MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  clwlkclwwlklem2a1 Structured version   Visualization version   GIF version

Theorem clwlkclwwlklem2a1 29245
Description: Lemma 1 for clwlkclwwlklem2a 29251. (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 14483 . . . . . . . . . . . 12 (𝑃 ∈ Word 𝑉 β†’ (β™―β€˜π‘ƒ) ∈ β„•0)
2 nn0cn 12482 . . . . . . . . . . . 12 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (β™―β€˜π‘ƒ) ∈ β„‚)
3 peano2cnm 11526 . . . . . . . . . . . . . . 15 ((β™―β€˜π‘ƒ) ∈ β„‚ β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ β„‚)
43subid1d 11560 . . . . . . . . . . . . . 14 ((β™―β€˜π‘ƒ) ∈ β„‚ β†’ (((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) = ((β™―β€˜π‘ƒ) βˆ’ 1))
54oveq1d 7424 . . . . . . . . . . . . 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 12464 . . . . . . . . . . . . 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 7425 . . . . . . . . 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 3326 . . . . . . . 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 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 14514 . . . . . . . . . . . . . . . 16 (𝑃 ∈ Word 𝑉 β†’ (lastSβ€˜π‘ƒ) = (π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 1)))
17 2m1e1 12338 . . . . . . . . . . . . . . . . . . . . 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 7425 . . . . . . . . . . . . . . . . . 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 12290 . . . . . . . . . . . . . . . . . . 19 (𝑃 ∈ Word 𝑉 β†’ 2 ∈ β„‚)
23 1cnd 11209 . . . . . . . . . . . . . . . . . . 19 (𝑃 ∈ Word 𝑉 β†’ 1 ∈ β„‚)
2421, 22, 23subsubd 11599 . . . . . . . . . . . . . . . . . 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 6896 . . . . . . . . . . . . . . . 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 4745 . . . . . . . . . . 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 7444 . . . . 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 6892 . . . . . . . 8 (𝑖 = ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ (π‘ƒβ€˜π‘–) = (π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)))
43 fvoveq1 7432 . . . . . . . 8 (𝑖 = ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ (π‘ƒβ€˜(𝑖 + 1)) = (π‘ƒβ€˜(((β™―β€˜π‘ƒ) βˆ’ 2) + 1)))
4442, 43preq12d 4746 . . . . . . 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 4895 . . . . 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 12339 . . . . . . . . . . 11 1 = (2 βˆ’ 1)
5049a1i 11 . . . . . . . . . 10 (𝑃 ∈ Word 𝑉 β†’ 1 = (2 βˆ’ 1))
5150oveq2d 7425 . . . . . . . . 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 7425 . . . . . . 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 12481 . . . . . . . . . . . . . 14 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (β™―β€˜π‘ƒ) ∈ ℝ)
56 2re 12286 . . . . . . . . . . . . . . 15 2 ∈ ℝ
5756a1i 11 . . . . . . . . . . . . . 14 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ 2 ∈ ℝ)
5855, 57subge0d 11804 . . . . . . . . . . . . 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 12583 . . . . . . . . . . . . 13 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (β™―β€˜π‘ƒ) ∈ β„€)
61 2z 12594 . . . . . . . . . . . . . 14 2 ∈ β„€
6261a1i 11 . . . . . . . . . . . . 13 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ 2 ∈ β„€)
6360, 62zsubcld 12671 . . . . . . . . . . . 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 12571 . . . . . . . . 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 12867 . . . . . . . 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 13740 . . . . . . 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 3326 . . 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 3062  Vcvv 3475   βˆͺ cun 3947  {csn 4629  {cpr 4631   class class class wbr 5149  ran crn 5678  β€˜cfv 6544  (class class class)co 7409  β„‚cc 11108  β„cr 11109  0cc0 11110  1c1 11111   + caddc 11113   ≀ cle 11249   βˆ’ cmin 11444  2c2 12267  β„•0cn0 12472  β„€cz 12558  β„€β‰₯cuz 12822  ..^cfzo 13627  β™―chash 14290  Word cword 14464  lastSclsw 14512
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-fz 13485  df-fzo 13628  df-hash 14291  df-word 14465  df-lsw 14513
This theorem is referenced by:  clwlkclwwlklem2a  29251
  Copyright terms: Public domain W3C validator