ 1)) = (π‘₯ prefix ((β™―β€˜π‘‘) βˆ’ 1)))))
6463ex 411 . . . . . . . . . . . . . . . . . 18 ((𝑑 prefix (𝑁 + 1)) = π‘Š β†’ ((π‘₯ prefix (𝑁 + 1)) = π‘Š β†’ ((β™―β€˜π‘‘) = (𝑁 + 2) β†’ (𝑁 ∈ β„•0 β†’ (𝑑 prefix ((β™―β€˜π‘‘) βˆ’ 1)) = (π‘₯ prefix ((β™―β€˜π‘‘) βˆ’ 1))))))
6564com23 86 . . . . . . . . . . . . . . . . 17 ((𝑑 prefix (𝑁 + 1)) = π‘Š β†’ ((β™―β€˜π‘‘) = (𝑁 + 2) β†’ ((π‘₯ prefix (𝑁 + 1)) = π‘Š β†’ (𝑁 ∈ β„•0 β†’ (𝑑 prefix ((β™―β€˜π‘‘) βˆ’ 1)) = (π‘₯ prefix ((β™―β€˜π‘‘) βˆ’ 1))))))
6665impcom 406 . . . . . . . . . . . . . . . 16 (((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š) β†’ ((π‘₯ prefix (𝑁 + 1)) = π‘Š β†’ (𝑁 ∈ β„•0 β†’ (𝑑 prefix ((β™―β€˜π‘‘) βˆ’ 1)) = (π‘₯ prefix ((β™―β€˜π‘‘) βˆ’ 1)))))
6766com12 32 . . . . . . . . . . . . . . 15 ((π‘₯ prefix (𝑁 + 1)) = π‘Š β†’ (((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š) β†’ (𝑁 ∈ β„•0 β†’ (𝑑 prefix ((β™―β€˜π‘‘) βˆ’ 1)) = (π‘₯ prefix ((β™―β€˜π‘‘) βˆ’ 1)))))
68673ad2ant2 1132 . . . . . . . . . . . . . 14 (((β™―β€˜π‘₯) = (𝑁 + 2) ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸) β†’ (((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š) β†’ (𝑁 ∈ β„•0 β†’ (𝑑 prefix ((β™―β€˜π‘‘) βˆ’ 1)) = (π‘₯ prefix ((β™―β€˜π‘‘) βˆ’ 1)))))
6968adantl 480 . . . . . . . . . . . . 13 ((π‘₯ ∈ Word 𝑉 ∧ ((β™―β€˜π‘₯) = (𝑁 + 2) ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸)) β†’ (((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š) β†’ (𝑁 ∈ β„•0 β†’ (𝑑 prefix ((β™―β€˜π‘‘) βˆ’ 1)) = (π‘₯ prefix ((β™―β€˜π‘‘) βˆ’ 1)))))
7069com12 32 . . . . . . . . . . . 12 (((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š) β†’ ((π‘₯ ∈ Word 𝑉 ∧ ((β™―β€˜π‘₯) = (𝑁 + 2) ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸)) β†’ (𝑁 ∈ β„•0 β†’ (𝑑 prefix ((β™―β€˜π‘‘) βˆ’ 1)) = (π‘₯ prefix ((β™―β€˜π‘‘) βˆ’ 1)))))
71703adant3 1130 . . . . . . . . . . 11 (((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸) β†’ ((π‘₯ ∈ Word 𝑉 ∧ ((β™―β€˜π‘₯) = (𝑁 + 2) ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸)) β†’ (𝑁 ∈ β„•0 β†’ (𝑑 prefix ((β™―β€˜π‘‘) βˆ’ 1)) = (π‘₯ prefix ((β™―β€˜π‘‘) βˆ’ 1)))))
7271adantl 480 . . . . . . . . . 10 ((𝑑 ∈ Word 𝑉 ∧ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)) β†’ ((π‘₯ ∈ Word 𝑉 ∧ ((β™―β€˜π‘₯) = (𝑁 + 2) ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸)) β†’ (𝑁 ∈ β„•0 β†’ (𝑑 prefix ((β™―β€˜π‘‘) βˆ’ 1)) = (π‘₯ prefix ((β™―β€˜π‘‘) βˆ’ 1)))))
7372imp31 416 . . . . . . . . 9 ((((𝑑 ∈ Word 𝑉 ∧ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)) ∧ (π‘₯ ∈ Word 𝑉 ∧ ((β™―β€˜π‘₯) = (𝑁 + 2) ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸))) ∧ 𝑁 ∈ β„•0) β†’ (𝑑 prefix ((β™―β€˜π‘‘) βˆ’ 1)) = (π‘₯ prefix ((β™―β€˜π‘‘) βˆ’ 1)))
7473adantr 479 . . . . . . . 8 (((((𝑑 ∈ Word 𝑉 ∧ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)) ∧ (π‘₯ ∈ Word 𝑉 ∧ ((β™―β€˜π‘₯) = (𝑁 + 2) ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸))) ∧ 𝑁 ∈ β„•0) ∧ (lastSβ€˜π‘‘) = (lastSβ€˜π‘₯)) β†’ (𝑑 prefix ((β™―β€˜π‘‘) βˆ’ 1)) = (π‘₯ prefix ((β™―β€˜π‘‘) βˆ’ 1)))
75 simpl 481 . . . . . . . . . . . . 13 ((𝑑 ∈ Word 𝑉 ∧ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)) β†’ 𝑑 ∈ Word 𝑉)
76 simpl 481 . . . . . . . . . . . . 13 ((π‘₯ ∈ Word 𝑉 ∧ ((β™―β€˜π‘₯) = (𝑁 + 2) ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸)) β†’ π‘₯ ∈ Word 𝑉)
7775, 76anim12i 611 . . . . . . . . . . . 12 (((𝑑 ∈ Word 𝑉 ∧ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)) ∧ (π‘₯ ∈ Word 𝑉 ∧ ((β™―β€˜π‘₯) = (𝑁 + 2) ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸))) β†’ (𝑑 ∈ Word 𝑉 ∧ π‘₯ ∈ Word 𝑉))
7877adantr 479 . . . . . . . . . . 11 ((((𝑑 ∈ Word 𝑉 ∧ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)) ∧ (π‘₯ ∈ Word 𝑉 ∧ ((β™―β€˜π‘₯) = (𝑁 + 2) ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸))) ∧ 𝑁 ∈ β„•0) β†’ (𝑑 ∈ Word 𝑉 ∧ π‘₯ ∈ Word 𝑉))
79 nn0re 12485 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ ℝ)
80 2re 12290 . . . . . . . . . . . . . . . . . . . . . 22 2 ∈ ℝ
8180a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ β„•0 β†’ 2 ∈ ℝ)
82 nn0ge0 12501 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ β„•0 β†’ 0 ≀ 𝑁)
83 2pos 12319 . . . . . . . . . . . . . . . . . . . . . 22 0 < 2
8483a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ β„•0 β†’ 0 < 2)
8579, 81, 82, 84addgegt0d 11791 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ β„•0 β†’ 0 < (𝑁 + 2))
8685adantl 480 . . . . . . . . . . . . . . . . . . 19 (((β™―β€˜π‘‘) = (𝑁 + 2) ∧ 𝑁 ∈ β„•0) β†’ 0 < (𝑁 + 2))
87 breq2 5151 . . . . . . . . . . . . . . . . . . . 20 ((β™―β€˜π‘‘) = (𝑁 + 2) β†’ (0 < (β™―β€˜π‘‘) ↔ 0 < (𝑁 + 2)))
8887adantr 479 . . . . . . . . . . . . . . . . . . 19 (((β™―β€˜π‘‘) = (𝑁 + 2) ∧ 𝑁 ∈ β„•0) β†’ (0 < (β™―β€˜π‘‘) ↔ 0 < (𝑁 + 2)))
8986, 88mpbird 256 . . . . . . . . . . . . . . . . . 18 (((β™―β€˜π‘‘) = (𝑁 + 2) ∧ 𝑁 ∈ β„•0) β†’ 0 < (β™―β€˜π‘‘))
90 hashgt0n0 14329 . . . . . . . . . . . . . . . . . 18 ((𝑑 ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘‘)) β†’ 𝑑 β‰  βˆ…)
9189, 90sylan2 591 . . . . . . . . . . . . . . . . 17 ((𝑑 ∈ Word 𝑉 ∧ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ 𝑁 ∈ β„•0)) β†’ 𝑑 β‰  βˆ…)
9291exp32 419 . . . . . . . . . . . . . . . 16 (𝑑 ∈ Word 𝑉 β†’ ((β™―β€˜π‘‘) = (𝑁 + 2) β†’ (𝑁 ∈ β„•0 β†’ 𝑑 β‰  βˆ…)))
9392com12 32 . . . . . . . . . . . . . . 15 ((β™―β€˜π‘‘) = (𝑁 + 2) β†’ (𝑑 ∈ Word 𝑉 β†’ (𝑁 ∈ β„•0 β†’ 𝑑 β‰  βˆ…)))
94933ad2ant1 1131 . . . . . . . . . . . . . 14 (((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸) β†’ (𝑑 ∈ Word 𝑉 β†’ (𝑁 ∈ β„•0 β†’ 𝑑 β‰  βˆ…)))
9594impcom 406 . . . . . . . . . . . . 13 ((𝑑 ∈ Word 𝑉 ∧ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)) β†’ (𝑁 ∈ β„•0 β†’ 𝑑 β‰  βˆ…))
9695adantr 479 . . . . . . . . . . . 12 (((𝑑 ∈ Word 𝑉 ∧ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)) ∧ (π‘₯ ∈ Word 𝑉 ∧ ((β™―β€˜π‘₯) = (𝑁 + 2) ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸))) β†’ (𝑁 ∈ β„•0 β†’ 𝑑 β‰  βˆ…))
9796imp 405 . . . . . . . . . . 11 ((((𝑑 ∈ Word 𝑉 ∧ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)) ∧ (π‘₯ ∈ Word 𝑉 ∧ ((β™―β€˜π‘₯) = (𝑁 + 2) ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸))) ∧ 𝑁 ∈ β„•0) β†’ 𝑑 β‰  βˆ…)
9885adantl 480 . . . . . . . . . . . . . . . . . . 19 (((β™―β€˜π‘₯) = (𝑁 + 2) ∧ 𝑁 ∈ β„•0) β†’ 0 < (𝑁 + 2))
99 breq2 5151 . . . . . . . . . . . . . . . . . . . 20 ((β™―β€˜π‘₯) = (𝑁 + 2) β†’ (0 < (β™―β€˜π‘₯) ↔ 0 < (𝑁 + 2)))
10099adantr 479 . . . . . . . . . . . . . . . . . . 19 (((β™―β€˜π‘₯) = (𝑁 + 2) ∧ 𝑁 ∈ β„•0) β†’ (0 < (β™―β€˜π‘₯) ↔ 0 < (𝑁 + 2)))
10198, 100mpbird 256 . . . . . . . . . . . . . . . . . 18 (((β™―β€˜π‘₯) = (𝑁 + 2) ∧ 𝑁 ∈ β„•0) β†’ 0 < (β™―β€˜π‘₯))
102 hashgt0n0 14329 . . . . . . . . . . . . . . . . . 18 ((π‘₯ ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘₯)) β†’ π‘₯ β‰  βˆ…)
103101, 102sylan2 591 . . . . . . . . . . . . . . . . 17 ((π‘₯ ∈ Word 𝑉 ∧ ((β™―β€˜π‘₯) = (𝑁 + 2) ∧ 𝑁 ∈ β„•0)) β†’ π‘₯ β‰  βˆ…)
104103exp32 419 . . . . . . . . . . . . . . . 16 (π‘₯ ∈ Word 𝑉 β†’ ((β™―β€˜π‘₯) = (𝑁 + 2) β†’ (𝑁 ∈ β„•0 β†’ π‘₯ β‰  βˆ…)))
105104com12 32 . . . . . . . . . . . . . . 15 ((β™―β€˜π‘₯) = (𝑁 + 2) β†’ (π‘₯ ∈ Word 𝑉 β†’ (𝑁 ∈ β„•0 β†’ π‘₯ β‰  βˆ…)))
1061053ad2ant1 1131 . . . . . . . . . . . . . 14 (((β™―β€˜π‘₯) = (𝑁 + 2) ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸) β†’ (π‘₯ ∈ Word 𝑉 β†’ (𝑁 ∈ β„•0 β†’ π‘₯ β‰  βˆ…)))
107106impcom 406 . . . . . . . . . . . . 13 ((π‘₯ ∈ Word 𝑉 ∧ ((β™―β€˜π‘₯) = (𝑁 + 2) ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸)) β†’ (𝑁 ∈ β„•0 β†’ π‘₯ β‰  βˆ…))
108107adantl 480 . . . . . . . . . . . 12 (((𝑑 ∈ Word 𝑉 ∧ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)) ∧ (π‘₯ ∈ Word 𝑉 ∧ ((β™―β€˜π‘₯) = (𝑁 + 2) ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸))) β†’ (𝑁 ∈ β„•0 β†’ π‘₯ β‰  βˆ…))
109108imp 405 . . . . . . . . . . 11 ((((𝑑 ∈ Word 𝑉 ∧ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)) ∧ (π‘₯ ∈ Word 𝑉 ∧ ((β™―β€˜π‘₯) = (𝑁 + 2) ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸))) ∧ 𝑁 ∈ β„•0) β†’ π‘₯ β‰  βˆ…)
11078, 97, 109jca32 514 . . . . . . . . . 10 ((((𝑑 ∈ Word 𝑉 ∧ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)) ∧ (π‘₯ ∈ Word 𝑉 ∧ ((β™―β€˜π‘₯) = (𝑁 + 2) ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸))) ∧ 𝑁 ∈ β„•0) β†’ ((𝑑 ∈ Word 𝑉 ∧ π‘₯ ∈ Word 𝑉) ∧ (𝑑 β‰  βˆ… ∧ π‘₯ β‰  βˆ…)))
111110adantr 479 . . . . . . . . 9 (((((𝑑 ∈ Word 𝑉 ∧ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)) ∧ (π‘₯ ∈ Word 𝑉 ∧ ((β™―β€˜π‘₯) = (𝑁 + 2) ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸))) ∧ 𝑁 ∈ β„•0) ∧ (lastSβ€˜π‘‘) = (lastSβ€˜π‘₯)) β†’ ((𝑑 ∈ Word 𝑉 ∧ π‘₯ ∈ Word 𝑉) ∧ (𝑑 β‰  βˆ… ∧ π‘₯ β‰  βˆ…)))
112 simpl 481 . . . . . . . . . . . 12 ((𝑑 ∈ Word 𝑉 ∧ π‘₯ ∈ Word 𝑉) β†’ 𝑑 ∈ Word 𝑉)
113112adantr 479 . . . . . . . . . . 11 (((𝑑 ∈ Word 𝑉 ∧ π‘₯ ∈ Word 𝑉) ∧ (𝑑 β‰  βˆ… ∧ π‘₯ β‰  βˆ…)) β†’ 𝑑 ∈ Word 𝑉)
114 simpr 483 . . . . . . . . . . . 12 ((𝑑 ∈ Word 𝑉 ∧ π‘₯ ∈ Word 𝑉) β†’ π‘₯ ∈ Word 𝑉)
115114adantr 479 . . . . . . . . . . 11 (((𝑑 ∈ Word 𝑉 ∧ π‘₯ ∈ Word 𝑉) ∧ (𝑑 β‰  βˆ… ∧ π‘₯ β‰  βˆ…)) β†’ π‘₯ ∈ Word 𝑉)
116 hashneq0 14328 . . . . . . . . . . . . . . . 16 (𝑑 ∈ Word 𝑉 β†’ (0 < (β™―β€˜π‘‘) ↔ 𝑑 β‰  βˆ…))
117116biimprd 247 . . . . . . . . . . . . . . 15 (𝑑 ∈ Word 𝑉 β†’ (𝑑 β‰  βˆ… β†’ 0 < (β™―β€˜π‘‘)))
118117adantr 479 . . . . . . . . . . . . . 14 ((𝑑 ∈ Word 𝑉 ∧ π‘₯ ∈ Word 𝑉) β†’ (𝑑 β‰  βˆ… β†’ 0 < (β™―β€˜π‘‘)))
119118com12 32 . . . . . . . . . . . . 13 (𝑑 β‰  βˆ… β†’ ((𝑑 ∈ Word 𝑉 ∧ π‘₯ ∈ Word 𝑉) β†’ 0 < (β™―β€˜π‘‘)))
120119adantr 479 . . . . . . . . . . . 12 ((𝑑 β‰  βˆ… ∧ π‘₯ β‰  βˆ…) β†’ ((𝑑 ∈ Word 𝑉 ∧ π‘₯ ∈ Word 𝑉) β†’ 0 < (β™―β€˜π‘‘)))
121120impcom 406 . . . . . . . . . . 11 (((𝑑 ∈ Word 𝑉 ∧ π‘₯ ∈ Word 𝑉) ∧ (𝑑 β‰  βˆ… ∧ π‘₯ β‰  βˆ…)) β†’ 0 < (β™―β€˜π‘‘))
122 pfxsuff1eqwrdeq 14653 . . . . . . . . . . 11 ((𝑑 ∈ Word 𝑉 ∧ π‘₯ ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘‘)) β†’ (𝑑 = π‘₯ ↔ ((β™―β€˜π‘‘) = (β™―β€˜π‘₯) ∧ ((𝑑 prefix ((β™―β€˜π‘‘) βˆ’ 1)) = (π‘₯ prefix ((β™―β€˜π‘‘) βˆ’ 1)) ∧ (lastSβ€˜π‘‘) = (lastSβ€˜π‘₯)))))
123113, 115, 121, 122syl3anc 1369 . . . . . . . . . 10 (((𝑑 ∈ Word 𝑉 ∧ π‘₯ ∈ Word 𝑉) ∧ (𝑑 β‰  βˆ… ∧ π‘₯ β‰  βˆ…)) β†’ (𝑑 = π‘₯ ↔ ((β™―β€˜π‘‘) = (β™―β€˜π‘₯) ∧ ((𝑑 prefix ((β™―β€˜π‘‘) βˆ’ 1)) = (π‘₯ prefix ((β™―β€˜π‘‘) βˆ’ 1)) ∧ (lastSβ€˜π‘‘) = (lastSβ€˜π‘₯)))))
124 ancom 459 . . . . . . . . . . . 12 (((𝑑 prefix ((β™―β€˜π‘‘) βˆ’ 1)) = (π‘₯ prefix ((β™―β€˜π‘‘) βˆ’ 1)) ∧ (lastSβ€˜π‘‘) = (lastSβ€˜π‘₯)) ↔ ((lastSβ€˜π‘‘) = (lastSβ€˜π‘₯) ∧ (𝑑 prefix ((β™―β€˜π‘‘) βˆ’ 1)) = (π‘₯ prefix ((β™―β€˜π‘‘) βˆ’ 1))))
125124anbi2i 621 . . . . . . . . . . 11 (((β™―β€˜π‘‘) = (β™―β€˜π‘₯) ∧ ((𝑑 prefix ((β™―β€˜π‘‘) βˆ’ 1)) = (π‘₯ prefix ((β™―β€˜π‘‘) βˆ’ 1)) ∧ (lastSβ€˜π‘‘) = (lastSβ€˜π‘₯))) ↔ ((β™―β€˜π‘‘) = (β™―β€˜π‘₯) ∧ ((lastSβ€˜π‘‘) = (lastSβ€˜π‘₯) ∧ (𝑑 prefix ((β™―β€˜π‘‘) βˆ’ 1)) = (π‘₯ prefix ((β™―β€˜π‘‘) βˆ’ 1)))))
126 3anass 1093 . . . . . . . . . . 11 (((β™―β€˜π‘‘) = (β™―β€˜π‘₯) ∧ (lastSβ€˜π‘‘) = (lastSβ€˜π‘₯) ∧ (𝑑 prefix ((β™―β€˜π‘‘) βˆ’ 1)) = (π‘₯ prefix ((β™―β€˜π‘‘) βˆ’ 1))) ↔ ((β™―β€˜π‘‘) = (β™―β€˜π‘₯) ∧ ((lastSβ€˜π‘‘) = (lastSβ€˜π‘₯) ∧ (𝑑 prefix ((β™―β€˜π‘‘) βˆ’ 1)) = (π‘₯ prefix ((β™―β€˜π‘‘) βˆ’ 1)))))
127125, 126bitr4i 277 . . . . . . . . . 10 (((β™―β€˜π‘‘) = (β™―β€˜π‘₯) ∧ ((𝑑 prefix ((β™―β€˜π‘‘) βˆ’ 1)) = (π‘₯ prefix ((β™―β€˜π‘‘) βˆ’ 1)) ∧ (lastSβ€˜π‘‘) = (lastSβ€˜π‘₯))) ↔ ((β™―β€˜π‘‘) = (β™―β€˜π‘₯) ∧ (lastSβ€˜π‘‘) = (lastSβ€˜π‘₯) ∧ (𝑑 prefix ((β™―β€˜π‘‘) βˆ’ 1)) = (π‘₯ prefix ((β™―β€˜π‘‘) βˆ’ 1))))
128123, 127bitrdi 286 . . . . . . . . 9 (((𝑑 ∈ Word 𝑉 ∧ π‘₯ ∈ Word 𝑉) ∧ (𝑑 β‰  βˆ… ∧ π‘₯ β‰  βˆ…)) β†’ (𝑑 = π‘₯ ↔ ((β™―β€˜π‘‘) = (β™―β€˜π‘₯) ∧ (lastSβ€˜π‘‘) = (lastSβ€˜π‘₯) ∧ (𝑑 prefix ((β™―β€˜π‘‘) βˆ’ 1)) = (π‘₯ prefix ((β™―β€˜π‘‘) βˆ’ 1)))))
129111, 128syl 17 . . . . . . . 8 (((((𝑑 ∈ Word 𝑉 ∧ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)) ∧ (π‘₯ ∈ Word 𝑉 ∧ ((β™―β€˜π‘₯) = (𝑁 + 2) ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸))) ∧ 𝑁 ∈ β„•0) ∧ (lastSβ€˜π‘‘) = (lastSβ€˜π‘₯)) β†’ (𝑑 = π‘₯ ↔ ((β™―β€˜π‘‘) = (β™―β€˜π‘₯) ∧ (lastSβ€˜π‘‘) = (lastSβ€˜π‘₯) ∧ (𝑑 prefix ((β™―β€˜π‘‘) βˆ’ 1)) = (π‘₯ prefix ((β™―β€˜π‘‘) βˆ’ 1)))))
13040, 41, 74, 129mpbir3and 1340 . . . . . . 7 (((((𝑑 ∈ Word 𝑉 ∧ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)) ∧ (π‘₯ ∈ Word 𝑉 ∧ ((β™―β€˜π‘₯) = (𝑁 + 2) ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸))) ∧ 𝑁 ∈ β„•0) ∧ (lastSβ€˜π‘‘) = (lastSβ€˜π‘₯)) β†’ 𝑑 = π‘₯)
131130exp31 418 . . . . . 6 (((𝑑 ∈ Word 𝑉 ∧ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)) ∧ (π‘₯ ∈ Word 𝑉 ∧ ((β™―β€˜π‘₯) = (𝑁 + 2) ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸))) β†’ (𝑁 ∈ β„•0 β†’ ((lastSβ€˜π‘‘) = (lastSβ€˜π‘₯) β†’ 𝑑 = π‘₯)))
13222, 30, 131syl2anb 596 . . . . 5 ((𝑑 ∈ 𝐷 ∧ π‘₯ ∈ 𝐷) β†’ (𝑁 ∈ β„•0 β†’ ((lastSβ€˜π‘‘) = (lastSβ€˜π‘₯) β†’ 𝑑 = π‘₯)))
133132impcom 406 . . . 4 ((𝑁 ∈ β„•0 ∧ (𝑑 ∈ 𝐷 ∧ π‘₯ ∈ 𝐷)) β†’ ((lastSβ€˜π‘‘) = (lastSβ€˜π‘₯) β†’ 𝑑 = π‘₯))
13414, 133sylbid 239 . . 3 ((𝑁 ∈ β„•0 ∧ (𝑑 ∈ 𝐷 ∧ π‘₯ ∈ 𝐷)) β†’ ((πΉβ€˜π‘‘) = (πΉβ€˜π‘₯) β†’ 𝑑 = π‘₯))
135134ralrimivva 3198 . 2 (𝑁 ∈ β„•0 β†’ βˆ€π‘‘ ∈ 𝐷 βˆ€π‘₯ ∈ 𝐷 ((πΉβ€˜π‘‘) = (πΉβ€˜π‘₯) β†’ 𝑑 = π‘₯))
136 dff13 7256 . 2 (𝐹:𝐷–1-1→𝑅 ↔ (𝐹:π·βŸΆπ‘… ∧ βˆ€π‘‘ ∈ 𝐷 βˆ€π‘₯ ∈ 𝐷 ((πΉβ€˜π‘‘) = (πΉβ€˜π‘₯) β†’ 𝑑 = π‘₯)))
1376, 135, 136sylanbrc 581 1 (𝑁 ∈ β„•0 β†’ 𝐹:𝐷–1-1→𝑅)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  βˆ€wral 3059  {crab 3430  βˆ…c0 4321  {cpr 4629   class class class wbr 5147   ↦ cmpt 5230  βŸΆwf 6538  β€“1-1β†’wf1 6539  β€˜cfv 6542  (class class class)co 7411  β„cr 11111  0cc0 11112  1c1 11113   + caddc 11115   < clt 11252   βˆ’ cmin 11448  2c2 12271  β„•0cn0 12476  β™―chash 14294  Word cword 14468  lastSclsw 14516   prefix cpfx 14624  Vtxcvtx 28523  Edgcedg 28574
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-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-2 12279  df-n0 12477  df-xnn0 12549  df-z 12563  df-uz 12827  df-fz 13489  df-fzo 13632  df-hash 14295  df-word 14469  df-lsw 14517  df-s1 14550  df-substr 14595  df-pfx 14625
This theorem is referenced by:  wwlksnextbij0  29422
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