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Theorem clwlkclwwlklem2a1 28985
Description: Lemma 1 for clwlkclwwlklem2a 28991. (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 14430 . . . . . . . . . . . 12 (𝑃 ∈ Word 𝑉 β†’ (β™―β€˜π‘ƒ) ∈ β„•0)
2 nn0cn 12431 . . . . . . . . . . . 12 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (β™―β€˜π‘ƒ) ∈ β„‚)
3 peano2cnm 11475 . . . . . . . . . . . . . . 15 ((β™―β€˜π‘ƒ) ∈ β„‚ β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ β„‚)
43subid1d 11509 . . . . . . . . . . . . . 14 ((β™―β€˜π‘ƒ) ∈ β„‚ β†’ (((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) = ((β™―β€˜π‘ƒ) βˆ’ 1))
54oveq1d 7376 . . . . . . . . . . . . 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 12413 . . . . . . . . . . . . 13 ((β™―β€˜π‘ƒ) ∈ β„‚ β†’ (((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 1) = ((β™―β€˜π‘ƒ) βˆ’ 2))
75, 6eqtrd 2773 . . . . . . . . . . . 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 482 . . . . . . . . . 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 7377 . . . . . . . . 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 3312 . . . . . . . 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 249 . . . . . . 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 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†’ ((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 483 . . . . 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 409 . . . 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 14461 . . . . . . . . . . . . . . . 16 (𝑃 ∈ Word 𝑉 β†’ (lastSβ€˜π‘ƒ) = (π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 1)))
17 2m1e1 12287 . . . . . . . . . . . . . . . . . . . . 21 (2 βˆ’ 1) = 1
1817a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑃 ∈ Word 𝑉 β†’ (2 βˆ’ 1) = 1)
1918eqcomd 2739 . . . . . . . . . . . . . . . . . . 19 (𝑃 ∈ Word 𝑉 β†’ 1 = (2 βˆ’ 1))
2019oveq2d 7377 . . . . . . . . . . . . . . . . . 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 12239 . . . . . . . . . . . . . . . . . . 19 (𝑃 ∈ Word 𝑉 β†’ 2 ∈ β„‚)
23 1cnd 11158 . . . . . . . . . . . . . . . . . . 19 (𝑃 ∈ Word 𝑉 β†’ 1 ∈ β„‚)
2421, 22, 23subsubd 11548 . . . . . . . . . . . . . . . . . 18 (𝑃 ∈ Word 𝑉 β†’ ((β™―β€˜π‘ƒ) βˆ’ (2 βˆ’ 1)) = (((β™―β€˜π‘ƒ) βˆ’ 2) + 1))
2520, 24eqtrd 2773 . . . . . . . . . . . . . . . . 17 (𝑃 ∈ Word 𝑉 β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) = (((β™―β€˜π‘ƒ) βˆ’ 2) + 1))
2625fveq2d 6850 . . . . . . . . . . . . . . . 16 (𝑃 ∈ Word 𝑉 β†’ (π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 1)) = (π‘ƒβ€˜(((β™―β€˜π‘ƒ) βˆ’ 2) + 1)))
2716, 26eqtrd 2773 . . . . . . . . . . . . . . 15 (𝑃 ∈ Word 𝑉 β†’ (lastSβ€˜π‘ƒ) = (π‘ƒβ€˜(((β™―β€˜π‘ƒ) βˆ’ 2) + 1)))
2827adantr 482 . . . . . . . . . . . . . 14 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (lastSβ€˜π‘ƒ) = (π‘ƒβ€˜(((β™―β€˜π‘ƒ) βˆ’ 2) + 1)))
2928adantr 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)))
30 eqeq1 2737 . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . 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 4705 . . . . . . . . . . 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 2819 . . . . . . . . . 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 414 . . . . . . . 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 483 . . . . . 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 409 . . . . 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 409 . . . 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 7396 . . . . 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 6846 . . . . . . . 8 (𝑖 = ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ (π‘ƒβ€˜π‘–) = (π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)))
43 fvoveq1 7384 . . . . . . . 8 (𝑖 = ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ (π‘ƒβ€˜(𝑖 + 1)) = (π‘ƒβ€˜(((β™―β€˜π‘ƒ) βˆ’ 2) + 1)))
4442, 43preq12d 4706 . . . . . . 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 2819 . . . . . 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 4855 . . . . 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 712 . . 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 12288 . . . . . . . . . . 11 1 = (2 βˆ’ 1)
5049a1i 11 . . . . . . . . . 10 (𝑃 ∈ Word 𝑉 β†’ 1 = (2 βˆ’ 1))
5150oveq2d 7377 . . . . . . . . 9 (𝑃 ∈ Word 𝑉 β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) = ((β™―β€˜π‘ƒ) βˆ’ (2 βˆ’ 1)))
5251, 24eqtrd 2773 . . . . . . . 8 (𝑃 ∈ Word 𝑉 β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) = (((β™―β€˜π‘ƒ) βˆ’ 2) + 1))
5352oveq2d 7377 . . . . . . 7 (𝑃 ∈ Word 𝑉 β†’ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) = (0..^(((β™―β€˜π‘ƒ) βˆ’ 2) + 1)))
5453adantr 482 . . . . . 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 12430 . . . . . . . . . . . . . 14 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (β™―β€˜π‘ƒ) ∈ ℝ)
56 2re 12235 . . . . . . . . . . . . . . 15 2 ∈ ℝ
5756a1i 11 . . . . . . . . . . . . . 14 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ 2 ∈ ℝ)
5855, 57subge0d 11753 . . . . . . . . . . . . 13 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (0 ≀ ((β™―β€˜π‘ƒ) βˆ’ 2) ↔ 2 ≀ (β™―β€˜π‘ƒ)))
5958biimprd 248 . . . . . . . . . . . 12 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (2 ≀ (β™―β€˜π‘ƒ) β†’ 0 ≀ ((β™―β€˜π‘ƒ) βˆ’ 2)))
60 nn0z 12532 . . . . . . . . . . . . 13 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (β™―β€˜π‘ƒ) ∈ β„€)
61 2z 12543 . . . . . . . . . . . . . 14 2 ∈ β„€
6261a1i 11 . . . . . . . . . . . . 13 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ 2 ∈ β„€)
6360, 62zsubcld 12620 . . . . . . . . . . . 12 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ ((β™―β€˜π‘ƒ) βˆ’ 2) ∈ β„€)
6459, 63jctild 527 . . . . . . . . . . 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 408 . . . . . . . . 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 12520 . . . . . . . . 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 12816 . . . . . . . 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 13689 . . . . . . 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 2773 . . . . 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 482 . . . 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 3312 . . 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 257 . 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 414 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  Vcvv 3447   βˆͺ cun 3912  {csn 4590  {cpr 4592   class class class wbr 5109  ran crn 5638  β€˜cfv 6500  (class class class)co 7361  β„‚cc 11057  β„cr 11058  0cc0 11059  1c1 11060   + caddc 11062   ≀ cle 11198   βˆ’ cmin 11393  2c2 12216  β„•0cn0 12421  β„€cz 12507  β„€β‰₯cuz 12771  ..^cfzo 13576  β™―chash 14239  Word cword 14411  lastSclsw 14459
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-er 8654  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-card 9883  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-2 12224  df-n0 12422  df-z 12508  df-uz 12772  df-fz 13434  df-fzo 13577  df-hash 14240  df-word 14412  df-lsw 14460
This theorem is referenced by:  clwlkclwwlklem2a  28991
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