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Theorem wwlksnextinj 28941
Description: Lemma for wwlksnextbij 28944. (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 28940 . 2 (𝑁 ∈ β„•0 β†’ 𝐹:π·βŸΆπ‘…)
7 fveq2 6862 . . . . . . 7 (𝑑 = 𝑑 β†’ (lastSβ€˜π‘‘) = (lastSβ€˜π‘‘))
8 fvex 6875 . . . . . . 7 (lastSβ€˜π‘‘) ∈ V
97, 5, 8fvmpt 6968 . . . . . 6 (𝑑 ∈ 𝐷 β†’ (πΉβ€˜π‘‘) = (lastSβ€˜π‘‘))
10 fveq2 6862 . . . . . . 7 (𝑑 = π‘₯ β†’ (lastSβ€˜π‘‘) = (lastSβ€˜π‘₯))
11 fvex 6875 . . . . . . 7 (lastSβ€˜π‘₯) ∈ V
1210, 5, 11fvmpt 6968 . . . . . 6 (π‘₯ ∈ 𝐷 β†’ (πΉβ€˜π‘₯) = (lastSβ€˜π‘₯))
139, 12eqeqan12d 2745 . . . . 5 ((𝑑 ∈ 𝐷 ∧ π‘₯ ∈ 𝐷) β†’ ((πΉβ€˜π‘‘) = (πΉβ€˜π‘₯) ↔ (lastSβ€˜π‘‘) = (lastSβ€˜π‘₯)))
1413adantl 482 . . . 4 ((𝑁 ∈ β„•0 ∧ (𝑑 ∈ 𝐷 ∧ π‘₯ ∈ 𝐷)) β†’ ((πΉβ€˜π‘‘) = (πΉβ€˜π‘₯) ↔ (lastSβ€˜π‘‘) = (lastSβ€˜π‘₯)))
15 fveqeq2 6871 . . . . . . . 8 (𝑀 = 𝑑 β†’ ((β™―β€˜π‘€) = (𝑁 + 2) ↔ (β™―β€˜π‘‘) = (𝑁 + 2)))
16 oveq1 7384 . . . . . . . . 9 (𝑀 = 𝑑 β†’ (𝑀 prefix (𝑁 + 1)) = (𝑑 prefix (𝑁 + 1)))
1716eqeq1d 2733 . . . . . . . 8 (𝑀 = 𝑑 β†’ ((𝑀 prefix (𝑁 + 1)) = π‘Š ↔ (𝑑 prefix (𝑁 + 1)) = π‘Š))
18 fveq2 6862 . . . . . . . . . 10 (𝑀 = 𝑑 β†’ (lastSβ€˜π‘€) = (lastSβ€˜π‘‘))
1918preq2d 4721 . . . . . . . . 9 (𝑀 = 𝑑 β†’ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} = {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)})
2019eleq1d 2817 . . . . . . . 8 (𝑀 = 𝑑 β†’ ({(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸 ↔ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸))
2115, 17, 203anbi123d 1436 . . . . . . 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 3666 . . . . . 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 6871 . . . . . . . 8 (𝑀 = π‘₯ β†’ ((β™―β€˜π‘€) = (𝑁 + 2) ↔ (β™―β€˜π‘₯) = (𝑁 + 2)))
24 oveq1 7384 . . . . . . . . 9 (𝑀 = π‘₯ β†’ (𝑀 prefix (𝑁 + 1)) = (π‘₯ prefix (𝑁 + 1)))
2524eqeq1d 2733 . . . . . . . 8 (𝑀 = π‘₯ β†’ ((𝑀 prefix (𝑁 + 1)) = π‘Š ↔ (π‘₯ prefix (𝑁 + 1)) = π‘Š))
26 fveq2 6862 . . . . . . . . . 10 (𝑀 = π‘₯ β†’ (lastSβ€˜π‘€) = (lastSβ€˜π‘₯))
2726preq2d 4721 . . . . . . . . 9 (𝑀 = π‘₯ β†’ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} = {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)})
2827eleq1d 2817 . . . . . . . 8 (𝑀 = π‘₯ β†’ ({(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸 ↔ {(lastSβ€˜π‘Š), (lastSβ€˜π‘₯)} ∈ 𝐸))
2923, 25, 283anbi123d 1436 . . . . . . 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 3666 . . . . . 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 2757 . . . . . . . . . . . . . . . . 17 (((β™―β€˜π‘‘) = (𝑁 + 2) ∧ (β™―β€˜π‘₯) = (𝑁 + 2)) β†’ (β™―β€˜π‘‘) = (β™―β€˜π‘₯))
3231expcom 414 . . . . . . . . . . . . . . . 16 ((β™―β€˜π‘₯) = (𝑁 + 2) β†’ ((β™―β€˜π‘‘) = (𝑁 + 2) β†’ (β™―β€˜π‘‘) = (β™―β€˜π‘₯)))
33323ad2ant1 1133 . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . 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 1133 . . . . . . . . . . . 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 482 . . . . . . . . . . 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 407 . . . . . . . . . 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 481 . . . . . . . . 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 481 . . . . . . . 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 485 . . . . . . . 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 2757 . . . . . . . . . . . . . . . . . . . 20 (((𝑑 prefix (𝑁 + 1)) = π‘Š ∧ (π‘₯ prefix (𝑁 + 1)) = π‘Š) β†’ (𝑑 prefix (𝑁 + 1)) = (π‘₯ prefix (𝑁 + 1)))
43 1e2m1 12304 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1 = (2 βˆ’ 1)
4443a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ β„•0 β†’ 1 = (2 βˆ’ 1))
4544oveq2d 7393 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) = (𝑁 + (2 βˆ’ 1)))
46 nn0cn 12447 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ β„‚)
47 2cnd 12255 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ β„•0 β†’ 2 ∈ β„‚)
48 1cnd 11174 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ β„•0 β†’ 1 ∈ β„‚)
4946, 47, 48addsubassd 11556 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ β„•0 β†’ ((𝑁 + 2) βˆ’ 1) = (𝑁 + (2 βˆ’ 1)))
5045, 49eqtr4d 2774 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) = ((𝑁 + 2) βˆ’ 1))
5150adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 ∈ β„•0 ∧ (β™―β€˜π‘‘) = (𝑁 + 2)) β†’ (𝑁 + 1) = ((𝑁 + 2) βˆ’ 1))
52 oveq1 7384 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((β™―β€˜π‘‘) = (𝑁 + 2) β†’ ((β™―β€˜π‘‘) βˆ’ 1) = ((𝑁 + 2) βˆ’ 1))
5352eqeq2d 2742 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((β™―β€˜π‘‘) = (𝑁 + 2) β†’ ((𝑁 + 1) = ((β™―β€˜π‘‘) βˆ’ 1) ↔ (𝑁 + 1) = ((𝑁 + 2) βˆ’ 1)))
5453adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . 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 7385 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 + 1) = ((β™―β€˜π‘‘) βˆ’ 1) β†’ (𝑑 prefix (𝑁 + 1)) = (𝑑 prefix ((β™―β€˜π‘‘) βˆ’ 1)))
57 oveq2 7385 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 + 1) = ((β™―β€˜π‘‘) βˆ’ 1) β†’ (π‘₯ prefix (𝑁 + 1)) = (π‘₯ prefix ((β™―β€˜π‘‘) βˆ’ 1)))
5856, 57eqeq12d 2747 . . . . . . . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . . . . 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 1134 . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . 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 1132 . . . . . . . . . . 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 482 . . . . . . . . . 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 418 . . . . . . . . 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 481 . . . . . . . 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 483 . . . . . . . . . . . . 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 483 . . . . . . . . . . . . 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 613 . . . . . . . . . . . 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 481 . . . . . . . . . . 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 12446 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ ℝ)
80 2re 12251 . . . . . . . . . . . . . . . . . . . . . 22 2 ∈ ℝ
8180a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ β„•0 β†’ 2 ∈ ℝ)
82 nn0ge0 12462 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ β„•0 β†’ 0 ≀ 𝑁)
83 2pos 12280 . . . . . . . . . . . . . . . . . . . . . 22 0 < 2
8483a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ β„•0 β†’ 0 < 2)
8579, 81, 82, 84addgegt0d 11752 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ β„•0 β†’ 0 < (𝑁 + 2))
8685adantl 482 . . . . . . . . . . . . . . . . . . 19 (((β™―β€˜π‘‘) = (𝑁 + 2) ∧ 𝑁 ∈ β„•0) β†’ 0 < (𝑁 + 2))
87 breq2 5129 . . . . . . . . . . . . . . . . . . . 20 ((β™―β€˜π‘‘) = (𝑁 + 2) β†’ (0 < (β™―β€˜π‘‘) ↔ 0 < (𝑁 + 2)))
8887adantr 481 . . . . . . . . . . . . . . . . . . 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 14290 . . . . . . . . . . . . . . . . . 18 ((𝑑 ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘‘)) β†’ 𝑑 β‰  βˆ…)
9189, 90sylan2 593 . . . . . . . . . . . . . . . . 17 ((𝑑 ∈ Word 𝑉 ∧ ((β™―β€˜π‘‘) = (𝑁 + 2) ∧ 𝑁 ∈ β„•0)) β†’ 𝑑 β‰  βˆ…)
9291exp32 421 . . . . . . . . . . . . . . . 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 1133 . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . 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 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†’ 𝑑 β‰  βˆ…))
9796imp 407 . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . 19 (((β™―β€˜π‘₯) = (𝑁 + 2) ∧ 𝑁 ∈ β„•0) β†’ 0 < (𝑁 + 2))
99 breq2 5129 . . . . . . . . . . . . . . . . . . . 20 ((β™―β€˜π‘₯) = (𝑁 + 2) β†’ (0 < (β™―β€˜π‘₯) ↔ 0 < (𝑁 + 2)))
10099adantr 481 . . . . . . . . . . . . . . . . . . 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 14290 . . . . . . . . . . . . . . . . . 18 ((π‘₯ ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘₯)) β†’ π‘₯ β‰  βˆ…)
103101, 102sylan2 593 . . . . . . . . . . . . . . . . 17 ((π‘₯ ∈ Word 𝑉 ∧ ((β™―β€˜π‘₯) = (𝑁 + 2) ∧ 𝑁 ∈ β„•0)) β†’ π‘₯ β‰  βˆ…)
104103exp32 421 . . . . . . . . . . . . . . . 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 1133 . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . 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 482 . . . . . . . . . . . 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 407 . . . . . . . . . . 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 516 . . . . . . . . . 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 481 . . . . . . . . 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 483 . . . . . . . . . . . 12 ((𝑑 ∈ Word 𝑉 ∧ π‘₯ ∈ Word 𝑉) β†’ 𝑑 ∈ Word 𝑉)
113112adantr 481 . . . . . . . . . . 11 (((𝑑 ∈ Word 𝑉 ∧ π‘₯ ∈ Word 𝑉) ∧ (𝑑 β‰  βˆ… ∧ π‘₯ β‰  βˆ…)) β†’ 𝑑 ∈ Word 𝑉)
114 simpr 485 . . . . . . . . . . . 12 ((𝑑 ∈ Word 𝑉 ∧ π‘₯ ∈ Word 𝑉) β†’ π‘₯ ∈ Word 𝑉)
115114adantr 481 . . . . . . . . . . 11 (((𝑑 ∈ Word 𝑉 ∧ π‘₯ ∈ Word 𝑉) ∧ (𝑑 β‰  βˆ… ∧ π‘₯ β‰  βˆ…)) β†’ π‘₯ ∈ Word 𝑉)
116 hashneq0 14289 . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . 12 ((𝑑 β‰  βˆ… ∧ π‘₯ β‰  βˆ…) β†’ ((𝑑 ∈ Word 𝑉 ∧ π‘₯ ∈ Word 𝑉) β†’ 0 < (β™―β€˜π‘‘)))
121120impcom 408 . . . . . . . . . . 11 (((𝑑 ∈ Word 𝑉 ∧ π‘₯ ∈ Word 𝑉) ∧ (𝑑 β‰  βˆ… ∧ π‘₯ β‰  βˆ…)) β†’ 0 < (β™―β€˜π‘‘))
122 pfxsuff1eqwrdeq 14614 . . . . . . . . . . 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 1371 . . . . . . . . . 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 461 . . . . . . . . . . . 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 623 . . . . . . . . . . 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 1095 . . . . . . . . . . 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 1342 . . . . . . 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 420 . . . . . 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 598 . . . . 5 ((𝑑 ∈ 𝐷 ∧ π‘₯ ∈ 𝐷) β†’ (𝑁 ∈ β„•0 β†’ ((lastSβ€˜π‘‘) = (lastSβ€˜π‘₯) β†’ 𝑑 = π‘₯)))
133132impcom 408 . . . 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 3199 . 2 (𝑁 ∈ β„•0 β†’ βˆ€π‘‘ ∈ 𝐷 βˆ€π‘₯ ∈ 𝐷 ((πΉβ€˜π‘‘) = (πΉβ€˜π‘₯) β†’ 𝑑 = π‘₯))
136 dff13 7222 . 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 583 1 (𝑁 ∈ β„•0 β†’ 𝐹:𝐷–1-1→𝑅)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2939  βˆ€wral 3060  {crab 3418  βˆ…c0 4302  {cpr 4608   class class class wbr 5125   ↦ cmpt 5208  βŸΆwf 6512  β€“1-1β†’wf1 6513  β€˜cfv 6516  (class class class)co 7377  β„cr 11074  0cc0 11075  1c1 11076   + caddc 11078   < clt 11213   βˆ’ cmin 11409  2c2 12232  β„•0cn0 12437  β™―chash 14255  Word cword 14429  lastSclsw 14477   prefix cpfx 14585  Vtxcvtx 28044  Edgcedg 28095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5262  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  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 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-pss 3947  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-int 4928  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-tr 5243  df-id 5551  df-eprel 5557  df-po 5565  df-so 5566  df-fr 5608  df-we 5610  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-pred 6273  df-ord 6340  df-on 6341  df-lim 6342  df-suc 6343  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7333  df-ov 7380  df-oprab 7381  df-mpo 7382  df-om 7823  df-1st 7941  df-2nd 7942  df-frecs 8232  df-wrecs 8263  df-recs 8337  df-rdg 8376  df-1o 8432  df-er 8670  df-en 8906  df-dom 8907  df-sdom 8908  df-fin 8909  df-card 9899  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11411  df-neg 11412  df-nn 12178  df-2 12240  df-n0 12438  df-xnn0 12510  df-z 12524  df-uz 12788  df-fz 13450  df-fzo 13593  df-hash 14256  df-word 14430  df-lsw 14478  df-s1 14511  df-substr 14556  df-pfx 14586
This theorem is referenced by:  wwlksnextbij0  28943
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