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Theorem 2swrd2eqwrdeq 14904
Description: Two words of length at least two are equal if and only if they have the same prefix and the same two single symbols suffix. (Contributed by AV, 24-Sep-2018.) (Revised by AV, 12-Oct-2022.)
Assertion
Ref Expression
2swrd2eqwrdeq ((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) β†’ (π‘Š = π‘ˆ ↔ ((β™―β€˜π‘Š) = (β™―β€˜π‘ˆ) ∧ ((π‘Š prefix ((β™―β€˜π‘Š) βˆ’ 2)) = (π‘ˆ prefix ((β™―β€˜π‘Š) βˆ’ 2)) ∧ (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2)) = (π‘ˆβ€˜((β™―β€˜π‘Š) βˆ’ 2)) ∧ (lastSβ€˜π‘Š) = (lastSβ€˜π‘ˆ)))))
Proof of Theorem 2swrd2eqwrdeq
StepHypRef Expression
1 lencl 14483 . . . . 5 (π‘Š ∈ Word 𝑉 β†’ (β™―β€˜π‘Š) ∈ β„•0)
2 1z 12592 . . . . . . . . . 10 1 ∈ β„€
3 nn0z 12583 . . . . . . . . . 10 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (β™―β€˜π‘Š) ∈ β„€)
4 zltp1le 12612 . . . . . . . . . 10 ((1 ∈ β„€ ∧ (β™―β€˜π‘Š) ∈ β„€) β†’ (1 < (β™―β€˜π‘Š) ↔ (1 + 1) ≀ (β™―β€˜π‘Š)))
52, 3, 4sylancr 588 . . . . . . . . 9 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (1 < (β™―β€˜π‘Š) ↔ (1 + 1) ≀ (β™―β€˜π‘Š)))
6 1p1e2 12337 . . . . . . . . . . . 12 (1 + 1) = 2
76a1i 11 . . . . . . . . . . 11 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (1 + 1) = 2)
87breq1d 5159 . . . . . . . . . 10 ((β™―β€˜π‘Š) ∈ β„•0 β†’ ((1 + 1) ≀ (β™―β€˜π‘Š) ↔ 2 ≀ (β™―β€˜π‘Š)))
98biimpd 228 . . . . . . . . 9 ((β™―β€˜π‘Š) ∈ β„•0 β†’ ((1 + 1) ≀ (β™―β€˜π‘Š) β†’ 2 ≀ (β™―β€˜π‘Š)))
105, 9sylbid 239 . . . . . . . 8 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (1 < (β™―β€˜π‘Š) β†’ 2 ≀ (β™―β€˜π‘Š)))
1110imp 408 . . . . . . 7 (((β™―β€˜π‘Š) ∈ β„•0 ∧ 1 < (β™―β€˜π‘Š)) β†’ 2 ≀ (β™―β€˜π‘Š))
12 2nn0 12489 . . . . . . . 8 2 ∈ β„•0
13 simpl 484 . . . . . . . 8 (((β™―β€˜π‘Š) ∈ β„•0 ∧ 1 < (β™―β€˜π‘Š)) β†’ (β™―β€˜π‘Š) ∈ β„•0)
14 nn0sub 12522 . . . . . . . 8 ((2 ∈ β„•0 ∧ (β™―β€˜π‘Š) ∈ β„•0) β†’ (2 ≀ (β™―β€˜π‘Š) ↔ ((β™―β€˜π‘Š) βˆ’ 2) ∈ β„•0))
1512, 13, 14sylancr 588 . . . . . . 7 (((β™―β€˜π‘Š) ∈ β„•0 ∧ 1 < (β™―β€˜π‘Š)) β†’ (2 ≀ (β™―β€˜π‘Š) ↔ ((β™―β€˜π‘Š) βˆ’ 2) ∈ β„•0))
1611, 15mpbid 231 . . . . . 6 (((β™―β€˜π‘Š) ∈ β„•0 ∧ 1 < (β™―β€˜π‘Š)) β†’ ((β™―β€˜π‘Š) βˆ’ 2) ∈ β„•0)
173adantr 482 . . . . . . 7 (((β™―β€˜π‘Š) ∈ β„•0 ∧ 1 < (β™―β€˜π‘Š)) β†’ (β™―β€˜π‘Š) ∈ β„€)
18 0red 11217 . . . . . . . . . 10 ((β™―β€˜π‘Š) ∈ β„•0 β†’ 0 ∈ ℝ)
19 1red 11215 . . . . . . . . . 10 ((β™―β€˜π‘Š) ∈ β„•0 β†’ 1 ∈ ℝ)
20 nn0re 12481 . . . . . . . . . 10 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (β™―β€˜π‘Š) ∈ ℝ)
2118, 19, 203jca 1129 . . . . . . . . 9 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (0 ∈ ℝ ∧ 1 ∈ ℝ ∧ (β™―β€˜π‘Š) ∈ ℝ))
22 0lt1 11736 . . . . . . . . 9 0 < 1
23 lttr 11290 . . . . . . . . . 10 ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ (β™―β€˜π‘Š) ∈ ℝ) β†’ ((0 < 1 ∧ 1 < (β™―β€˜π‘Š)) β†’ 0 < (β™―β€˜π‘Š)))
2423expd 417 . . . . . . . . 9 ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ (β™―β€˜π‘Š) ∈ ℝ) β†’ (0 < 1 β†’ (1 < (β™―β€˜π‘Š) β†’ 0 < (β™―β€˜π‘Š))))
2521, 22, 24mpisyl 21 . . . . . . . 8 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (1 < (β™―β€˜π‘Š) β†’ 0 < (β™―β€˜π‘Š)))
2625imp 408 . . . . . . 7 (((β™―β€˜π‘Š) ∈ β„•0 ∧ 1 < (β™―β€˜π‘Š)) β†’ 0 < (β™―β€˜π‘Š))
27 elnnz 12568 . . . . . . 7 ((β™―β€˜π‘Š) ∈ β„• ↔ ((β™―β€˜π‘Š) ∈ β„€ ∧ 0 < (β™―β€˜π‘Š)))
2817, 26, 27sylanbrc 584 . . . . . 6 (((β™―β€˜π‘Š) ∈ β„•0 ∧ 1 < (β™―β€˜π‘Š)) β†’ (β™―β€˜π‘Š) ∈ β„•)
29 2rp 12979 . . . . . . . . 9 2 ∈ ℝ+
3029a1i 11 . . . . . . . 8 ((β™―β€˜π‘Š) ∈ β„•0 β†’ 2 ∈ ℝ+)
3120, 30ltsubrpd 13048 . . . . . . 7 ((β™―β€˜π‘Š) ∈ β„•0 β†’ ((β™―β€˜π‘Š) βˆ’ 2) < (β™―β€˜π‘Š))
3231adantr 482 . . . . . 6 (((β™―β€˜π‘Š) ∈ β„•0 ∧ 1 < (β™―β€˜π‘Š)) β†’ ((β™―β€˜π‘Š) βˆ’ 2) < (β™―β€˜π‘Š))
33 elfzo0 13673 . . . . . 6 (((β™―β€˜π‘Š) βˆ’ 2) ∈ (0..^(β™―β€˜π‘Š)) ↔ (((β™―β€˜π‘Š) βˆ’ 2) ∈ β„•0 ∧ (β™―β€˜π‘Š) ∈ β„• ∧ ((β™―β€˜π‘Š) βˆ’ 2) < (β™―β€˜π‘Š)))
3416, 28, 32, 33syl3anbrc 1344 . . . . 5 (((β™―β€˜π‘Š) ∈ β„•0 ∧ 1 < (β™―β€˜π‘Š)) β†’ ((β™―β€˜π‘Š) βˆ’ 2) ∈ (0..^(β™―β€˜π‘Š)))
351, 34sylan 581 . . . 4 ((π‘Š ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) β†’ ((β™―β€˜π‘Š) βˆ’ 2) ∈ (0..^(β™―β€˜π‘Š)))
36353adant2 1132 . . 3 ((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) β†’ ((β™―β€˜π‘Š) βˆ’ 2) ∈ (0..^(β™―β€˜π‘Š)))
37 pfxsuffeqwrdeq 14648 . . 3 ((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ ((β™―β€˜π‘Š) βˆ’ 2) ∈ (0..^(β™―β€˜π‘Š))) β†’ (π‘Š = π‘ˆ ↔ ((β™―β€˜π‘Š) = (β™―β€˜π‘ˆ) ∧ ((π‘Š prefix ((β™―β€˜π‘Š) βˆ’ 2)) = (π‘ˆ prefix ((β™―β€˜π‘Š) βˆ’ 2)) ∧ (π‘Š substr ⟨((β™―β€˜π‘Š) βˆ’ 2), (β™―β€˜π‘Š)⟩) = (π‘ˆ substr ⟨((β™―β€˜π‘Š) βˆ’ 2), (β™―β€˜π‘Š)⟩)))))
3836, 37syld3an3 1410 . 2 ((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) β†’ (π‘Š = π‘ˆ ↔ ((β™―β€˜π‘Š) = (β™―β€˜π‘ˆ) ∧ ((π‘Š prefix ((β™―β€˜π‘Š) βˆ’ 2)) = (π‘ˆ prefix ((β™―β€˜π‘Š) βˆ’ 2)) ∧ (π‘Š substr ⟨((β™―β€˜π‘Š) βˆ’ 2), (β™―β€˜π‘Š)⟩) = (π‘ˆ substr ⟨((β™―β€˜π‘Š) βˆ’ 2), (β™―β€˜π‘Š)⟩)))))
39 swrd2lsw 14903 . . . . . . . . 9 ((π‘Š ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) β†’ (π‘Š substr ⟨((β™―β€˜π‘Š) βˆ’ 2), (β™―β€˜π‘Š)⟩) = βŸ¨β€œ(π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2))(lastSβ€˜π‘Š)β€βŸ©)
40393adant2 1132 . . . . . . . 8 ((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) β†’ (π‘Š substr ⟨((β™―β€˜π‘Š) βˆ’ 2), (β™―β€˜π‘Š)⟩) = βŸ¨β€œ(π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2))(lastSβ€˜π‘Š)β€βŸ©)
4140adantr 482 . . . . . . 7 (((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) ∧ (β™―β€˜π‘Š) = (β™―β€˜π‘ˆ)) β†’ (π‘Š substr ⟨((β™―β€˜π‘Š) βˆ’ 2), (β™―β€˜π‘Š)⟩) = βŸ¨β€œ(π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2))(lastSβ€˜π‘Š)β€βŸ©)
42 breq2 5153 . . . . . . . . . . 11 ((β™―β€˜π‘Š) = (β™―β€˜π‘ˆ) β†’ (1 < (β™―β€˜π‘Š) ↔ 1 < (β™―β€˜π‘ˆ)))
43423anbi3d 1443 . . . . . . . . . 10 ((β™―β€˜π‘Š) = (β™―β€˜π‘ˆ) β†’ ((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) ↔ (π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘ˆ))))
44 swrd2lsw 14903 . . . . . . . . . . 11 ((π‘ˆ ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘ˆ)) β†’ (π‘ˆ substr ⟨((β™―β€˜π‘ˆ) βˆ’ 2), (β™―β€˜π‘ˆ)⟩) = βŸ¨β€œ(π‘ˆβ€˜((β™―β€˜π‘ˆ) βˆ’ 2))(lastSβ€˜π‘ˆ)β€βŸ©)
45443adant1 1131 . . . . . . . . . 10 ((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘ˆ)) β†’ (π‘ˆ substr ⟨((β™―β€˜π‘ˆ) βˆ’ 2), (β™―β€˜π‘ˆ)⟩) = βŸ¨β€œ(π‘ˆβ€˜((β™―β€˜π‘ˆ) βˆ’ 2))(lastSβ€˜π‘ˆ)β€βŸ©)
4643, 45syl6bi 253 . . . . . . . . 9 ((β™―β€˜π‘Š) = (β™―β€˜π‘ˆ) β†’ ((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) β†’ (π‘ˆ substr ⟨((β™―β€˜π‘ˆ) βˆ’ 2), (β™―β€˜π‘ˆ)⟩) = βŸ¨β€œ(π‘ˆβ€˜((β™―β€˜π‘ˆ) βˆ’ 2))(lastSβ€˜π‘ˆ)β€βŸ©))
4746impcom 409 . . . . . . . 8 (((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) ∧ (β™―β€˜π‘Š) = (β™―β€˜π‘ˆ)) β†’ (π‘ˆ substr ⟨((β™―β€˜π‘ˆ) βˆ’ 2), (β™―β€˜π‘ˆ)⟩) = βŸ¨β€œ(π‘ˆβ€˜((β™―β€˜π‘ˆ) βˆ’ 2))(lastSβ€˜π‘ˆ)β€βŸ©)
48 oveq1 7416 . . . . . . . . . . . 12 ((β™―β€˜π‘Š) = (β™―β€˜π‘ˆ) β†’ ((β™―β€˜π‘Š) βˆ’ 2) = ((β™―β€˜π‘ˆ) βˆ’ 2))
49 id 22 . . . . . . . . . . . 12 ((β™―β€˜π‘Š) = (β™―β€˜π‘ˆ) β†’ (β™―β€˜π‘Š) = (β™―β€˜π‘ˆ))
5048, 49opeq12d 4882 . . . . . . . . . . 11 ((β™―β€˜π‘Š) = (β™―β€˜π‘ˆ) β†’ ⟨((β™―β€˜π‘Š) βˆ’ 2), (β™―β€˜π‘Š)⟩ = ⟨((β™―β€˜π‘ˆ) βˆ’ 2), (β™―β€˜π‘ˆ)⟩)
5150oveq2d 7425 . . . . . . . . . 10 ((β™―β€˜π‘Š) = (β™―β€˜π‘ˆ) β†’ (π‘ˆ substr ⟨((β™―β€˜π‘Š) βˆ’ 2), (β™―β€˜π‘Š)⟩) = (π‘ˆ substr ⟨((β™―β€˜π‘ˆ) βˆ’ 2), (β™―β€˜π‘ˆ)⟩))
5251eqeq1d 2735 . . . . . . . . 9 ((β™―β€˜π‘Š) = (β™―β€˜π‘ˆ) β†’ ((π‘ˆ substr ⟨((β™―β€˜π‘Š) βˆ’ 2), (β™―β€˜π‘Š)⟩) = βŸ¨β€œ(π‘ˆβ€˜((β™―β€˜π‘ˆ) βˆ’ 2))(lastSβ€˜π‘ˆ)β€βŸ© ↔ (π‘ˆ substr ⟨((β™―β€˜π‘ˆ) βˆ’ 2), (β™―β€˜π‘ˆ)⟩) = βŸ¨β€œ(π‘ˆβ€˜((β™―β€˜π‘ˆ) βˆ’ 2))(lastSβ€˜π‘ˆ)β€βŸ©))
5352adantl 483 . . . . . . . 8 (((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) ∧ (β™―β€˜π‘Š) = (β™―β€˜π‘ˆ)) β†’ ((π‘ˆ substr ⟨((β™―β€˜π‘Š) βˆ’ 2), (β™―β€˜π‘Š)⟩) = βŸ¨β€œ(π‘ˆβ€˜((β™―β€˜π‘ˆ) βˆ’ 2))(lastSβ€˜π‘ˆ)β€βŸ© ↔ (π‘ˆ substr ⟨((β™―β€˜π‘ˆ) βˆ’ 2), (β™―β€˜π‘ˆ)⟩) = βŸ¨β€œ(π‘ˆβ€˜((β™―β€˜π‘ˆ) βˆ’ 2))(lastSβ€˜π‘ˆ)β€βŸ©))
5447, 53mpbird 257 . . . . . . 7 (((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) ∧ (β™―β€˜π‘Š) = (β™―β€˜π‘ˆ)) β†’ (π‘ˆ substr ⟨((β™―β€˜π‘Š) βˆ’ 2), (β™―β€˜π‘Š)⟩) = βŸ¨β€œ(π‘ˆβ€˜((β™―β€˜π‘ˆ) βˆ’ 2))(lastSβ€˜π‘ˆ)β€βŸ©)
5541, 54eqeq12d 2749 . . . . . 6 (((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) ∧ (β™―β€˜π‘Š) = (β™―β€˜π‘ˆ)) β†’ ((π‘Š substr ⟨((β™―β€˜π‘Š) βˆ’ 2), (β™―β€˜π‘Š)⟩) = (π‘ˆ substr ⟨((β™―β€˜π‘Š) βˆ’ 2), (β™―β€˜π‘Š)⟩) ↔ βŸ¨β€œ(π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2))(lastSβ€˜π‘Š)β€βŸ© = βŸ¨β€œ(π‘ˆβ€˜((β™―β€˜π‘ˆ) βˆ’ 2))(lastSβ€˜π‘ˆ)β€βŸ©))
56 fvexd 6907 . . . . . . 7 (((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) ∧ (β™―β€˜π‘Š) = (β™―β€˜π‘ˆ)) β†’ (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2)) ∈ V)
57 fvexd 6907 . . . . . . 7 (((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) ∧ (β™―β€˜π‘Š) = (β™―β€˜π‘ˆ)) β†’ (lastSβ€˜π‘Š) ∈ V)
58 fvexd 6907 . . . . . . 7 (((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) ∧ (β™―β€˜π‘Š) = (β™―β€˜π‘ˆ)) β†’ (π‘ˆβ€˜((β™―β€˜π‘ˆ) βˆ’ 2)) ∈ V)
59 fvexd 6907 . . . . . . 7 (((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) ∧ (β™―β€˜π‘Š) = (β™―β€˜π‘ˆ)) β†’ (lastSβ€˜π‘ˆ) ∈ V)
60 s2eq2s1eq 14887 . . . . . . 7 ((((π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2)) ∈ V ∧ (lastSβ€˜π‘Š) ∈ V) ∧ ((π‘ˆβ€˜((β™―β€˜π‘ˆ) βˆ’ 2)) ∈ V ∧ (lastSβ€˜π‘ˆ) ∈ V)) β†’ (βŸ¨β€œ(π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2))(lastSβ€˜π‘Š)β€βŸ© = βŸ¨β€œ(π‘ˆβ€˜((β™―β€˜π‘ˆ) βˆ’ 2))(lastSβ€˜π‘ˆ)β€βŸ© ↔ (βŸ¨β€œ(π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2))β€βŸ© = βŸ¨β€œ(π‘ˆβ€˜((β™―β€˜π‘ˆ) βˆ’ 2))β€βŸ© ∧ βŸ¨β€œ(lastSβ€˜π‘Š)β€βŸ© = βŸ¨β€œ(lastSβ€˜π‘ˆ)β€βŸ©)))
6156, 57, 58, 59, 60syl22anc 838 . . . . . 6 (((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) ∧ (β™―β€˜π‘Š) = (β™―β€˜π‘ˆ)) β†’ (βŸ¨β€œ(π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2))(lastSβ€˜π‘Š)β€βŸ© = βŸ¨β€œ(π‘ˆβ€˜((β™―β€˜π‘ˆ) βˆ’ 2))(lastSβ€˜π‘ˆ)β€βŸ© ↔ (βŸ¨β€œ(π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2))β€βŸ© = βŸ¨β€œ(π‘ˆβ€˜((β™―β€˜π‘ˆ) βˆ’ 2))β€βŸ© ∧ βŸ¨β€œ(lastSβ€˜π‘Š)β€βŸ© = βŸ¨β€œ(lastSβ€˜π‘ˆ)β€βŸ©)))
62 fvex 6905 . . . . . . . . 9 (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2)) ∈ V
63 s111 14565 . . . . . . . . 9 (((π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2)) ∈ V ∧ (π‘ˆβ€˜((β™―β€˜π‘ˆ) βˆ’ 2)) ∈ V) β†’ (βŸ¨β€œ(π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2))β€βŸ© = βŸ¨β€œ(π‘ˆβ€˜((β™―β€˜π‘ˆ) βˆ’ 2))β€βŸ© ↔ (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2)) = (π‘ˆβ€˜((β™―β€˜π‘ˆ) βˆ’ 2))))
6462, 58, 63sylancr 588 . . . . . . . 8 (((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) ∧ (β™―β€˜π‘Š) = (β™―β€˜π‘ˆ)) β†’ (βŸ¨β€œ(π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2))β€βŸ© = βŸ¨β€œ(π‘ˆβ€˜((β™―β€˜π‘ˆ) βˆ’ 2))β€βŸ© ↔ (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2)) = (π‘ˆβ€˜((β™―β€˜π‘ˆ) βˆ’ 2))))
65 fvoveq1 7432 . . . . . . . . . . 11 ((β™―β€˜π‘ˆ) = (β™―β€˜π‘Š) β†’ (π‘ˆβ€˜((β™―β€˜π‘ˆ) βˆ’ 2)) = (π‘ˆβ€˜((β™―β€˜π‘Š) βˆ’ 2)))
6665eqcoms 2741 . . . . . . . . . 10 ((β™―β€˜π‘Š) = (β™―β€˜π‘ˆ) β†’ (π‘ˆβ€˜((β™―β€˜π‘ˆ) βˆ’ 2)) = (π‘ˆβ€˜((β™―β€˜π‘Š) βˆ’ 2)))
6766adantl 483 . . . . . . . . 9 (((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) ∧ (β™―β€˜π‘Š) = (β™―β€˜π‘ˆ)) β†’ (π‘ˆβ€˜((β™―β€˜π‘ˆ) βˆ’ 2)) = (π‘ˆβ€˜((β™―β€˜π‘Š) βˆ’ 2)))
6867eqeq2d 2744 . . . . . . . 8 (((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) ∧ (β™―β€˜π‘Š) = (β™―β€˜π‘ˆ)) β†’ ((π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2)) = (π‘ˆβ€˜((β™―β€˜π‘ˆ) βˆ’ 2)) ↔ (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2)) = (π‘ˆβ€˜((β™―β€˜π‘Š) βˆ’ 2))))
6964, 68bitrd 279 . . . . . . 7 (((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) ∧ (β™―β€˜π‘Š) = (β™―β€˜π‘ˆ)) β†’ (βŸ¨β€œ(π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2))β€βŸ© = βŸ¨β€œ(π‘ˆβ€˜((β™―β€˜π‘ˆ) βˆ’ 2))β€βŸ© ↔ (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2)) = (π‘ˆβ€˜((β™―β€˜π‘Š) βˆ’ 2))))
70 fvex 6905 . . . . . . . 8 (lastSβ€˜π‘Š) ∈ V
71 s111 14565 . . . . . . . 8 (((lastSβ€˜π‘Š) ∈ V ∧ (lastSβ€˜π‘ˆ) ∈ V) β†’ (βŸ¨β€œ(lastSβ€˜π‘Š)β€βŸ© = βŸ¨β€œ(lastSβ€˜π‘ˆ)β€βŸ© ↔ (lastSβ€˜π‘Š) = (lastSβ€˜π‘ˆ)))
7270, 59, 71sylancr 588 . . . . . . 7 (((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) ∧ (β™―β€˜π‘Š) = (β™―β€˜π‘ˆ)) β†’ (βŸ¨β€œ(lastSβ€˜π‘Š)β€βŸ© = βŸ¨β€œ(lastSβ€˜π‘ˆ)β€βŸ© ↔ (lastSβ€˜π‘Š) = (lastSβ€˜π‘ˆ)))
7369, 72anbi12d 632 . . . . . 6 (((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) ∧ (β™―β€˜π‘Š) = (β™―β€˜π‘ˆ)) β†’ ((βŸ¨β€œ(π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2))β€βŸ© = βŸ¨β€œ(π‘ˆβ€˜((β™―β€˜π‘ˆ) βˆ’ 2))β€βŸ© ∧ βŸ¨β€œ(lastSβ€˜π‘Š)β€βŸ© = βŸ¨β€œ(lastSβ€˜π‘ˆ)β€βŸ©) ↔ ((π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2)) = (π‘ˆβ€˜((β™―β€˜π‘Š) βˆ’ 2)) ∧ (lastSβ€˜π‘Š) = (lastSβ€˜π‘ˆ))))
7455, 61, 733bitrd 305 . . . . 5 (((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) ∧ (β™―β€˜π‘Š) = (β™―β€˜π‘ˆ)) β†’ ((π‘Š substr ⟨((β™―β€˜π‘Š) βˆ’ 2), (β™―β€˜π‘Š)⟩) = (π‘ˆ substr ⟨((β™―β€˜π‘Š) βˆ’ 2), (β™―β€˜π‘Š)⟩) ↔ ((π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2)) = (π‘ˆβ€˜((β™―β€˜π‘Š) βˆ’ 2)) ∧ (lastSβ€˜π‘Š) = (lastSβ€˜π‘ˆ))))
7574anbi2d 630 . . . 4 (((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) ∧ (β™―β€˜π‘Š) = (β™―β€˜π‘ˆ)) β†’ (((π‘Š prefix ((β™―β€˜π‘Š) βˆ’ 2)) = (π‘ˆ prefix ((β™―β€˜π‘Š) βˆ’ 2)) ∧ (π‘Š substr ⟨((β™―β€˜π‘Š) βˆ’ 2), (β™―β€˜π‘Š)⟩) = (π‘ˆ substr ⟨((β™―β€˜π‘Š) βˆ’ 2), (β™―β€˜π‘Š)⟩)) ↔ ((π‘Š prefix ((β™―β€˜π‘Š) βˆ’ 2)) = (π‘ˆ prefix ((β™―β€˜π‘Š) βˆ’ 2)) ∧ ((π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2)) = (π‘ˆβ€˜((β™―β€˜π‘Š) βˆ’ 2)) ∧ (lastSβ€˜π‘Š) = (lastSβ€˜π‘ˆ)))))
76 3anass 1096 . . . 4 (((π‘Š prefix ((β™―β€˜π‘Š) βˆ’ 2)) = (π‘ˆ prefix ((β™―β€˜π‘Š) βˆ’ 2)) ∧ (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2)) = (π‘ˆβ€˜((β™―β€˜π‘Š) βˆ’ 2)) ∧ (lastSβ€˜π‘Š) = (lastSβ€˜π‘ˆ)) ↔ ((π‘Š prefix ((β™―β€˜π‘Š) βˆ’ 2)) = (π‘ˆ prefix ((β™―β€˜π‘Š) βˆ’ 2)) ∧ ((π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2)) = (π‘ˆβ€˜((β™―β€˜π‘Š) βˆ’ 2)) ∧ (lastSβ€˜π‘Š) = (lastSβ€˜π‘ˆ))))
7775, 76bitr4di 289 . . 3 (((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) ∧ (β™―β€˜π‘Š) = (β™―β€˜π‘ˆ)) β†’ (((π‘Š prefix ((β™―β€˜π‘Š) βˆ’ 2)) = (π‘ˆ prefix ((β™―β€˜π‘Š) βˆ’ 2)) ∧ (π‘Š substr ⟨((β™―β€˜π‘Š) βˆ’ 2), (β™―β€˜π‘Š)⟩) = (π‘ˆ substr ⟨((β™―β€˜π‘Š) βˆ’ 2), (β™―β€˜π‘Š)⟩)) ↔ ((π‘Š prefix ((β™―β€˜π‘Š) βˆ’ 2)) = (π‘ˆ prefix ((β™―β€˜π‘Š) βˆ’ 2)) ∧ (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2)) = (π‘ˆβ€˜((β™―β€˜π‘Š) βˆ’ 2)) ∧ (lastSβ€˜π‘Š) = (lastSβ€˜π‘ˆ))))
7877pm5.32da 580 . 2 ((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) β†’ (((β™―β€˜π‘Š) = (β™―β€˜π‘ˆ) ∧ ((π‘Š prefix ((β™―β€˜π‘Š) βˆ’ 2)) = (π‘ˆ prefix ((β™―β€˜π‘Š) βˆ’ 2)) ∧ (π‘Š substr ⟨((β™―β€˜π‘Š) βˆ’ 2), (β™―β€˜π‘Š)⟩) = (π‘ˆ substr ⟨((β™―β€˜π‘Š) βˆ’ 2), (β™―β€˜π‘Š)⟩))) ↔ ((β™―β€˜π‘Š) = (β™―β€˜π‘ˆ) ∧ ((π‘Š prefix ((β™―β€˜π‘Š) βˆ’ 2)) = (π‘ˆ prefix ((β™―β€˜π‘Š) βˆ’ 2)) ∧ (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2)) = (π‘ˆβ€˜((β™―β€˜π‘Š) βˆ’ 2)) ∧ (lastSβ€˜π‘Š) = (lastSβ€˜π‘ˆ)))))
7938, 78bitrd 279 1 ((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) β†’ (π‘Š = π‘ˆ ↔ ((β™―β€˜π‘Š) = (β™―β€˜π‘ˆ) ∧ ((π‘Š prefix ((β™―β€˜π‘Š) βˆ’ 2)) = (π‘ˆ prefix ((β™―β€˜π‘Š) βˆ’ 2)) ∧ (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2)) = (π‘ˆβ€˜((β™―β€˜π‘Š) βˆ’ 2)) ∧ (lastSβ€˜π‘Š) = (lastSβ€˜π‘ˆ)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  Vcvv 3475  βŸ¨cop 4635   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409  β„cr 11109  0cc0 11110  1c1 11111   + caddc 11113   < clt 11248   ≀ cle 11249   βˆ’ cmin 11444  β„•cn 12212  2c2 12267  β„•0cn0 12472  β„€cz 12558  β„+crp 12974  ..^cfzo 13627  β™―chash 14290  Word cword 14464  lastSclsw 14512  βŸ¨β€œcs1 14545   substr csubstr 14590   prefix cpfx 14620  βŸ¨β€œcs2 14792
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-n0 12473  df-z 12559  df-uz 12823  df-rp 12975  df-fz 13485  df-fzo 13628  df-hash 14291  df-word 14465  df-lsw 14513  df-concat 14521  df-s1 14546  df-substr 14591  df-pfx 14621  df-s2 14799
This theorem is referenced by:  numclwwlk1lem2f1  29610
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