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Theorem clwlkclwwlklem2 29797
Description: Lemma 2 for clwlkclwwlk 29799. (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 6788 . . . 4 (𝐸:dom 𝐸–1-1→𝑅 β†’ 𝐸 Fn dom 𝐸)
2 dffn3 6729 . . . 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 14507 . . . . . . . . 9 (𝐹 ∈ Word dom 𝐸 β†’ (β™―β€˜πΉ) ∈ β„•0)
5 ffn 6716 . . . . . . . . 9 (𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ β†’ 𝑃 Fn (0...(β™―β€˜πΉ)))
6 fnfz0hash 14429 . . . . . . . . 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 595 . . . . . . . 8 ((𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰) β†’ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1))
8 ffz0iswrd 14515 . . . . . . . . . . . 12 (𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ β†’ 𝑃 ∈ Word 𝑉)
9 lsw 14538 . . . . . . . . . . . . . . . . 17 (𝑃 ∈ Word 𝑉 β†’ (lastSβ€˜π‘ƒ) = (π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 1)))
109ad6antr 735 . . . . . . . . . . . . . . . 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 7437 . . . . . . . . . . . . . . . . 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 732 . . . . . . . . . . . . . . . 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 2734 . . . . . . . . . . . . . . . . . . 19 ((π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)) ↔ (π‘ƒβ€˜(β™―β€˜πΉ)) = (π‘ƒβ€˜0))
14 nn0cn 12504 . . . . . . . . . . . . . . . . . . . . . . . 24 ((β™―β€˜πΉ) ∈ β„•0 β†’ (β™―β€˜πΉ) ∈ β„‚)
15 1cnd 11231 . . . . . . . . . . . . . . . . . . . . . . . 24 ((β™―β€˜πΉ) ∈ β„•0 β†’ 1 ∈ β„‚)
1614, 15pncand 11594 . . . . . . . . . . . . . . . . . . . . . . 23 ((β™―β€˜πΉ) ∈ β„•0 β†’ (((β™―β€˜πΉ) + 1) βˆ’ 1) = (β™―β€˜πΉ))
1716eqcomd 2733 . . . . . . . . . . . . . . . . . . . . . 22 ((β™―β€˜πΉ) ∈ β„•0 β†’ (β™―β€˜πΉ) = (((β™―β€˜πΉ) + 1) βˆ’ 1))
1817ad4antlr 732 . . . . . . . . . . . . . . . . . . . . 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 6899 . . . . . . . . . . . . . . . . . . . 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 490 . . . . . . . . . . . . . . . . 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 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) + 1) βˆ’ 1)) = (π‘ƒβ€˜0))
2410, 12, 233eqtrd 2771 . . . . . . . . . . . . . . 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 12605 . . . . . . . . . . . . . . . . . . . . . 22 ((β™―β€˜πΉ) ∈ β„•0 β†’ (β™―β€˜πΉ) ∈ β„€)
26 peano2zm 12627 . . . . . . . . . . . . . . . . . . . . . 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 12503 . . . . . . . . . . . . . . . . . . . . . 22 ((β™―β€˜πΉ) ∈ β„•0 β†’ (β™―β€˜πΉ) ∈ ℝ)
2928lem1d 12169 . . . . . . . . . . . . . . . . . . . . 21 ((β™―β€˜πΉ) ∈ β„•0 β†’ ((β™―β€˜πΉ) βˆ’ 1) ≀ (β™―β€˜πΉ))
30 eluz2 12850 . . . . . . . . . . . . . . . . . . . . 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 1341 . . . . . . . . . . . . . . . . . . . 20 ((β™―β€˜πΉ) ∈ β„•0 β†’ (β™―β€˜πΉ) ∈ (β„€β‰₯β€˜((β™―β€˜πΉ) βˆ’ 1)))
3231ad4antlr 732 . . . . . . . . . . . . . . . . . . 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 13684 . . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . 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 14493 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐹 ∈ Word dom 𝐸 β†’ 𝐹:(0..^(β™―β€˜πΉ))⟢dom 𝐸)
39 simpll 766 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 13685 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3978 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7089 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 419 . . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . . . . . . . . . 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 729 . . . . . . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . . . . . . . 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 7089 . . . . . . . . . . . . . . . . . . . 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 2734 . . . . . . . . . . . . . . . . . . . . . 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 2813 . . . . . . . . . . . . . . . . . . . 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 3162 . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . 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 5146 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1) β†’ (2 ≀ (β™―β€˜π‘ƒ) ↔ 2 ≀ ((β™―β€˜πΉ) + 1)))
6160adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 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 12308 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2 ∈ ℝ
6362a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((β™―β€˜πΉ) ∈ β„•0 β†’ 2 ∈ ℝ)
64 1red 11237 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((β™―β€˜πΉ) ∈ β„•0 β†’ 1 ∈ ℝ)
6563, 64, 28lesubaddd 11833 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((β™―β€˜πΉ) ∈ β„•0 β†’ ((2 βˆ’ 1) ≀ (β™―β€˜πΉ) ↔ 2 ≀ ((β™―β€˜πΉ) + 1)))
66 2m1e1 12360 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (2 βˆ’ 1) = 1
6766breq1i 5149 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((2 βˆ’ 1) ≀ (β™―β€˜πΉ) ↔ 1 ≀ (β™―β€˜πΉ))
68 elnnnn0c 12539 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((β™―β€˜πΉ) ∈ β„• ↔ ((β™―β€˜πΉ) ∈ β„•0 ∧ 1 ≀ (β™―β€˜πΉ)))
6968simplbi2 500 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 260 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((β™―β€˜πΉ) ∈ β„•0 β†’ (2 ≀ ((β™―β€˜πΉ) + 1) β†’ (β™―β€˜πΉ) ∈ β„•))
7271adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 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 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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 406 . . . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . . 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 13696 . . . . . . . . . . . . . . . . . . . . . 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 13747 . . . . . . . . . . . . . . . . . . . . 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 6896 . . . . . . . . . . . . . . . . . . . . . 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 6891 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 = ((β™―β€˜πΉ) βˆ’ 1) β†’ (π‘ƒβ€˜π‘–) = (π‘ƒβ€˜((β™―β€˜πΉ) βˆ’ 1)))
83 fvoveq1 7437 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 = ((β™―β€˜πΉ) βˆ’ 1) β†’ (π‘ƒβ€˜(𝑖 + 1)) = (π‘ƒβ€˜(((β™―β€˜πΉ) βˆ’ 1) + 1)))
8482, 83preq12d 4741 . . . . . . . . . . . . . . . . . . . . . 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 2743 . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . 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 11597 . . . . . . . . . . . . . . . . . . . . . . . 24 ((β™―β€˜πΉ) ∈ β„•0 β†’ (((β™―β€˜πΉ) βˆ’ 1) + 1) = (β™―β€˜πΉ))
8988ad4antlr 732 . . . . . . . . . . . . . . . . . . . . . . 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 6895 . . . . . . . . . . . . . . . . . . . . . 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 4740 . . . . . . . . . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . . . . . . . . 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 732 . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 417 . . . . . . . . . . . . . . . . . . . . . . . . . 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 7089 . . . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . . 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 7089 . . . . . . . . . . . . . . . . . . . . 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 2734 . . . . . . . . . . . . . . . . . . . . . . 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 2813 . . . . . . . . . . . . . . . . . . . . 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 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†’ ((((((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 407 . . . . . . . . . . . . . . . 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 4734 . . . . . . . . . . . . . . . . . . 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 2813 . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . . . 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 1126 . . . . . . . . . . . . . 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 434 . . . . . . . . . . . . 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 434 . . . . . . . . . . . 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 406 . . . . . . . 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 413 . . . . . 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 406 . . . 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 411 . . 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 579 . 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 1109 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 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  βˆ€wral 3056   βŠ† wss 3944  {cpr 4626   class class class wbr 5142  dom cdm 5672  ran crn 5673   Fn wfn 6537  βŸΆwf 6538  β€“1-1β†’wf1 6539  β€˜cfv 6542  (class class class)co 7414  β„cr 11129  0cc0 11130  1c1 11131   + caddc 11133   ≀ cle 11271   βˆ’ cmin 11466  β„•cn 12234  2c2 12289  β„•0cn0 12494  β„€cz 12580  β„€β‰₯cuz 12844  ...cfz 13508  ..^cfzo 13651  β™―chash 14313  Word cword 14488  lastSclsw 14536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-card 9954  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-2 12297  df-n0 12495  df-z 12581  df-uz 12845  df-fz 13509  df-fzo 13652  df-hash 14314  df-word 14489  df-lsw 14537
This theorem is referenced by:  clwlkclwwlklem3  29798
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