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Theorem clwlkclwwlklem2 29866
Description: Lemma 2 for clwlkclwwlk 29868. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 11-Apr-2021.)
Assertion
Ref Expression
clwlkclwwlklem2 (((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)))) β†’ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜πΉ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜0)} ∈ ran 𝐸))
Distinct variable groups:   𝑖,𝐸   𝑃,𝑖   𝑅,𝑖   𝑖,𝑉   𝑖,𝐹
Proof of Theorem clwlkclwwlklem2
StepHypRef Expression
1 f1fn 6792 . . . 4 (𝐸:dom 𝐸–1-1→𝑅 β†’ 𝐸 Fn dom 𝐸)
2 dffn3 6733 . . . 4 (𝐸 Fn dom 𝐸 ↔ 𝐸:dom 𝐸⟢ran 𝐸)
31, 2sylib 217 . . 3 (𝐸:dom 𝐸–1-1→𝑅 β†’ 𝐸:dom 𝐸⟢ran 𝐸)
4 lencl 14515 . . . . . . . . 9 (𝐹 ∈ Word dom 𝐸 β†’ (β™―β€˜πΉ) ∈ β„•0)
5 ffn 6721 . . . . . . . . 9 (𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ β†’ 𝑃 Fn (0...(β™―β€˜πΉ)))
6 fnfz0hash 14437 . . . . . . . . 9 (((β™―β€˜πΉ) ∈ β„•0 ∧ 𝑃 Fn (0...(β™―β€˜πΉ))) β†’ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1))
74, 5, 6syl2an 594 . . . . . . . 8 ((𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰) β†’ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1))
8 ffz0iswrd 14523 . . . . . . . . . . . 12 (𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ β†’ 𝑃 ∈ Word 𝑉)
9 lsw 14546 . . . . . . . . . . . . . . . . 17 (𝑃 ∈ Word 𝑉 β†’ (lastSβ€˜π‘ƒ) = (π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 1)))
109ad6antr 734 . . . . . . . . . . . . . . . 16 (((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) ∧ (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)))) β†’ (lastSβ€˜π‘ƒ) = (π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 1)))
11 fvoveq1 7440 . . . . . . . . . . . . . . . . 17 ((β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1) β†’ (π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 1)) = (π‘ƒβ€˜(((β™―β€˜πΉ) + 1) βˆ’ 1)))
1211ad4antlr 731 . . . . . . . . . . . . . . . 16 (((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) ∧ (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)))) β†’ (π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 1)) = (π‘ƒβ€˜(((β™―β€˜πΉ) + 1) βˆ’ 1)))
13 eqcom 2732 . . . . . . . . . . . . . . . . . . 19 ((π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)) ↔ (π‘ƒβ€˜(β™―β€˜πΉ)) = (π‘ƒβ€˜0))
14 nn0cn 12512 . . . . . . . . . . . . . . . . . . . . . . . 24 ((β™―β€˜πΉ) ∈ β„•0 β†’ (β™―β€˜πΉ) ∈ β„‚)
15 1cnd 11239 . . . . . . . . . . . . . . . . . . . . . . . 24 ((β™―β€˜πΉ) ∈ β„•0 β†’ 1 ∈ β„‚)
1614, 15pncand 11602 . . . . . . . . . . . . . . . . . . . . . . 23 ((β™―β€˜πΉ) ∈ β„•0 β†’ (((β™―β€˜πΉ) + 1) βˆ’ 1) = (β™―β€˜πΉ))
1716eqcomd 2731 . . . . . . . . . . . . . . . . . . . . . 22 ((β™―β€˜πΉ) ∈ β„•0 β†’ (β™―β€˜πΉ) = (((β™―β€˜πΉ) + 1) βˆ’ 1))
1817ad4antlr 731 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) β†’ (β™―β€˜πΉ) = (((β™―β€˜πΉ) + 1) βˆ’ 1))
1918fveqeq2d 6902 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) β†’ ((π‘ƒβ€˜(β™―β€˜πΉ)) = (π‘ƒβ€˜0) ↔ (π‘ƒβ€˜(((β™―β€˜πΉ) + 1) βˆ’ 1)) = (π‘ƒβ€˜0)))
2019biimpd 228 . . . . . . . . . . . . . . . . . . 19 ((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) β†’ ((π‘ƒβ€˜(β™―β€˜πΉ)) = (π‘ƒβ€˜0) β†’ (π‘ƒβ€˜(((β™―β€˜πΉ) + 1) βˆ’ 1)) = (π‘ƒβ€˜0)))
2113, 20biimtrid 241 . . . . . . . . . . . . . . . . . 18 ((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) β†’ ((π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)) β†’ (π‘ƒβ€˜(((β™―β€˜πΉ) + 1) βˆ’ 1)) = (π‘ƒβ€˜0)))
2221adantld 489 . . . . . . . . . . . . . . . . 17 ((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) β†’ ((βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))) β†’ (π‘ƒβ€˜(((β™―β€˜πΉ) + 1) βˆ’ 1)) = (π‘ƒβ€˜0)))
2322imp 405 . . . . . . . . . . . . . . . 16 (((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) ∧ (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)))) β†’ (π‘ƒβ€˜(((β™―β€˜πΉ) + 1) βˆ’ 1)) = (π‘ƒβ€˜0))
2410, 12, 233eqtrd 2769 . . . . . . . . . . . . . . 15 (((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) ∧ (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)))) β†’ (lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0))
25 nn0z 12613 . . . . . . . . . . . . . . . . . . . . . 22 ((β™―β€˜πΉ) ∈ β„•0 β†’ (β™―β€˜πΉ) ∈ β„€)
26 peano2zm 12635 . . . . . . . . . . . . . . . . . . . . . 22 ((β™―β€˜πΉ) ∈ β„€ β†’ ((β™―β€˜πΉ) βˆ’ 1) ∈ β„€)
2725, 26syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((β™―β€˜πΉ) ∈ β„•0 β†’ ((β™―β€˜πΉ) βˆ’ 1) ∈ β„€)
28 nn0re 12511 . . . . . . . . . . . . . . . . . . . . . 22 ((β™―β€˜πΉ) ∈ β„•0 β†’ (β™―β€˜πΉ) ∈ ℝ)
2928lem1d 12177 . . . . . . . . . . . . . . . . . . . . 21 ((β™―β€˜πΉ) ∈ β„•0 β†’ ((β™―β€˜πΉ) βˆ’ 1) ≀ (β™―β€˜πΉ))
30 eluz2 12858 . . . . . . . . . . . . . . . . . . . . 21 ((β™―β€˜πΉ) ∈ (β„€β‰₯β€˜((β™―β€˜πΉ) βˆ’ 1)) ↔ (((β™―β€˜πΉ) βˆ’ 1) ∈ β„€ ∧ (β™―β€˜πΉ) ∈ β„€ ∧ ((β™―β€˜πΉ) βˆ’ 1) ≀ (β™―β€˜πΉ)))
3127, 25, 29, 30syl3anbrc 1340 . . . . . . . . . . . . . . . . . . . 20 ((β™―β€˜πΉ) ∈ β„•0 β†’ (β™―β€˜πΉ) ∈ (β„€β‰₯β€˜((β™―β€˜πΉ) βˆ’ 1)))
3231ad4antlr 731 . . . . . . . . . . . . . . . . . . 19 ((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) β†’ (β™―β€˜πΉ) ∈ (β„€β‰₯β€˜((β™―β€˜πΉ) βˆ’ 1)))
33 fzoss2 13692 . . . . . . . . . . . . . . . . . . 19 ((β™―β€˜πΉ) ∈ (β„€β‰₯β€˜((β™―β€˜πΉ) βˆ’ 1)) β†’ (0..^((β™―β€˜πΉ) βˆ’ 1)) βŠ† (0..^(β™―β€˜πΉ)))
34 ssralv 4046 . . . . . . . . . . . . . . . . . . 19 ((0..^((β™―β€˜πΉ) βˆ’ 1)) βŠ† (0..^(β™―β€˜πΉ)) β†’ (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} β†’ βˆ€π‘– ∈ (0..^((β™―β€˜πΉ) βˆ’ 1))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))
3532, 33, 343syl 18 . . . . . . . . . . . . . . . . . 18 ((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) β†’ (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} β†’ βˆ€π‘– ∈ (0..^((β™―β€˜πΉ) βˆ’ 1))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))
36 simpr 483 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) β†’ 𝐸:dom 𝐸⟢ran 𝐸)
3736adantr 479 . . . . . . . . . . . . . . . . . . . . 21 (((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) ∧ 𝑖 ∈ (0..^((β™―β€˜πΉ) βˆ’ 1))) β†’ 𝐸:dom 𝐸⟢ran 𝐸)
38 wrdf 14501 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐹 ∈ Word dom 𝐸 β†’ 𝐹:(0..^(β™―β€˜πΉ))⟢dom 𝐸)
39 simpll 765 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝐹:(0..^(β™―β€˜πΉ))⟢dom 𝐸 ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ 𝑖 ∈ (0..^((β™―β€˜πΉ) βˆ’ 1))) β†’ 𝐹:(0..^(β™―β€˜πΉ))⟢dom 𝐸)
40 fzossrbm1 13693 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((β™―β€˜πΉ) ∈ β„€ β†’ (0..^((β™―β€˜πΉ) βˆ’ 1)) βŠ† (0..^(β™―β€˜πΉ)))
4125, 40syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((β™―β€˜πΉ) ∈ β„•0 β†’ (0..^((β™―β€˜πΉ) βˆ’ 1)) βŠ† (0..^(β™―β€˜πΉ)))
4241adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐹:(0..^(β™―β€˜πΉ))⟢dom 𝐸 ∧ (β™―β€˜πΉ) ∈ β„•0) β†’ (0..^((β™―β€˜πΉ) βˆ’ 1)) βŠ† (0..^(β™―β€˜πΉ)))
4342sselda 3977 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝐹:(0..^(β™―β€˜πΉ))⟢dom 𝐸 ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ 𝑖 ∈ (0..^((β™―β€˜πΉ) βˆ’ 1))) β†’ 𝑖 ∈ (0..^(β™―β€˜πΉ)))
4439, 43ffvelcdmd 7092 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐹:(0..^(β™―β€˜πΉ))⟢dom 𝐸 ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ 𝑖 ∈ (0..^((β™―β€˜πΉ) βˆ’ 1))) β†’ (πΉβ€˜π‘–) ∈ dom 𝐸)
4544exp31 418 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐹:(0..^(β™―β€˜πΉ))⟢dom 𝐸 β†’ ((β™―β€˜πΉ) ∈ β„•0 β†’ (𝑖 ∈ (0..^((β™―β€˜πΉ) βˆ’ 1)) β†’ (πΉβ€˜π‘–) ∈ dom 𝐸)))
4638, 45syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐹 ∈ Word dom 𝐸 β†’ ((β™―β€˜πΉ) ∈ β„•0 β†’ (𝑖 ∈ (0..^((β™―β€˜πΉ) βˆ’ 1)) β†’ (πΉβ€˜π‘–) ∈ dom 𝐸)))
4746adantl 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) β†’ ((β™―β€˜πΉ) ∈ β„•0 β†’ (𝑖 ∈ (0..^((β™―β€˜πΉ) βˆ’ 1)) β†’ (πΉβ€˜π‘–) ∈ dom 𝐸)))
4847imp 405 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) β†’ (𝑖 ∈ (0..^((β™―β€˜πΉ) βˆ’ 1)) β†’ (πΉβ€˜π‘–) ∈ dom 𝐸))
4948ad3antrrr 728 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) β†’ (𝑖 ∈ (0..^((β™―β€˜πΉ) βˆ’ 1)) β†’ (πΉβ€˜π‘–) ∈ dom 𝐸))
5049imp 405 . . . . . . . . . . . . . . . . . . . . 21 (((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) ∧ 𝑖 ∈ (0..^((β™―β€˜πΉ) βˆ’ 1))) β†’ (πΉβ€˜π‘–) ∈ dom 𝐸)
5137, 50ffvelcdmd 7092 . . . . . . . . . . . . . . . . . . . 20 (((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) ∧ 𝑖 ∈ (0..^((β™―β€˜πΉ) βˆ’ 1))) β†’ (πΈβ€˜(πΉβ€˜π‘–)) ∈ ran 𝐸)
52 eqcom 2732 . . . . . . . . . . . . . . . . . . . . . 22 ((πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ↔ {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} = (πΈβ€˜(πΉβ€˜π‘–)))
5352biimpi 215 . . . . . . . . . . . . . . . . . . . . 21 ((πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} β†’ {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} = (πΈβ€˜(πΉβ€˜π‘–)))
5453eleq1d 2810 . . . . . . . . . . . . . . . . . . . 20 ((πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} β†’ ({(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ↔ (πΈβ€˜(πΉβ€˜π‘–)) ∈ ran 𝐸))
5551, 54syl5ibrcom 246 . . . . . . . . . . . . . . . . . . 19 (((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) ∧ 𝑖 ∈ (0..^((β™―β€˜πΉ) βˆ’ 1))) β†’ ((πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} β†’ {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸))
5655ralimdva 3157 . . . . . . . . . . . . . . . . . 18 ((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜πΉ) βˆ’ 1))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} β†’ βˆ€π‘– ∈ (0..^((β™―β€˜πΉ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸))
5735, 56syldc 48 . . . . . . . . . . . . . . . . 17 (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} β†’ ((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) β†’ βˆ€π‘– ∈ (0..^((β™―β€˜πΉ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸))
5857adantr 479 . . . . . . . . . . . . . . . 16 ((βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))) β†’ ((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) β†’ βˆ€π‘– ∈ (0..^((β™―β€˜πΉ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸))
5958impcom 406 . . . . . . . . . . . . . . 15 (((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) ∧ (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)))) β†’ βˆ€π‘– ∈ (0..^((β™―β€˜πΉ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸)
60 breq2 5152 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1) β†’ (2 ≀ (β™―β€˜π‘ƒ) ↔ 2 ≀ ((β™―β€˜πΉ) + 1)))
6160adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) β†’ (2 ≀ (β™―β€˜π‘ƒ) ↔ 2 ≀ ((β™―β€˜πΉ) + 1)))
62 2re 12316 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2 ∈ ℝ
6362a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((β™―β€˜πΉ) ∈ β„•0 β†’ 2 ∈ ℝ)
64 1red 11245 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((β™―β€˜πΉ) ∈ β„•0 β†’ 1 ∈ ℝ)
6563, 64, 28lesubaddd 11841 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((β™―β€˜πΉ) ∈ β„•0 β†’ ((2 βˆ’ 1) ≀ (β™―β€˜πΉ) ↔ 2 ≀ ((β™―β€˜πΉ) + 1)))
66 2m1e1 12368 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (2 βˆ’ 1) = 1
6766breq1i 5155 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((2 βˆ’ 1) ≀ (β™―β€˜πΉ) ↔ 1 ≀ (β™―β€˜πΉ))
68 elnnnn0c 12547 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((β™―β€˜πΉ) ∈ β„• ↔ ((β™―β€˜πΉ) ∈ β„•0 ∧ 1 ≀ (β™―β€˜πΉ)))
6968simplbi2 499 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((β™―β€˜πΉ) ∈ β„•0 β†’ (1 ≀ (β™―β€˜πΉ) β†’ (β™―β€˜πΉ) ∈ β„•))
7067, 69biimtrid 241 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((β™―β€˜πΉ) ∈ β„•0 β†’ ((2 βˆ’ 1) ≀ (β™―β€˜πΉ) β†’ (β™―β€˜πΉ) ∈ β„•))
7165, 70sylbird 259 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((β™―β€˜πΉ) ∈ β„•0 β†’ (2 ≀ ((β™―β€˜πΉ) + 1) β†’ (β™―β€˜πΉ) ∈ β„•))
7271adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) β†’ (2 ≀ ((β™―β€˜πΉ) + 1) β†’ (β™―β€˜πΉ) ∈ β„•))
7372adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) β†’ (2 ≀ ((β™―β€˜πΉ) + 1) β†’ (β™―β€˜πΉ) ∈ β„•))
7461, 73sylbid 239 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) β†’ (2 ≀ (β™―β€˜π‘ƒ) β†’ (β™―β€˜πΉ) ∈ β„•))
7574imp 405 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (β™―β€˜πΉ) ∈ β„•)
7675adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) β†’ (β™―β€˜πΉ) ∈ β„•)
77 lbfzo0 13704 . . . . . . . . . . . . . . . . . . . . . 22 (0 ∈ (0..^(β™―β€˜πΉ)) ↔ (β™―β€˜πΉ) ∈ β„•)
7876, 77sylibr 233 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) β†’ 0 ∈ (0..^(β™―β€˜πΉ)))
79 fzoend 13755 . . . . . . . . . . . . . . . . . . . . 21 (0 ∈ (0..^(β™―β€˜πΉ)) β†’ ((β™―β€˜πΉ) βˆ’ 1) ∈ (0..^(β™―β€˜πΉ)))
8078, 79syl 17 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) β†’ ((β™―β€˜πΉ) βˆ’ 1) ∈ (0..^(β™―β€˜πΉ)))
81 2fveq3 6899 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 = ((β™―β€˜πΉ) βˆ’ 1) β†’ (πΈβ€˜(πΉβ€˜π‘–)) = (πΈβ€˜(πΉβ€˜((β™―β€˜πΉ) βˆ’ 1))))
82 fveq2 6894 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 = ((β™―β€˜πΉ) βˆ’ 1) β†’ (π‘ƒβ€˜π‘–) = (π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)))
83 fvoveq1 7440 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 = ((β™―β€˜πΉ) βˆ’ 1) β†’ (π‘ƒβ€˜(𝑖 + 1)) = (π‘ƒβ€˜(((β™―β€˜πΉ) βˆ’ 1) + 1)))
8482, 83preq12d 4746 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 = ((β™―β€˜πΉ) βˆ’ 1) β†’ {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} = {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜(((β™―β€˜πΉ) βˆ’ 1) + 1))})
8581, 84eqeq12d 2741 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = ((β™―β€˜πΉ) βˆ’ 1) β†’ ((πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ↔ (πΈβ€˜(πΉβ€˜((β™―β€˜πΉ) βˆ’ 1))) = {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜(((β™―β€˜πΉ) βˆ’ 1) + 1))}))
8685adantl 480 . . . . . . . . . . . . . . . . . . . 20 (((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) ∧ 𝑖 = ((β™―β€˜πΉ) βˆ’ 1)) β†’ ((πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ↔ (πΈβ€˜(πΉβ€˜((β™―β€˜πΉ) βˆ’ 1))) = {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜(((β™―β€˜πΉ) βˆ’ 1) + 1))}))
8780, 86rspcdv 3599 . . . . . . . . . . . . . . . . . . 19 ((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) β†’ (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} β†’ (πΈβ€˜(πΉβ€˜((β™―β€˜πΉ) βˆ’ 1))) = {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜(((β™―β€˜πΉ) βˆ’ 1) + 1))}))
8814, 15npcand 11605 . . . . . . . . . . . . . . . . . . . . . . . 24 ((β™―β€˜πΉ) ∈ β„•0 β†’ (((β™―β€˜πΉ) βˆ’ 1) + 1) = (β™―β€˜πΉ))
8988ad4antlr 731 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) β†’ (((β™―β€˜πΉ) βˆ’ 1) + 1) = (β™―β€˜πΉ))
9089fveq2d 6898 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) β†’ (π‘ƒβ€˜(((β™―β€˜πΉ) βˆ’ 1) + 1)) = (π‘ƒβ€˜(β™―β€˜πΉ)))
9190preq2d 4745 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) β†’ {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜(((β™―β€˜πΉ) βˆ’ 1) + 1))} = {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜(β™―β€˜πΉ))})
9291eqeq2d 2736 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) β†’ ((πΈβ€˜(πΉβ€˜((β™―β€˜πΉ) βˆ’ 1))) = {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜(((β™―β€˜πΉ) βˆ’ 1) + 1))} ↔ (πΈβ€˜(πΉβ€˜((β™―β€˜πΉ) βˆ’ 1))) = {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜(β™―β€˜πΉ))}))
9338ad4antlr 731 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ 𝐹:(0..^(β™―β€˜πΉ))⟢dom 𝐸)
9471com12 32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (2 ≀ ((β™―β€˜πΉ) + 1) β†’ ((β™―β€˜πΉ) ∈ β„•0 β†’ (β™―β€˜πΉ) ∈ β„•))
9560, 94biimtrdi 252 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1) β†’ (2 ≀ (β™―β€˜π‘ƒ) β†’ ((β™―β€˜πΉ) ∈ β„•0 β†’ (β™―β€˜πΉ) ∈ β„•)))
9695com3r 87 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((β™―β€˜πΉ) ∈ β„•0 β†’ ((β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1) β†’ (2 ≀ (β™―β€˜π‘ƒ) β†’ (β™―β€˜πΉ) ∈ β„•)))
9796adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) β†’ ((β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1) β†’ (2 ≀ (β™―β€˜π‘ƒ) β†’ (β™―β€˜πΉ) ∈ β„•)))
9897imp31 416 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (β™―β€˜πΉ) ∈ β„•)
9998, 77sylibr 233 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ 0 ∈ (0..^(β™―β€˜πΉ)))
10099, 79syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((β™―β€˜πΉ) βˆ’ 1) ∈ (0..^(β™―β€˜πΉ)))
10193, 100ffvelcdmd 7092 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (πΉβ€˜((β™―β€˜πΉ) βˆ’ 1)) ∈ dom 𝐸)
102101adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) β†’ (πΉβ€˜((β™―β€˜πΉ) βˆ’ 1)) ∈ dom 𝐸)
10336, 102ffvelcdmd 7092 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) β†’ (πΈβ€˜(πΉβ€˜((β™―β€˜πΉ) βˆ’ 1))) ∈ ran 𝐸)
104 eqcom 2732 . . . . . . . . . . . . . . . . . . . . . . 23 ((πΈβ€˜(πΉβ€˜((β™―β€˜πΉ) βˆ’ 1))) = {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜(β™―β€˜πΉ))} ↔ {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜(β™―β€˜πΉ))} = (πΈβ€˜(πΉβ€˜((β™―β€˜πΉ) βˆ’ 1))))
105104biimpi 215 . . . . . . . . . . . . . . . . . . . . . 22 ((πΈβ€˜(πΉβ€˜((β™―β€˜πΉ) βˆ’ 1))) = {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜(β™―β€˜πΉ))} β†’ {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜(β™―β€˜πΉ))} = (πΈβ€˜(πΉβ€˜((β™―β€˜πΉ) βˆ’ 1))))
106105eleq1d 2810 . . . . . . . . . . . . . . . . . . . . 21 ((πΈβ€˜(πΉβ€˜((β™―β€˜πΉ) βˆ’ 1))) = {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜(β™―β€˜πΉ))} β†’ ({(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜(β™―β€˜πΉ))} ∈ ran 𝐸 ↔ (πΈβ€˜(πΉβ€˜((β™―β€˜πΉ) βˆ’ 1))) ∈ ran 𝐸))
107103, 106syl5ibrcom 246 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) β†’ ((πΈβ€˜(πΉβ€˜((β™―β€˜πΉ) βˆ’ 1))) = {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜(β™―β€˜πΉ))} β†’ {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜(β™―β€˜πΉ))} ∈ ran 𝐸))
10892, 107sylbid 239 . . . . . . . . . . . . . . . . . . 19 ((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) β†’ ((πΈβ€˜(πΉβ€˜((β™―β€˜πΉ) βˆ’ 1))) = {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜(((β™―β€˜πΉ) βˆ’ 1) + 1))} β†’ {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜(β™―β€˜πΉ))} ∈ ran 𝐸))
10987, 108syldc 48 . . . . . . . . . . . . . . . . . 18 (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} β†’ ((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) β†’ {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜(β™―β€˜πΉ))} ∈ ran 𝐸))
110109adantr 479 . . . . . . . . . . . . . . . . 17 ((βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))) β†’ ((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) β†’ {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜(β™―β€˜πΉ))} ∈ ran 𝐸))
111110impcom 406 . . . . . . . . . . . . . . . 16 (((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) ∧ (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)))) β†’ {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜(β™―β€˜πΉ))} ∈ ran 𝐸)
112 preq2 4739 . . . . . . . . . . . . . . . . . . 19 ((π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)) β†’ {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜0)} = {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜(β™―β€˜πΉ))})
113112eleq1d 2810 . . . . . . . . . . . . . . . . . 18 ((π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)) β†’ ({(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜0)} ∈ ran 𝐸 ↔ {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜(β™―β€˜πΉ))} ∈ ran 𝐸))
114113adantl 480 . . . . . . . . . . . . . . . . 17 ((βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))) β†’ ({(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜0)} ∈ ran 𝐸 ↔ {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜(β™―β€˜πΉ))} ∈ ran 𝐸))
115114adantl 480 . . . . . . . . . . . . . . . 16 (((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) ∧ (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)))) β†’ ({(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜0)} ∈ ran 𝐸 ↔ {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜(β™―β€˜πΉ))} ∈ ran 𝐸))
116111, 115mpbird 256 . . . . . . . . . . . . . . 15 (((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) ∧ (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)))) β†’ {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜0)} ∈ ran 𝐸)
11724, 59, 1163jca 1125 . . . . . . . . . . . . . 14 (((((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ 𝐸:dom 𝐸⟢ran 𝐸) ∧ (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)))) β†’ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜πΉ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜0)} ∈ ran 𝐸))
118117exp41 433 . . . . . . . . . . . . 13 ((((𝑃 ∈ Word 𝑉 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (β™―β€˜πΉ) ∈ β„•0) ∧ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1)) β†’ (2 ≀ (β™―β€˜π‘ƒ) β†’ (𝐸:dom 𝐸⟢ran 𝐸 β†’ ((βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))) β†’ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜πΉ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜0)} ∈ ran 𝐸)))))
119118exp41 433 . . . . . . . . . . . 12 (𝑃 ∈ Word 𝑉 β†’ (𝐹 ∈ Word dom 𝐸 β†’ ((β™―β€˜πΉ) ∈ β„•0 β†’ ((β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1) β†’ (2 ≀ (β™―β€˜π‘ƒ) β†’ (𝐸:dom 𝐸⟢ran 𝐸 β†’ ((βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))) β†’ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜πΉ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜0)} ∈ ran 𝐸))))))))
1208, 119syl 17 . . . . . . . . . . 11 (𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ β†’ (𝐹 ∈ Word dom 𝐸 β†’ ((β™―β€˜πΉ) ∈ β„•0 β†’ ((β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1) β†’ (2 ≀ (β™―β€˜π‘ƒ) β†’ (𝐸:dom 𝐸⟢ran 𝐸 β†’ ((βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))) β†’ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜πΉ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜0)} ∈ ran 𝐸))))))))
121120com13 88 . . . . . . . . . 10 ((β™―β€˜πΉ) ∈ β„•0 β†’ (𝐹 ∈ Word dom 𝐸 β†’ (𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ β†’ ((β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1) β†’ (2 ≀ (β™―β€˜π‘ƒ) β†’ (𝐸:dom 𝐸⟢ran 𝐸 β†’ ((βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))) β†’ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜πΉ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜0)} ∈ ran 𝐸))))))))
1224, 121mpcom 38 . . . . . . . . 9 (𝐹 ∈ Word dom 𝐸 β†’ (𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ β†’ ((β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1) β†’ (2 ≀ (β™―β€˜π‘ƒ) β†’ (𝐸:dom 𝐸⟢ran 𝐸 β†’ ((βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))) β†’ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜πΉ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜0)} ∈ ran 𝐸)))))))
123122imp 405 . . . . . . . 8 ((𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰) β†’ ((β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1) β†’ (2 ≀ (β™―β€˜π‘ƒ) β†’ (𝐸:dom 𝐸⟢ran 𝐸 β†’ ((βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))) β†’ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜πΉ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜0)} ∈ ran 𝐸))))))
1247, 123mpd 15 . . . . . . 7 ((𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰) β†’ (2 ≀ (β™―β€˜π‘ƒ) β†’ (𝐸:dom 𝐸⟢ran 𝐸 β†’ ((βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))) β†’ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜πΉ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜0)} ∈ ran 𝐸)))))
125124expcom 412 . . . . . 6 (𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ β†’ (𝐹 ∈ Word dom 𝐸 β†’ (2 ≀ (β™―β€˜π‘ƒ) β†’ (𝐸:dom 𝐸⟢ran 𝐸 β†’ ((βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))) β†’ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜πΉ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜0)} ∈ ran 𝐸))))))
126125com14 96 . . . . 5 (𝐸:dom 𝐸⟢ran 𝐸 β†’ (𝐹 ∈ Word dom 𝐸 β†’ (2 ≀ (β™―β€˜π‘ƒ) β†’ (𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ β†’ ((βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))) β†’ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜πΉ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜0)} ∈ ran 𝐸))))))
127126imp 405 . . . 4 ((𝐸:dom 𝐸⟢ran 𝐸 ∧ 𝐹 ∈ Word dom 𝐸) β†’ (2 ≀ (β™―β€˜π‘ƒ) β†’ (𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ β†’ ((βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))) β†’ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜πΉ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜0)} ∈ ran 𝐸)))))
128127impcomd 410 . . 3 ((𝐸:dom 𝐸⟢ran 𝐸 ∧ 𝐹 ∈ Word dom 𝐸) β†’ ((𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))) β†’ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜πΉ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜0)} ∈ ran 𝐸))))
1293, 128sylan 578 . 2 ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝐹 ∈ Word dom 𝐸) β†’ ((𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))) β†’ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜πΉ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜0)} ∈ ran 𝐸))))
1301293imp 1108 1 (((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)))) β†’ ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜πΉ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ∧ {(π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)), (π‘ƒβ€˜0)} ∈ ran 𝐸))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3051   βŠ† wss 3945  {cpr 4631   class class class wbr 5148  dom cdm 5677  ran crn 5678   Fn wfn 6542  βŸΆwf 6543  β€“1-1β†’wf1 6544  β€˜cfv 6547  (class class class)co 7417  β„cr 11137  0cc0 11138  1c1 11139   + caddc 11141   ≀ cle 11279   βˆ’ cmin 11474  β„•cn 12242  2c2 12297  β„•0cn0 12502  β„€cz 12588  β„€β‰₯cuz 12852  ...cfz 13516  ..^cfzo 13659  β™―chash 14321  Word cword 14496  lastSclsw 14544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5364  ax-pr 5428  ax-un 7739  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3965  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  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 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6499  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-riota 7373  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7870  df-1st 7992  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8723  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-card 9962  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-nn 12243  df-2 12305  df-n0 12503  df-z 12589  df-uz 12853  df-fz 13517  df-fzo 13660  df-hash 14322  df-word 14497  df-lsw 14545
This theorem is referenced by:  clwlkclwwlklem3  29867
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