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Theorem clwlkclwwlklem2a1 29242
Description: Lemma 1 for clwlkclwwlklem2a 29248. (Contributed by Alexander van der Vekens, 21-Jun-2018.) (Revised by AV, 11-Apr-2021.)
Assertion
Ref Expression
clwlkclwwlklem2a1 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ (βˆ€π‘– ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜0)} ∈ ran 𝐸)) β†’ βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸))
Distinct variable groups:   𝑖,𝐸   𝑃,𝑖
Allowed substitution hint:   𝑉(𝑖)

Proof of Theorem clwlkclwwlklem2a1
StepHypRef Expression
1 lencl 14482 . . . . . . . . . . . 12 (𝑃 ∈ Word 𝑉 β†’ (β™―β€˜π‘ƒ) ∈ β„•0)
2 nn0cn 12481 . . . . . . . . . . . 12 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (β™―β€˜π‘ƒ) ∈ β„‚)
3 peano2cnm 11525 . . . . . . . . . . . . . . 15 ((β™―β€˜π‘ƒ) ∈ β„‚ β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ β„‚)
43subid1d 11559 . . . . . . . . . . . . . 14 ((β™―β€˜π‘ƒ) ∈ β„‚ β†’ (((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) = ((β™―β€˜π‘ƒ) βˆ’ 1))
54oveq1d 7423 . . . . . . . . . . . . 13 ((β™―β€˜π‘ƒ) ∈ β„‚ β†’ ((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1) = (((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 1))
6 sub1m1 12463 . . . . . . . . . . . . 13 ((β™―β€˜π‘ƒ) ∈ β„‚ β†’ (((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 1) = ((β™―β€˜π‘ƒ) βˆ’ 2))
75, 6eqtrd 2772 . . . . . . . . . . . 12 ((β™―β€˜π‘ƒ) ∈ β„‚ β†’ ((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1) = ((β™―β€˜π‘ƒ) βˆ’ 2))
81, 2, 73syl 18 . . . . . . . . . . 11 (𝑃 ∈ Word 𝑉 β†’ ((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1) = ((β™―β€˜π‘ƒ) βˆ’ 2))
98adantr 481 . . . . . . . . . 10 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1) = ((β™―β€˜π‘ƒ) βˆ’ 2))
109oveq2d 7424 . . . . . . . . 9 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)) = (0..^((β™―β€˜π‘ƒ) βˆ’ 2)))
1110raleqdv 3325 . . . . . . . 8 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (βˆ€π‘– ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ↔ βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 2)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸))
1211biimpcd 248 . . . . . . 7 (βˆ€π‘– ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 β†’ ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 2)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸))
1312adantr 481 . . . . . 6 ((βˆ€π‘– ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜0)} ∈ ran 𝐸) β†’ ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 2)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸))
1413adantl 482 . . . . 5 (((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ (βˆ€π‘– ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜0)} ∈ ran 𝐸)) β†’ ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 2)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸))
1514impcom 408 . . . 4 (((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ (βˆ€π‘– ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜0)} ∈ ran 𝐸))) β†’ βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 2)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸)
16 lsw 14513 . . . . . . . . . . . . . . . 16 (𝑃 ∈ Word 𝑉 β†’ (lastSβ€˜π‘ƒ) = (π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 1)))
17 2m1e1 12337 . . . . . . . . . . . . . . . . . . . . 21 (2 βˆ’ 1) = 1
1817a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑃 ∈ Word 𝑉 β†’ (2 βˆ’ 1) = 1)
1918eqcomd 2738 . . . . . . . . . . . . . . . . . . 19 (𝑃 ∈ Word 𝑉 β†’ 1 = (2 βˆ’ 1))
2019oveq2d 7424 . . . . . . . . . . . . . . . . . 18 (𝑃 ∈ Word 𝑉 β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) = ((β™―β€˜π‘ƒ) βˆ’ (2 βˆ’ 1)))
211, 2syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑃 ∈ Word 𝑉 β†’ (β™―β€˜π‘ƒ) ∈ β„‚)
22 2cnd 12289 . . . . . . . . . . . . . . . . . . 19 (𝑃 ∈ Word 𝑉 β†’ 2 ∈ β„‚)
23 1cnd 11208 . . . . . . . . . . . . . . . . . . 19 (𝑃 ∈ Word 𝑉 β†’ 1 ∈ β„‚)
2421, 22, 23subsubd 11598 . . . . . . . . . . . . . . . . . 18 (𝑃 ∈ Word 𝑉 β†’ ((β™―β€˜π‘ƒ) βˆ’ (2 βˆ’ 1)) = (((β™―β€˜π‘ƒ) βˆ’ 2) + 1))
2520, 24eqtrd 2772 . . . . . . . . . . . . . . . . 17 (𝑃 ∈ Word 𝑉 β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) = (((β™―β€˜π‘ƒ) βˆ’ 2) + 1))
2625fveq2d 6895 . . . . . . . . . . . . . . . 16 (𝑃 ∈ Word 𝑉 β†’ (π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 1)) = (π‘ƒβ€˜(((β™―β€˜π‘ƒ) βˆ’ 2) + 1)))
2716, 26eqtrd 2772 . . . . . . . . . . . . . . 15 (𝑃 ∈ Word 𝑉 β†’ (lastSβ€˜π‘ƒ) = (π‘ƒβ€˜(((β™―β€˜π‘ƒ) βˆ’ 2) + 1)))
2827adantr 481 . . . . . . . . . . . . . 14 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (lastSβ€˜π‘ƒ) = (π‘ƒβ€˜(((β™―β€˜π‘ƒ) βˆ’ 2) + 1)))
2928adantr 481 . . . . . . . . . . . . 13 (((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ (lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0)) β†’ (lastSβ€˜π‘ƒ) = (π‘ƒβ€˜(((β™―β€˜π‘ƒ) βˆ’ 2) + 1)))
30 eqeq1 2736 . . . . . . . . . . . . . 14 ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) β†’ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜(((β™―β€˜π‘ƒ) βˆ’ 2) + 1)) ↔ (π‘ƒβ€˜0) = (π‘ƒβ€˜(((β™―β€˜π‘ƒ) βˆ’ 2) + 1))))
3130adantl 482 . . . . . . . . . . . . 13 (((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ (lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0)) β†’ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜(((β™―β€˜π‘ƒ) βˆ’ 2) + 1)) ↔ (π‘ƒβ€˜0) = (π‘ƒβ€˜(((β™―β€˜π‘ƒ) βˆ’ 2) + 1))))
3229, 31mpbid 231 . . . . . . . . . . . 12 (((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ (lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0)) β†’ (π‘ƒβ€˜0) = (π‘ƒβ€˜(((β™―β€˜π‘ƒ) βˆ’ 2) + 1)))
3332preq2d 4744 . . . . . . . . . . 11 (((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ (lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0)) β†’ {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜0)} = {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜(((β™―β€˜π‘ƒ) βˆ’ 2) + 1))})
3433eleq1d 2818 . . . . . . . . . 10 (((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ (lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0)) β†’ ({(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜0)} ∈ ran 𝐸 ↔ {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜(((β™―β€˜π‘ƒ) βˆ’ 2) + 1))} ∈ ran 𝐸))
3534biimpd 228 . . . . . . . . 9 (((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ (lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0)) β†’ ({(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜0)} ∈ ran 𝐸 β†’ {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜(((β™―β€˜π‘ƒ) βˆ’ 2) + 1))} ∈ ran 𝐸))
3635ex 413 . . . . . . . 8 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) β†’ ({(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜0)} ∈ ran 𝐸 β†’ {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜(((β™―β€˜π‘ƒ) βˆ’ 2) + 1))} ∈ ran 𝐸)))
3736com13 88 . . . . . . 7 ({(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜0)} ∈ ran 𝐸 β†’ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) β†’ ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜(((β™―β€˜π‘ƒ) βˆ’ 2) + 1))} ∈ ran 𝐸)))
3837adantl 482 . . . . . 6 ((βˆ€π‘– ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜0)} ∈ ran 𝐸) β†’ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) β†’ ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜(((β™―β€˜π‘ƒ) βˆ’ 2) + 1))} ∈ ran 𝐸)))
3938impcom 408 . . . . 5 (((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ (βˆ€π‘– ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜0)} ∈ ran 𝐸)) β†’ ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜(((β™―β€˜π‘ƒ) βˆ’ 2) + 1))} ∈ ran 𝐸))
4039impcom 408 . . . 4 (((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ (βˆ€π‘– ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜0)} ∈ ran 𝐸))) β†’ {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜(((β™―β€˜π‘ƒ) βˆ’ 2) + 1))} ∈ ran 𝐸)
41 ovexd 7443 . . . . 5 (((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ (βˆ€π‘– ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜0)} ∈ ran 𝐸))) β†’ ((β™―β€˜π‘ƒ) βˆ’ 2) ∈ V)
42 fveq2 6891 . . . . . . . 8 (𝑖 = ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ (π‘ƒβ€˜π‘–) = (π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)))
43 fvoveq1 7431 . . . . . . . 8 (𝑖 = ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ (π‘ƒβ€˜(𝑖 + 1)) = (π‘ƒβ€˜(((β™―β€˜π‘ƒ) βˆ’ 2) + 1)))
4442, 43preq12d 4745 . . . . . . 7 (𝑖 = ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} = {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜(((β™―β€˜π‘ƒ) βˆ’ 2) + 1))})
4544eleq1d 2818 . . . . . 6 (𝑖 = ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ ({(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ↔ {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜(((β™―β€˜π‘ƒ) βˆ’ 2) + 1))} ∈ ran 𝐸))
4645ralunsn 4894 . . . . 5 (((β™―β€˜π‘ƒ) βˆ’ 2) ∈ V β†’ (βˆ€π‘– ∈ ((0..^((β™―β€˜π‘ƒ) βˆ’ 2)) βˆͺ {((β™―β€˜π‘ƒ) βˆ’ 2)}){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ↔ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 2)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜(((β™―β€˜π‘ƒ) βˆ’ 2) + 1))} ∈ ran 𝐸)))
4741, 46syl 17 . . . 4 (((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ (βˆ€π‘– ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜0)} ∈ ran 𝐸))) β†’ (βˆ€π‘– ∈ ((0..^((β™―β€˜π‘ƒ) βˆ’ 2)) βˆͺ {((β™―β€˜π‘ƒ) βˆ’ 2)}){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ↔ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 2)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜(((β™―β€˜π‘ƒ) βˆ’ 2) + 1))} ∈ ran 𝐸)))
4815, 40, 47mpbir2and 711 . . 3 (((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ (βˆ€π‘– ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜0)} ∈ ran 𝐸))) β†’ βˆ€π‘– ∈ ((0..^((β™―β€˜π‘ƒ) βˆ’ 2)) βˆͺ {((β™―β€˜π‘ƒ) βˆ’ 2)}){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸)
49 1e2m1 12338 . . . . . . . . . . 11 1 = (2 βˆ’ 1)
5049a1i 11 . . . . . . . . . 10 (𝑃 ∈ Word 𝑉 β†’ 1 = (2 βˆ’ 1))
5150oveq2d 7424 . . . . . . . . 9 (𝑃 ∈ Word 𝑉 β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) = ((β™―β€˜π‘ƒ) βˆ’ (2 βˆ’ 1)))
5251, 24eqtrd 2772 . . . . . . . 8 (𝑃 ∈ Word 𝑉 β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) = (((β™―β€˜π‘ƒ) βˆ’ 2) + 1))
5352oveq2d 7424 . . . . . . 7 (𝑃 ∈ Word 𝑉 β†’ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) = (0..^(((β™―β€˜π‘ƒ) βˆ’ 2) + 1)))
5453adantr 481 . . . . . 6 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) = (0..^(((β™―β€˜π‘ƒ) βˆ’ 2) + 1)))
55 nn0re 12480 . . . . . . . . . . . . . 14 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (β™―β€˜π‘ƒ) ∈ ℝ)
56 2re 12285 . . . . . . . . . . . . . . 15 2 ∈ ℝ
5756a1i 11 . . . . . . . . . . . . . 14 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ 2 ∈ ℝ)
5855, 57subge0d 11803 . . . . . . . . . . . . 13 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (0 ≀ ((β™―β€˜π‘ƒ) βˆ’ 2) ↔ 2 ≀ (β™―β€˜π‘ƒ)))
5958biimprd 247 . . . . . . . . . . . 12 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (2 ≀ (β™―β€˜π‘ƒ) β†’ 0 ≀ ((β™―β€˜π‘ƒ) βˆ’ 2)))
60 nn0z 12582 . . . . . . . . . . . . 13 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (β™―β€˜π‘ƒ) ∈ β„€)
61 2z 12593 . . . . . . . . . . . . . 14 2 ∈ β„€
6261a1i 11 . . . . . . . . . . . . 13 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ 2 ∈ β„€)
6360, 62zsubcld 12670 . . . . . . . . . . . 12 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ ((β™―β€˜π‘ƒ) βˆ’ 2) ∈ β„€)
6459, 63jctild 526 . . . . . . . . . . 11 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (2 ≀ (β™―β€˜π‘ƒ) β†’ (((β™―β€˜π‘ƒ) βˆ’ 2) ∈ β„€ ∧ 0 ≀ ((β™―β€˜π‘ƒ) βˆ’ 2))))
651, 64syl 17 . . . . . . . . . 10 (𝑃 ∈ Word 𝑉 β†’ (2 ≀ (β™―β€˜π‘ƒ) β†’ (((β™―β€˜π‘ƒ) βˆ’ 2) ∈ β„€ ∧ 0 ≀ ((β™―β€˜π‘ƒ) βˆ’ 2))))
6665imp 407 . . . . . . . . 9 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (((β™―β€˜π‘ƒ) βˆ’ 2) ∈ β„€ ∧ 0 ≀ ((β™―β€˜π‘ƒ) βˆ’ 2)))
67 elnn0z 12570 . . . . . . . . 9 (((β™―β€˜π‘ƒ) βˆ’ 2) ∈ β„•0 ↔ (((β™―β€˜π‘ƒ) βˆ’ 2) ∈ β„€ ∧ 0 ≀ ((β™―β€˜π‘ƒ) βˆ’ 2)))
6866, 67sylibr 233 . . . . . . . 8 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((β™―β€˜π‘ƒ) βˆ’ 2) ∈ β„•0)
69 elnn0uz 12866 . . . . . . . 8 (((β™―β€˜π‘ƒ) βˆ’ 2) ∈ β„•0 ↔ ((β™―β€˜π‘ƒ) βˆ’ 2) ∈ (β„€β‰₯β€˜0))
7068, 69sylib 217 . . . . . . 7 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((β™―β€˜π‘ƒ) βˆ’ 2) ∈ (β„€β‰₯β€˜0))
71 fzosplitsn 13739 . . . . . . 7 (((β™―β€˜π‘ƒ) βˆ’ 2) ∈ (β„€β‰₯β€˜0) β†’ (0..^(((β™―β€˜π‘ƒ) βˆ’ 2) + 1)) = ((0..^((β™―β€˜π‘ƒ) βˆ’ 2)) βˆͺ {((β™―β€˜π‘ƒ) βˆ’ 2)}))
7270, 71syl 17 . . . . . 6 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (0..^(((β™―β€˜π‘ƒ) βˆ’ 2) + 1)) = ((0..^((β™―β€˜π‘ƒ) βˆ’ 2)) βˆͺ {((β™―β€˜π‘ƒ) βˆ’ 2)}))
7354, 72eqtrd 2772 . . . . 5 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) = ((0..^((β™―β€˜π‘ƒ) βˆ’ 2)) βˆͺ {((β™―β€˜π‘ƒ) βˆ’ 2)}))
7473adantr 481 . . . 4 (((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ (βˆ€π‘– ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜0)} ∈ ran 𝐸))) β†’ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) = ((0..^((β™―β€˜π‘ƒ) βˆ’ 2)) βˆͺ {((β™―β€˜π‘ƒ) βˆ’ 2)}))
7574raleqdv 3325 . . 3 (((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ (βˆ€π‘– ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜0)} ∈ ran 𝐸))) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ↔ βˆ€π‘– ∈ ((0..^((β™―β€˜π‘ƒ) βˆ’ 2)) βˆͺ {((β™―β€˜π‘ƒ) βˆ’ 2)}){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸))
7648, 75mpbird 256 . 2 (((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ (βˆ€π‘– ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜0)} ∈ ran 𝐸))) β†’ βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸)
7776ex 413 1 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ (βˆ€π‘– ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜0)} ∈ ran 𝐸)) β†’ βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  Vcvv 3474   βˆͺ cun 3946  {csn 4628  {cpr 4630   class class class wbr 5148  ran crn 5677  β€˜cfv 6543  (class class class)co 7408  β„‚cc 11107  β„cr 11108  0cc0 11109  1c1 11110   + caddc 11112   ≀ cle 11248   βˆ’ cmin 11443  2c2 12266  β„•0cn0 12471  β„€cz 12557  β„€β‰₯cuz 12821  ..^cfzo 13626  β™―chash 14289  Word cword 14463  lastSclsw 14511
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-card 9933  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-n0 12472  df-z 12558  df-uz 12822  df-fz 13484  df-fzo 13627  df-hash 14290  df-word 14464  df-lsw 14512
This theorem is referenced by:  clwlkclwwlklem2a  29248
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