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Theorem wwlksnextinj 29684
Description: Lemma for wwlksnextbij 29687. (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 29683 . 2 (𝑁 ∈ β„•0 β†’ 𝐹:π·βŸΆπ‘…)
7 fveq2 6891 . . . . . . 7 (𝑑 = 𝑑 β†’ (lastSβ€˜π‘‘) = (lastSβ€˜π‘‘))
8 fvex 6904 . . . . . . 7 (lastSβ€˜π‘‘) ∈ V
97, 5, 8fvmpt 6999 . . . . . 6 (𝑑 ∈ 𝐷 β†’ (πΉβ€˜π‘‘) = (lastSβ€˜π‘‘))
10 fveq2 6891 . . . . . . 7 (𝑑 = π‘₯ β†’ (lastSβ€˜π‘‘) = (lastSβ€˜π‘₯))
11 fvex 6904 . . . . . . 7 (lastSβ€˜π‘₯) ∈ V
1210, 5, 11fvmpt 6999 . . . . . 6 (π‘₯ ∈ 𝐷 β†’ (πΉβ€˜π‘₯) = (lastSβ€˜π‘₯))
139, 12eqeqan12d 2741 . . . . 5 ((𝑑 ∈ 𝐷 ∧ π‘₯ ∈ 𝐷) β†’ ((πΉβ€˜π‘‘) = (πΉβ€˜π‘₯) ↔ (lastSβ€˜π‘‘) = (lastSβ€˜π‘₯)))
1413adantl 481 . . . 4 ((𝑁 ∈ β„•0 ∧ (𝑑 ∈ 𝐷 ∧ π‘₯ ∈ 𝐷)) β†’ ((πΉβ€˜π‘‘) = (πΉβ€˜π‘₯) ↔ (lastSβ€˜π‘‘) = (lastSβ€˜π‘₯)))
15 fveqeq2 6900 . . . . . . . 8 (𝑀 = 𝑑 β†’ ((β™―β€˜π‘€) = (𝑁 + 2) ↔ (β™―β€˜π‘‘) = (𝑁 + 2)))
16 oveq1 7421 . . . . . . . . 9 (𝑀 = 𝑑 β†’ (𝑀 prefix (𝑁 + 1)) = (𝑑 prefix (𝑁 + 1)))
1716eqeq1d 2729 . . . . . . . 8 (𝑀 = 𝑑 β†’ ((𝑀 prefix (𝑁 + 1)) = π‘Š ↔ (𝑑 prefix (𝑁 + 1)) = π‘Š))
18 fveq2 6891 . . . . . . . . . 10 (𝑀 = 𝑑 β†’ (lastSβ€˜π‘€) = (lastSβ€˜π‘‘))
1918preq2d 4740 . . . . . . . . 9 (𝑀 = 𝑑 β†’ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} = {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)})
2019eleq1d 2813 . . . . . . . 8 (𝑀 = 𝑑 β†’ ({(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸 ↔ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸))
2115, 17, 203anbi123d 1433 . . . . . . 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 3683 . . . . . 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 6900 . . . . . . . 8 (𝑀 = π‘₯ β†’ ((β™―β€˜π‘€) = (𝑁 + 2) ↔ (β™―β€˜π‘₯) = (𝑁 + 2)))
24 oveq1 7421 . . . . . . . . 9 (𝑀 = π‘₯ β†’ (𝑀 prefix (𝑁 + 1)) = (π‘₯ prefix (𝑁 + 1)))
2524eqeq1d 2729 . . . . . . . 8 (𝑀 = π‘₯ β†’ ((𝑀 prefix (𝑁 + 1)) = π‘Š ↔ (π‘₯ prefix (𝑁 + 1)) = π‘Š))
26 fveq2 6891 . . . . . . . . . 10 (𝑀 = π‘₯ β†’ (lastSβ€˜π‘€) = (lastSβ€˜π‘₯))
2726preq2d 4740 . . . . . . . . 9 (𝑀 = π‘₯ β†’ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} = {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)})
2827eleq1d 2813 . . . . . . . 8 (𝑀 = π‘₯ β†’ ({(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸 ↔ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸))
2923, 25, 283anbi123d 1433 . . . . . . 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 3683 . . . . . 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 2753 . . . . . . . . . . . . . . . . 17 (((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (β™―β€˜π‘₯) = (𝑁 + 2)) β†’ (β™―β€˜π‘‘) = (β™―β€˜π‘₯))
3231expcom 413 . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 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 481 . . . . . . . . . . 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 406 . . . . . . . . . 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 480 . . . . . . . . 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 480 . . . . . . . 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 484 . . . . . . . 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 2753 . . . . . . . . . . . . . . . . . . . 20 (((𝑑 prefix (𝑁 + 1)) = π‘Š ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š) β†’ (𝑑 prefix (𝑁 + 1)) = (π‘₯ prefix (𝑁 + 1)))
43 1e2m1 12355 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1 = (2 βˆ’ 1)
4443a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ β„•0 β†’ 1 = (2 βˆ’ 1))
4544oveq2d 7430 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) = (𝑁 + (2 βˆ’ 1)))
46 nn0cn 12498 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ β„‚)
47 2cnd 12306 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ β„•0 β†’ 2 ∈ β„‚)
48 1cnd 11225 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ β„•0 β†’ 1 ∈ β„‚)
4946, 47, 48addsubassd 11607 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ β„•0 β†’ ((𝑁 + 2) βˆ’ 1) = (𝑁 + (2 βˆ’ 1)))
5045, 49eqtr4d 2770 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) = ((𝑁 + 2) βˆ’ 1))
5150adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 ∈ β„•0 ∧ (β™―β€˜π‘‘) = (𝑁 + 2)) β†’ (𝑁 + 1) = ((𝑁 + 2) βˆ’ 1))
52 oveq1 7421 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((β™―β€˜π‘‘) = (𝑁 + 2) β†’ ((β™―β€˜π‘‘) βˆ’ 1) = ((𝑁 + 2) βˆ’ 1))
5352eqeq2d 2738 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((β™―β€˜π‘‘) = (𝑁 + 2) β†’ ((𝑁 + 1) = ((β™―β€˜π‘‘) βˆ’ 1) ↔ (𝑁 + 1) = ((𝑁 + 2) βˆ’ 1)))
5453adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 ∈ β„•0 ∧ (β™―β€˜π‘‘) = (𝑁 + 2)) β†’ (𝑁 + 1) = ((β™―β€˜π‘‘) βˆ’ 1))
56 oveq2 7422 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 + 1) = ((β™―β€˜π‘‘) βˆ’ 1) β†’ (𝑑 prefix (𝑁 + 1)) = (𝑑 prefix ((β™―β€˜π‘‘) βˆ’ 1)))
57 oveq2 7422 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 + 1) = ((β™―β€˜π‘‘) βˆ’ 1) β†’ (π‘₯ prefix (𝑁 + 1)) = (π‘₯ prefix ((β™―β€˜π‘‘) βˆ’ 1)))
5856, 57eqeq12d 2743 . . . . . . . . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . 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 481 . . . . . . . . . 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 417 . . . . . . . . 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 480 . . . . . . . 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 482 . . . . . . . . . . . . 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 482 . . . . . . . . . . . . 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 612 . . . . . . . . . . . 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 480 . . . . . . . . . . 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 12497 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ ℝ)
80 2re 12302 . . . . . . . . . . . . . . . . . . . . . 22 2 ∈ ℝ
8180a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ β„•0 β†’ 2 ∈ ℝ)
82 nn0ge0 12513 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ β„•0 β†’ 0 ≀ 𝑁)
83 2pos 12331 . . . . . . . . . . . . . . . . . . . . . 22 0 < 2
8483a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ β„•0 β†’ 0 < 2)
8579, 81, 82, 84addgegt0d 11803 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ β„•0 β†’ 0 < (𝑁 + 2))
8685adantl 481 . . . . . . . . . . . . . . . . . . 19 (((β™―β€˜π‘‘) = (𝑁 + 2) ∧ 𝑁 ∈ β„•0) β†’ 0 < (𝑁 + 2))
87 breq2 5146 . . . . . . . . . . . . . . . . . . . 20 ((β™―β€˜π‘‘) = (𝑁 + 2) β†’ (0 < (β™―β€˜π‘‘) ↔ 0 < (𝑁 + 2)))
8887adantr 480 . . . . . . . . . . . . . . . . . . 19 (((β™―β€˜π‘‘) = (𝑁 + 2) ∧ 𝑁 ∈ β„•0) β†’ (0 < (β™―β€˜π‘‘) ↔ 0 < (𝑁 + 2)))
8986, 88mpbird 257 . . . . . . . . . . . . . . . . . 18 (((β™―β€˜π‘‘) = (𝑁 + 2) ∧ 𝑁 ∈ β„•0) β†’ 0 < (β™―β€˜π‘‘))
90 hashgt0n0 14342 . . . . . . . . . . . . . . . . . 18 ((𝑑 ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘‘)) β†’ 𝑑 β‰  βˆ…)
9189, 90sylan2 592 . . . . . . . . . . . . . . . . 17 ((𝑑 ∈ Word 𝑉 ∧ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ 𝑁 ∈ β„•0)) β†’ 𝑑 β‰  βˆ…)
9291exp32 420 . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . 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 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†’ 𝑑 β‰  βˆ…))
9796imp 406 . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . 19 (((β™―β€˜π‘₯) = (𝑁 + 2) ∧ 𝑁 ∈ β„•0) β†’ 0 < (𝑁 + 2))
99 breq2 5146 . . . . . . . . . . . . . . . . . . . 20 ((β™―β€˜π‘₯) = (𝑁 + 2) β†’ (0 < (β™―β€˜π‘₯) ↔ 0 < (𝑁 + 2)))
10099adantr 480 . . . . . . . . . . . . . . . . . . 19 (((β™―β€˜π‘₯) = (𝑁 + 2) ∧ 𝑁 ∈ β„•0) β†’ (0 < (β™―β€˜π‘₯) ↔ 0 < (𝑁 + 2)))
10198, 100mpbird 257 . . . . . . . . . . . . . . . . . 18 (((β™―β€˜π‘₯) = (𝑁 + 2) ∧ 𝑁 ∈ β„•0) β†’ 0 < (β™―β€˜π‘₯))
102 hashgt0n0 14342 . . . . . . . . . . . . . . . . . 18 ((π‘₯ ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘₯)) β†’ π‘₯ β‰  βˆ…)
103101, 102sylan2 592 . . . . . . . . . . . . . . . . 17 ((π‘₯ ∈ Word 𝑉 ∧ ((β™―β€˜π‘₯) = (𝑁 + 2) ∧ 𝑁 ∈ β„•0)) β†’ π‘₯ β‰  βˆ…)
104103exp32 420 . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . 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 481 . . . . . . . . . . . 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 406 . . . . . . . . . . 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 515 . . . . . . . . . 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 480 . . . . . . . . 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 482 . . . . . . . . . . . 12 ((𝑑 ∈ Word 𝑉 ∧ π‘₯ ∈ Word 𝑉) β†’ 𝑑 ∈ Word 𝑉)
113112adantr 480 . . . . . . . . . . 11 (((𝑑 ∈ Word 𝑉 ∧ π‘₯ ∈ Word 𝑉) ∧ (𝑑 β‰  βˆ… ∧ π‘₯ β‰  βˆ…)) β†’ 𝑑 ∈ Word 𝑉)
114 simpr 484 . . . . . . . . . . . 12 ((𝑑 ∈ Word 𝑉 ∧ π‘₯ ∈ Word 𝑉) β†’ π‘₯ ∈ Word 𝑉)
115114adantr 480 . . . . . . . . . . 11 (((𝑑 ∈ Word 𝑉 ∧ π‘₯ ∈ Word 𝑉) ∧ (𝑑 β‰  βˆ… ∧ π‘₯ β‰  βˆ…)) β†’ π‘₯ ∈ Word 𝑉)
116 hashneq0 14341 . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . 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 480 . . . . . . . . . . . 12 ((𝑑 β‰  βˆ… ∧ π‘₯ β‰  βˆ…) β†’ ((𝑑 ∈ Word 𝑉 ∧ π‘₯ ∈ Word 𝑉) β†’ 0 < (β™―β€˜π‘‘)))
121120impcom 407 . . . . . . . . . . 11 (((𝑑 ∈ Word 𝑉 ∧ π‘₯ ∈ Word 𝑉) ∧ (𝑑 β‰  βˆ… ∧ π‘₯ β‰  βˆ…)) β†’ 0 < (β™―β€˜π‘‘))
122 pfxsuff1eqwrdeq 14667 . . . . . . . . . . 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 460 . . . . . . . . . . . 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 622 . . . . . . . . . . 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 278 . . . . . . . . . 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 287 . . . . . . . . 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 419 . . . . . 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 597 . . . . 5 ((𝑑 ∈ 𝐷 ∧ π‘₯ ∈ 𝐷) β†’ (𝑁 ∈ β„•0 β†’ ((lastSβ€˜π‘‘) = (lastSβ€˜π‘₯) β†’ 𝑑 = π‘₯)))
133132impcom 407 . . . 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 3195 . 2 (𝑁 ∈ β„•0 β†’ βˆ€π‘‘ ∈ 𝐷 βˆ€π‘₯ ∈ 𝐷 ((πΉβ€˜π‘‘) = (πΉβ€˜π‘₯) β†’ 𝑑 = π‘₯))
136 dff13 7259 . 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 582 1 (𝑁 ∈ β„•0 β†’ 𝐹:𝐷–1-1→𝑅)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   β‰  wne 2935  βˆ€wral 3056  {crab 3427  βˆ…c0 4318  {cpr 4626   class class class wbr 5142   ↦ cmpt 5225  βŸΆwf 6538  β€“1-1β†’wf1 6539  β€˜cfv 6542  (class class class)co 7414  β„cr 11123  0cc0 11124  1c1 11125   + caddc 11127   < clt 11264   βˆ’ cmin 11460  2c2 12283  β„•0cn0 12488  β™―chash 14307  Word cword 14482  lastSclsw 14530   prefix cpfx 14638  Vtxcvtx 28783  Edgcedg 28834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  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 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7863  df-1st 7985  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-er 8716  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-card 9948  df-pnf 11266  df-mnf 11267  df-xr 11268  df-ltxr 11269  df-le 11270  df-sub 11462  df-neg 11463  df-nn 12229  df-2 12291  df-n0 12489  df-xnn0 12561  df-z 12575  df-uz 12839  df-fz 13503  df-fzo 13646  df-hash 14308  df-word 14483  df-lsw 14531  df-s1 14564  df-substr 14609  df-pfx 14639
This theorem is referenced by:  wwlksnextbij0  29686
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