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Theorem clwlkclwwlklem2a 29864
Description: Lemma for clwlkclwwlklem2 29866. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 11-Apr-2021.)
Hypothesis
Ref Expression
clwlkclwwlklem2.f 𝐹 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ if(π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2), (β—‘πΈβ€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}), (β—‘πΈβ€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜0)})))
Assertion
Ref Expression
clwlkclwwlklem2a ((𝐸:dom 𝐸–1-1→𝑅 ∧ 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†’ ((𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ 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)))))
Distinct variable groups:   π‘₯,𝑃   π‘₯,𝐸   π‘₯,𝑉   𝑖,𝐸   𝑖,𝐹   𝑃,𝑖   𝑅,𝑖,π‘₯   𝑖,𝑉
Allowed substitution hint:   𝐹(π‘₯)
Proof of Theorem clwlkclwwlklem2a
StepHypRef Expression
1 simpl 481 . . . . . . . . . 10 ((π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ (((𝐸:dom 𝐸–1-1→𝑅 ∧ 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))) ∧ π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)))) β†’ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2))
2 f1f1orn 6847 . . . . . . . . . . . . . 14 (𝐸:dom 𝐸–1-1→𝑅 β†’ 𝐸:dom 𝐸–1-1-ontoβ†’ran 𝐸)
323ad2ant1 1130 . . . . . . . . . . . . 13 ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ 𝐸:dom 𝐸–1-1-ontoβ†’ran 𝐸)
43adantr 479 . . . . . . . . . . . 12 (((𝐸:dom 𝐸–1-1→𝑅 ∧ 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†’ 𝐸:dom 𝐸–1-1-ontoβ†’ran 𝐸)
54ad2antrl 726 . . . . . . . . . . 11 ((π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ (((𝐸:dom 𝐸–1-1→𝑅 ∧ 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))) ∧ π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)))) β†’ 𝐸:dom 𝐸–1-1-ontoβ†’ran 𝐸)
6 elfzo0 13705 . . . . . . . . . . . . . . . . . . . . 21 (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↔ (π‘₯ ∈ β„•0 ∧ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ β„• ∧ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 1)))
7 lencl 14515 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑃 ∈ Word 𝑉 β†’ (β™―β€˜π‘ƒ) ∈ β„•0)
8 simpl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((π‘₯ ∈ β„•0 ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ π‘₯ ∈ β„•0)
98adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((π‘₯ ∈ β„•0 ∧ (β™―β€˜π‘ƒ) ∈ β„•0) ∧ (π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ 2 ≀ (β™―β€˜π‘ƒ))) β†’ π‘₯ ∈ β„•0)
10 elnn0z 12601 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (π‘₯ ∈ β„•0 ↔ (π‘₯ ∈ β„€ ∧ 0 ≀ π‘₯))
11 0red 11247 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((π‘₯ ∈ β„€ ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ 0 ∈ ℝ)
12 zre 12592 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (π‘₯ ∈ β„€ β†’ π‘₯ ∈ ℝ)
1312adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((π‘₯ ∈ β„€ ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ π‘₯ ∈ ℝ)
14 nn0re 12511 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (β™―β€˜π‘ƒ) ∈ ℝ)
15 2re 12316 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2 ∈ ℝ
1615a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ 2 ∈ ℝ)
1714, 16resubcld 11672 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ ((β™―β€˜π‘ƒ) βˆ’ 2) ∈ ℝ)
1817adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((π‘₯ ∈ β„€ ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ ((β™―β€˜π‘ƒ) βˆ’ 2) ∈ ℝ)
19 lelttr 11334 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((0 ∈ ℝ ∧ π‘₯ ∈ ℝ ∧ ((β™―β€˜π‘ƒ) βˆ’ 2) ∈ ℝ) β†’ ((0 ≀ π‘₯ ∧ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2)) β†’ 0 < ((β™―β€˜π‘ƒ) βˆ’ 2)))
2011, 13, 18, 19syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((π‘₯ ∈ β„€ ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ ((0 ≀ π‘₯ ∧ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2)) β†’ 0 < ((β™―β€˜π‘ƒ) βˆ’ 2)))
21 nn0z 12613 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (β™―β€˜π‘ƒ) ∈ β„€)
22 2z 12624 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2 ∈ β„€
2322a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ 2 ∈ β„€)
2421, 23zsubcld 12701 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ ((β™―β€˜π‘ƒ) βˆ’ 2) ∈ β„€)
2524anim1i 613 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 (((β™―β€˜π‘ƒ) ∈ β„•0 ∧ 0 < ((β™―β€˜π‘ƒ) βˆ’ 2)) β†’ (((β™―β€˜π‘ƒ) βˆ’ 2) ∈ β„€ ∧ 0 < ((β™―β€˜π‘ƒ) βˆ’ 2)))
26 elnnz 12598 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 (((β™―β€˜π‘ƒ) βˆ’ 2) ∈ β„• ↔ (((β™―β€˜π‘ƒ) βˆ’ 2) ∈ β„€ ∧ 0 < ((β™―β€˜π‘ƒ) βˆ’ 2)))
2725, 26sylibr 233 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 (((β™―β€˜π‘ƒ) ∈ β„•0 ∧ 0 < ((β™―β€˜π‘ƒ) βˆ’ 2)) β†’ ((β™―β€˜π‘ƒ) βˆ’ 2) ∈ β„•)
28 nn0cn 12512 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (β™―β€˜π‘ƒ) ∈ β„‚)
29 peano2cnm 11556 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ((β™―β€˜π‘ƒ) ∈ β„‚ β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ β„‚)
3028, 29syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ β„‚)
3130subid1d 11590 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) = ((β™―β€˜π‘ƒ) βˆ’ 1))
3231oveq1d 7432 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ ((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1) = (((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 1))
33 1cnd 11239 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ 1 ∈ β„‚)
3428, 33, 33subsub4d 11632 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 1) = ((β™―β€˜π‘ƒ) βˆ’ (1 + 1)))
35 1p1e2 12367 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 (1 + 1) = 2
3635a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (1 + 1) = 2)
3736oveq2d 7433 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ ((β™―β€˜π‘ƒ) βˆ’ (1 + 1)) = ((β™―β€˜π‘ƒ) βˆ’ 2))
3834, 37eqtrd 2765 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 1) = ((β™―β€˜π‘ƒ) βˆ’ 2))
3932, 38eqtrd 2765 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ ((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1) = ((β™―β€˜π‘ƒ) βˆ’ 2))
4039eleq1d 2810 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1) ∈ β„• ↔ ((β™―β€˜π‘ƒ) βˆ’ 2) ∈ β„•))
4140adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 (((β™―β€˜π‘ƒ) ∈ β„•0 ∧ 0 < ((β™―β€˜π‘ƒ) βˆ’ 2)) β†’ (((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1) ∈ β„• ↔ ((β™―β€˜π‘ƒ) βˆ’ 2) ∈ β„•))
4227, 41mpbird 256 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (((β™―β€˜π‘ƒ) ∈ β„•0 ∧ 0 < ((β™―β€˜π‘ƒ) βˆ’ 2)) β†’ ((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1) ∈ β„•)
4342ex 411 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (0 < ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ ((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1) ∈ β„•))
4443adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((π‘₯ ∈ β„€ ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ (0 < ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ ((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1) ∈ β„•))
4520, 44syld 47 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((π‘₯ ∈ β„€ ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ ((0 ≀ π‘₯ ∧ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2)) β†’ ((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1) ∈ β„•))
4645exp4b 429 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (π‘₯ ∈ β„€ β†’ ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (0 ≀ π‘₯ β†’ (π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ ((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1) ∈ β„•))))
4746com23 86 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (π‘₯ ∈ β„€ β†’ (0 ≀ π‘₯ β†’ ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ ((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1) ∈ β„•))))
4847imp 405 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((π‘₯ ∈ β„€ ∧ 0 ≀ π‘₯) β†’ ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ ((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1) ∈ β„•)))
4910, 48sylbi 216 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (π‘₯ ∈ β„•0 β†’ ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ ((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1) ∈ β„•)))
5049imp 405 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((π‘₯ ∈ β„•0 ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ (π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ ((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1) ∈ β„•))
5150com12 32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ ((π‘₯ ∈ β„•0 ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ ((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1) ∈ β„•))
5251adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((π‘₯ ∈ β„•0 ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ ((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1) ∈ β„•))
5352impcom 406 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((π‘₯ ∈ β„•0 ∧ (β™―β€˜π‘ƒ) ∈ β„•0) ∧ (π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ 2 ≀ (β™―β€˜π‘ƒ))) β†’ ((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1) ∈ β„•)
54 df-2 12305 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2 = (1 + 1)
5554a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ 2 = (1 + 1))
5655oveq2d 7433 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ ((β™―β€˜π‘ƒ) βˆ’ 2) = ((β™―β€˜π‘ƒ) βˆ’ (1 + 1)))
5731eqcomd 2731 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) = (((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0))
5857oveq1d 7432 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 1) = ((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1))
5956, 34, 583eqtr2d 2771 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ ((β™―β€˜π‘ƒ) βˆ’ 2) = ((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1))
6059adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((π‘₯ ∈ β„•0 ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ ((β™―β€˜π‘ƒ) βˆ’ 2) = ((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1))
6160breq2d 5160 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((π‘₯ ∈ β„•0 ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ (π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ↔ π‘₯ < ((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)))
6261biimpcd 248 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ ((π‘₯ ∈ β„•0 ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ π‘₯ < ((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)))
6362adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((π‘₯ ∈ β„•0 ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ π‘₯ < ((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)))
6463impcom 406 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((π‘₯ ∈ β„•0 ∧ (β™―β€˜π‘ƒ) ∈ β„•0) ∧ (π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ 2 ≀ (β™―β€˜π‘ƒ))) β†’ π‘₯ < ((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1))
65 elfzo0 13705 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (π‘₯ ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)) ↔ (π‘₯ ∈ β„•0 ∧ ((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1) ∈ β„• ∧ π‘₯ < ((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)))
669, 53, 64, 65syl3anbrc 1340 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((π‘₯ ∈ β„•0 ∧ (β™―β€˜π‘ƒ) ∈ β„•0) ∧ (π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ 2 ≀ (β™―β€˜π‘ƒ))) β†’ π‘₯ ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)))
6766exp32 419 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((π‘₯ ∈ β„•0 ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ (π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ (2 ≀ (β™―β€˜π‘ƒ) β†’ π‘₯ ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)))))
6867a1d 25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((π‘₯ ∈ β„•0 ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ (π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 1) β†’ (π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ (2 ≀ (β™―β€˜π‘ƒ) β†’ π‘₯ ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1))))))
6968com24 95 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((π‘₯ ∈ β„•0 ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ (2 ≀ (β™―β€˜π‘ƒ) β†’ (π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ (π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 1) β†’ π‘₯ ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1))))))
7069ex 411 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (π‘₯ ∈ β„•0 β†’ ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (2 ≀ (β™―β€˜π‘ƒ) β†’ (π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ (π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 1) β†’ π‘₯ ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)))))))
7170com25 99 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (π‘₯ ∈ β„•0 β†’ (π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 1) β†’ (2 ≀ (β™―β€˜π‘ƒ) β†’ (π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ π‘₯ ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)))))))
7271imp 405 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((π‘₯ ∈ β„•0 ∧ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 1)) β†’ (2 ≀ (β™―β€˜π‘ƒ) β†’ (π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ π‘₯ ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1))))))
73723adant2 1128 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((π‘₯ ∈ β„•0 ∧ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ β„• ∧ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 1)) β†’ (2 ≀ (β™―β€˜π‘ƒ) β†’ (π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ π‘₯ ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1))))))
7473com14 96 . . . . . . . . . . . . . . . . . . . . . . . 24 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (2 ≀ (β™―β€˜π‘ƒ) β†’ (π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ ((π‘₯ ∈ β„•0 ∧ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ β„• ∧ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 1)) β†’ π‘₯ ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1))))))
757, 74syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑃 ∈ Word 𝑉 β†’ (2 ≀ (β™―β€˜π‘ƒ) β†’ (π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ ((π‘₯ ∈ β„•0 ∧ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ β„• ∧ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 1)) β†’ π‘₯ ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1))))))
7675imp 405 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ ((π‘₯ ∈ β„•0 ∧ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ β„• ∧ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 1)) β†’ π‘₯ ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)))))
77763adant1 1127 . . . . . . . . . . . . . . . . . . . . 21 ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ ((π‘₯ ∈ β„•0 ∧ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ β„• ∧ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 1)) β†’ π‘₯ ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)))))
786, 77syl7bi 254 . . . . . . . . . . . . . . . . . . . 20 ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) β†’ π‘₯ ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)))))
7978com13 88 . . . . . . . . . . . . . . . . . . 19 (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) β†’ (π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ π‘₯ ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)))))
8079imp31 416 . . . . . . . . . . . . . . . . . 18 (((π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ∧ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2)) ∧ (𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ))) β†’ π‘₯ ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)))
81 fveq2 6894 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = π‘₯ β†’ (π‘ƒβ€˜π‘–) = (π‘ƒβ€˜π‘₯))
82 fvoveq1 7440 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = π‘₯ β†’ (π‘ƒβ€˜(𝑖 + 1)) = (π‘ƒβ€˜(π‘₯ + 1)))
8381, 82preq12d 4746 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = π‘₯ β†’ {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} = {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})
8483eleq1d 2810 . . . . . . . . . . . . . . . . . . 19 (𝑖 = π‘₯ β†’ ({(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ↔ {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))} ∈ ran 𝐸))
8584adantl 480 . . . . . . . . . . . . . . . . . 18 ((((π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ∧ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2)) ∧ (𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ))) ∧ 𝑖 = π‘₯) β†’ ({(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ↔ {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))} ∈ ran 𝐸))
8680, 85rspcdv 3599 . . . . . . . . . . . . . . . . 17 (((π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ∧ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2)) ∧ (𝐸:dom 𝐸–1-1→𝑅 ∧ 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 β†’ {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))} ∈ ran 𝐸))
8786ex 411 . . . . . . . . . . . . . . . 16 ((π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ∧ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2)) β†’ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 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 β†’ {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))} ∈ ran 𝐸)))
8887com13 88 . . . . . . . . . . . . . . 15 (βˆ€π‘– ∈ (0..^((((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 β†’ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ∧ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2)) β†’ {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))} ∈ ran 𝐸)))
8988ad2antrl 726 . . . . . . . . . . . . . 14 (((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†’ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ∧ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2)) β†’ {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))} ∈ ran 𝐸)))
9089impcom 406 . . . . . . . . . . . . 13 (((𝐸:dom 𝐸–1-1→𝑅 ∧ 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‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ∧ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2)) β†’ {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))} ∈ ran 𝐸))
9190expdimp 451 . . . . . . . . . . . 12 ((((𝐸:dom 𝐸–1-1→𝑅 ∧ 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))) ∧ π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1))) β†’ (π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))} ∈ ran 𝐸))
9291impcom 406 . . . . . . . . . . 11 ((π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ (((𝐸:dom 𝐸–1-1→𝑅 ∧ 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))) ∧ π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)))) β†’ {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))} ∈ ran 𝐸)
93 f1ocnvdm 7292 . . . . . . . . . . 11 ((𝐸:dom 𝐸–1-1-ontoβ†’ran 𝐸 ∧ {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))} ∈ ran 𝐸) β†’ (β—‘πΈβ€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}) ∈ dom 𝐸)
945, 92, 93syl2anc 582 . . . . . . . . . 10 ((π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ (((𝐸:dom 𝐸–1-1→𝑅 ∧ 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))) ∧ π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)))) β†’ (β—‘πΈβ€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}) ∈ dom 𝐸)
951, 94jca 510 . . . . . . . . 9 ((π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ (((𝐸:dom 𝐸–1-1→𝑅 ∧ 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))) ∧ π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)))) β†’ (π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ (β—‘πΈβ€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}) ∈ dom 𝐸))
9695orcd 871 . . . . . . . 8 ((π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ (((𝐸:dom 𝐸–1-1→𝑅 ∧ 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))) ∧ π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)))) β†’ ((π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ (β—‘πΈβ€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}) ∈ dom 𝐸) ∨ (Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ (β—‘πΈβ€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜0)}) ∈ dom 𝐸)))
97 simpl 481 . . . . . . . . . 10 ((Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ (((𝐸:dom 𝐸–1-1→𝑅 ∧ 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))) ∧ π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)))) β†’ Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2))
984ad2antrl 726 . . . . . . . . . . 11 ((Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ (((𝐸:dom 𝐸–1-1→𝑅 ∧ 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))) ∧ π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)))) β†’ 𝐸:dom 𝐸–1-1-ontoβ†’ran 𝐸)
99 nn0z 12613 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (π‘₯ ∈ β„•0 β†’ π‘₯ ∈ β„€)
100 peano2zm 12635 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((β™―β€˜π‘ƒ) ∈ β„€ β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ β„€)
10121, 100syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ β„€)
10299, 101anim12i 611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((π‘₯ ∈ β„•0 ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ (π‘₯ ∈ β„€ ∧ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ β„€))
103 zltlem1 12645 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((π‘₯ ∈ β„€ ∧ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ β„€) β†’ (π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 1) ↔ π‘₯ ≀ (((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 1)))
104102, 103syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((π‘₯ ∈ β„•0 ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ (π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 1) ↔ π‘₯ ≀ (((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 1)))
10538adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((π‘₯ ∈ β„•0 ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ (((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 1) = ((β™―β€˜π‘ƒ) βˆ’ 2))
106105breq2d 5160 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((π‘₯ ∈ β„•0 ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ (π‘₯ ≀ (((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 1) ↔ π‘₯ ≀ ((β™―β€˜π‘ƒ) βˆ’ 2)))
107106biimpd 228 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((π‘₯ ∈ β„•0 ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ (π‘₯ ≀ (((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 1) β†’ π‘₯ ≀ ((β™―β€˜π‘ƒ) βˆ’ 2)))
108104, 107sylbid 239 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((π‘₯ ∈ β„•0 ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ (π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 1) β†’ π‘₯ ≀ ((β™―β€˜π‘ƒ) βˆ’ 2)))
109108impancom 450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((π‘₯ ∈ β„•0 ∧ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 1)) β†’ ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ π‘₯ ≀ ((β™―β€˜π‘ƒ) βˆ’ 2)))
110109imp 405 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((π‘₯ ∈ β„•0 ∧ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 1)) ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ π‘₯ ≀ ((β™―β€˜π‘ƒ) βˆ’ 2))
111 nn0re 12511 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (π‘₯ ∈ β„•0 β†’ π‘₯ ∈ ℝ)
112111adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((π‘₯ ∈ β„•0 ∧ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 1)) β†’ π‘₯ ∈ ℝ)
113112, 17anim12i 611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((π‘₯ ∈ β„•0 ∧ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 1)) ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ (π‘₯ ∈ ℝ ∧ ((β™―β€˜π‘ƒ) βˆ’ 2) ∈ ℝ))
114 lenlt 11322 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((π‘₯ ∈ ℝ ∧ ((β™―β€˜π‘ƒ) βˆ’ 2) ∈ ℝ) β†’ (π‘₯ ≀ ((β™―β€˜π‘ƒ) βˆ’ 2) ↔ Β¬ ((β™―β€˜π‘ƒ) βˆ’ 2) < π‘₯))
115113, 114syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((π‘₯ ∈ β„•0 ∧ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 1)) ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ (π‘₯ ≀ ((β™―β€˜π‘ƒ) βˆ’ 2) ↔ Β¬ ((β™―β€˜π‘ƒ) βˆ’ 2) < π‘₯))
116110, 115mpbid 231 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((π‘₯ ∈ β„•0 ∧ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 1)) ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ Β¬ ((β™―β€˜π‘ƒ) βˆ’ 2) < π‘₯)
117116anim1i 613 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((π‘₯ ∈ β„•0 ∧ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 1)) ∧ (β™―β€˜π‘ƒ) ∈ β„•0) ∧ Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2)) β†’ (Β¬ ((β™―β€˜π‘ƒ) βˆ’ 2) < π‘₯ ∧ Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2)))
118113ancomd 460 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((π‘₯ ∈ β„•0 ∧ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 1)) ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ (((β™―β€˜π‘ƒ) βˆ’ 2) ∈ ℝ ∧ π‘₯ ∈ ℝ))
119118adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((π‘₯ ∈ β„•0 ∧ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 1)) ∧ (β™―β€˜π‘ƒ) ∈ β„•0) ∧ Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2)) β†’ (((β™―β€˜π‘ƒ) βˆ’ 2) ∈ ℝ ∧ π‘₯ ∈ ℝ))
120 lttri3 11327 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((β™―β€˜π‘ƒ) βˆ’ 2) ∈ ℝ ∧ π‘₯ ∈ ℝ) β†’ (((β™―β€˜π‘ƒ) βˆ’ 2) = π‘₯ ↔ (Β¬ ((β™―β€˜π‘ƒ) βˆ’ 2) < π‘₯ ∧ Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2))))
121119, 120syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((π‘₯ ∈ β„•0 ∧ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 1)) ∧ (β™―β€˜π‘ƒ) ∈ β„•0) ∧ Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2)) β†’ (((β™―β€˜π‘ƒ) βˆ’ 2) = π‘₯ ↔ (Β¬ ((β™―β€˜π‘ƒ) βˆ’ 2) < π‘₯ ∧ Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2))))
122117, 121mpbird 256 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((π‘₯ ∈ β„•0 ∧ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 1)) ∧ (β™―β€˜π‘ƒ) ∈ β„•0) ∧ Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2)) β†’ ((β™―β€˜π‘ƒ) βˆ’ 2) = π‘₯)
123122exp31 418 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((π‘₯ ∈ β„•0 ∧ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 1)) β†’ ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ ((β™―β€˜π‘ƒ) βˆ’ 2) = π‘₯)))
124123com23 86 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((π‘₯ ∈ β„•0 ∧ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 1)) β†’ (Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ ((β™―β€˜π‘ƒ) βˆ’ 2) = π‘₯)))
1251243adant2 1128 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((π‘₯ ∈ β„•0 ∧ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ β„• ∧ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 1)) β†’ (Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ ((β™―β€˜π‘ƒ) βˆ’ 2) = π‘₯)))
1266, 125sylbi 216 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) β†’ (Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ ((β™―β€˜π‘ƒ) βˆ’ 2) = π‘₯)))
127126impcom 406 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1))) β†’ ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ ((β™―β€˜π‘ƒ) βˆ’ 2) = π‘₯))
1287, 127syl5com 31 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑃 ∈ Word 𝑉 β†’ ((Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1))) β†’ ((β™―β€˜π‘ƒ) βˆ’ 2) = π‘₯))
1291283ad2ant2 1131 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1))) β†’ ((β™―β€˜π‘ƒ) βˆ’ 2) = π‘₯))
130129imp 405 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ (Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)))) β†’ ((β™―β€˜π‘ƒ) βˆ’ 2) = π‘₯)
131130fveq2d 6898 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ (Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)))) β†’ (π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)) = (π‘ƒβ€˜π‘₯))
132131preq1d 4744 . . . . . . . . . . . . . . . . . . . . 21 (((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ (Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)))) β†’ {(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜0)} = {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜0)})
133132eleq1d 2810 . . . . . . . . . . . . . . . . . . . 20 (((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ (Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)))) β†’ ({(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜0)} ∈ ran 𝐸 ↔ {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜0)} ∈ ran 𝐸))
134133biimpd 228 . . . . . . . . . . . . . . . . . . 19 (((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ (Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)))) β†’ ({(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜0)} ∈ ran 𝐸 β†’ {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜0)} ∈ ran 𝐸))
135134exp32 419 . . . . . . . . . . . . . . . . . 18 ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) β†’ ({(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜0)} ∈ ran 𝐸 β†’ {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜0)} ∈ ran 𝐸))))
136135com12 32 . . . . . . . . . . . . . . . . 17 (Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) β†’ ({(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜0)} ∈ ran 𝐸 β†’ {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜0)} ∈ ran 𝐸))))
137136com14 96 . . . . . . . . . . . . . . . 16 ({(π‘ƒβ€˜((β™―β€˜π‘ƒ) βˆ’ 2)), (π‘ƒβ€˜0)} ∈ ran 𝐸 β†’ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) β†’ (Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜0)} ∈ ran 𝐸))))
138137adantl 480 . . . . . . . . . . . . . . 15 ((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†’ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) β†’ (Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜0)} ∈ ran 𝐸))))
139138adantl 480 . . . . . . . . . . . . . 14 (((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†’ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) β†’ (Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜0)} ∈ ran 𝐸))))
140139com12 32 . . . . . . . . . . . . 13 ((𝐸:dom 𝐸–1-1→𝑅 ∧ 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†’ (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) β†’ (Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜0)} ∈ ran 𝐸))))
141140imp31 416 . . . . . . . . . . . 12 ((((𝐸:dom 𝐸–1-1→𝑅 ∧ 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))) ∧ π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1))) β†’ (Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) β†’ {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜0)} ∈ ran 𝐸))
142141impcom 406 . . . . . . . . . . 11 ((Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ (((𝐸:dom 𝐸–1-1→𝑅 ∧ 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))) ∧ π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)))) β†’ {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜0)} ∈ ran 𝐸)
143 f1ocnvdm 7292 . . . . . . . . . . 11 ((𝐸:dom 𝐸–1-1-ontoβ†’ran 𝐸 ∧ {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜0)} ∈ ran 𝐸) β†’ (β—‘πΈβ€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜0)}) ∈ dom 𝐸)
14498, 142, 143syl2anc 582 . . . . . . . . . 10 ((Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ (((𝐸:dom 𝐸–1-1→𝑅 ∧ 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))) ∧ π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)))) β†’ (β—‘πΈβ€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜0)}) ∈ dom 𝐸)
14597, 144jca 510 . . . . . . . . 9 ((Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ (((𝐸:dom 𝐸–1-1→𝑅 ∧ 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))) ∧ π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)))) β†’ (Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ (β—‘πΈβ€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜0)}) ∈ dom 𝐸))
146145olcd 872 . . . . . . . 8 ((Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ (((𝐸:dom 𝐸–1-1→𝑅 ∧ 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))) ∧ π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)))) β†’ ((π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ (β—‘πΈβ€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}) ∈ dom 𝐸) ∨ (Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ (β—‘πΈβ€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜0)}) ∈ dom 𝐸)))
14796, 146pm2.61ian 810 . . . . . . 7 ((((𝐸:dom 𝐸–1-1→𝑅 ∧ 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))) ∧ π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1))) β†’ ((π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ (β—‘πΈβ€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}) ∈ dom 𝐸) ∨ (Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ (β—‘πΈβ€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜0)}) ∈ dom 𝐸)))
148 ifel 4573 . . . . . . 7 (if(π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2), (β—‘πΈβ€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}), (β—‘πΈβ€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜0)})) ∈ dom 𝐸 ↔ ((π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ (β—‘πΈβ€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}) ∈ dom 𝐸) ∨ (Β¬ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2) ∧ (β—‘πΈβ€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜0)}) ∈ dom 𝐸)))
149147, 148sylibr 233 . . . . . 6 ((((𝐸:dom 𝐸–1-1→𝑅 ∧ 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))) ∧ π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1))) β†’ if(π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2), (β—‘πΈβ€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}), (β—‘πΈβ€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜0)})) ∈ dom 𝐸)
150 clwlkclwwlklem2.f . . . . . 6 𝐹 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ if(π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 2), (β—‘πΈβ€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}), (β—‘πΈβ€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜0)})))
151149, 150fmptd 7121 . . . . 5 (((𝐸:dom 𝐸–1-1→𝑅 ∧ 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†’ 𝐹:(0..^((β™―β€˜π‘ƒ) βˆ’ 1))⟢dom 𝐸)
152 iswrdi 14500 . . . . 5 (𝐹:(0..^((β™―β€˜π‘ƒ) βˆ’ 1))⟢dom 𝐸 β†’ 𝐹 ∈ Word dom 𝐸)
153151, 152syl 17 . . . 4 (((𝐸:dom 𝐸–1-1→𝑅 ∧ 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†’ 𝐹 ∈ Word dom 𝐸)
154 wrdf 14501 . . . . . . . 8 (𝑃 ∈ Word 𝑉 β†’ 𝑃:(0..^(β™―β€˜π‘ƒ))βŸΆπ‘‰)
155154adantr 479 . . . . . . 7 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ 𝑃:(0..^(β™―β€˜π‘ƒ))βŸΆπ‘‰)
156150clwlkclwwlklem2a2 29859 . . . . . . . . 9 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (β™―β€˜πΉ) = ((β™―β€˜π‘ƒ) βˆ’ 1))
157 fzoval 13665 . . . . . . . . . . 11 ((β™―β€˜π‘ƒ) ∈ β„€ β†’ (0..^(β™―β€˜π‘ƒ)) = (0...((β™―β€˜π‘ƒ) βˆ’ 1)))
1587, 21, 1573syl 18 . . . . . . . . . 10 (𝑃 ∈ Word 𝑉 β†’ (0..^(β™―β€˜π‘ƒ)) = (0...((β™―β€˜π‘ƒ) βˆ’ 1)))
159 oveq2 7425 . . . . . . . . . . 11 (((β™―β€˜π‘ƒ) βˆ’ 1) = (β™―β€˜πΉ) β†’ (0...((β™―β€˜π‘ƒ) βˆ’ 1)) = (0...(β™―β€˜πΉ)))
160159eqcoms 2733 . . . . . . . . . 10 ((β™―β€˜πΉ) = ((β™―β€˜π‘ƒ) βˆ’ 1) β†’ (0...((β™―β€˜π‘ƒ) βˆ’ 1)) = (0...(β™―β€˜πΉ)))
161158, 160sylan9eq 2785 . . . . . . . . 9 ((𝑃 ∈ Word 𝑉 ∧ (β™―β€˜πΉ) = ((β™―β€˜π‘ƒ) βˆ’ 1)) β†’ (0..^(β™―β€˜π‘ƒ)) = (0...(β™―β€˜πΉ)))
162156, 161syldan 589 . . . . . . . 8 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (0..^(β™―β€˜π‘ƒ)) = (0...(β™―β€˜πΉ)))
163162feq2d 6707 . . . . . . 7 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (𝑃:(0..^(β™―β€˜π‘ƒ))βŸΆπ‘‰ ↔ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰))
164155, 163mpbid 231 . . . . . 6 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰)
1651643adant1 1127 . . . . 5 ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰)
166165adantr 479 . . . 4 (((𝐸:dom 𝐸–1-1→𝑅 ∧ 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†’ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰)
167 clwlkclwwlklem2a1 29858 . . . . . . 7 ((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))
1681673adant1 1127 . . . . . 6 ((𝐸:dom 𝐸–1-1→𝑅 ∧ 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))
169168imp 405 . . . . 5 (((𝐸:dom 𝐸–1-1→𝑅 ∧ 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)
1701563adant1 1127 . . . . . . . 8 ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (β™―β€˜πΉ) = ((β™―β€˜π‘ƒ) βˆ’ 1))
171170adantr 479 . . . . . . 7 (((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ (lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0)) β†’ (β™―β€˜πΉ) = ((β™―β€˜π‘ƒ) βˆ’ 1))
172150clwlkclwwlklem2a4 29863 . . . . . . . . . 10 ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) ∧ 𝑖 ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1))) β†’ ({(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 β†’ (πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})))
173172impl 454 . . . . . . . . 9 ((((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ (lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0)) ∧ 𝑖 ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1))) β†’ ({(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 β†’ (πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))
174173ralimdva 3157 . . . . . . . 8 (((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ (lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0)) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 β†’ βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))
175 oveq2 7425 . . . . . . . . . 10 ((β™―β€˜πΉ) = ((β™―β€˜π‘ƒ) βˆ’ 1) β†’ (0..^(β™―β€˜πΉ)) = (0..^((β™―β€˜π‘ƒ) βˆ’ 1)))
176175raleqdv 3315 . . . . . . . . 9 ((β™―β€˜πΉ) = ((β™―β€˜π‘ƒ) βˆ’ 1) β†’ (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ↔ βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))
177176imbi2d 339 . . . . . . . 8 ((β™―β€˜πΉ) = ((β™―β€˜π‘ƒ) βˆ’ 1) β†’ ((βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) ↔ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 β†’ βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})))
178174, 177imbitrrid 245 . . . . . . 7 ((β™―β€˜πΉ) = ((β™―β€˜π‘ƒ) βˆ’ 1) β†’ (((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ (lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0)) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})))
179171, 178mpcom 38 . . . . . 6 (((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) ∧ (lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0)) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))
180179adantrr 715 . . . . 5 (((𝐸:dom 𝐸–1-1→𝑅 ∧ 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 β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))
181169, 180mpd 15 . . . 4 (((𝐸:dom 𝐸–1-1→𝑅 ∧ 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Ή))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})
182153, 166, 1813jca 1125 . . 3 (((𝐸:dom 𝐸–1-1→𝑅 ∧ 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†’ (𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))
183150clwlkclwwlklem2a3 29860 . . . . . . . . . 10 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (π‘ƒβ€˜(β™―β€˜πΉ)) = (lastSβ€˜π‘ƒ))
1841833adant1 1127 . . . . . . . . 9 ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (π‘ƒβ€˜(β™―β€˜πΉ)) = (lastSβ€˜π‘ƒ))
185184eqcomd 2731 . . . . . . . 8 ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (lastSβ€˜π‘ƒ) = (π‘ƒβ€˜(β™―β€˜πΉ)))
186185eqeq2d 2736 . . . . . . 7 ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ ((π‘ƒβ€˜0) = (lastSβ€˜π‘ƒ) ↔ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))))
187186biimpcd 248 . . . . . 6 ((π‘ƒβ€˜0) = (lastSβ€˜π‘ƒ) β†’ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))))
188187eqcoms 2733 . . . . 5 ((lastSβ€˜π‘ƒ) = (π‘ƒβ€˜0) β†’ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))))
189188adantr 479 . . . 4 (((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†’ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))))
190189impcom 406 . . 3 (((𝐸:dom 𝐸–1-1→𝑅 ∧ 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€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)))
191182, 190jca 510 . 2 (((𝐸:dom 𝐸–1-1→𝑅 ∧ 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†’ ((𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ 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))))
192191ex 411 1 ((𝐸:dom 𝐸–1-1→𝑅 ∧ 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†’ ((𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ 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)))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 845   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3051  ifcif 4529  {cpr 4631   class class class wbr 5148   ↦ cmpt 5231  β—‘ccnv 5676  dom cdm 5677  ran crn 5678  βŸΆwf 6543  β€“1-1β†’wf1 6544  β€“1-1-ontoβ†’wf1o 6546  β€˜cfv 6547  (class class class)co 7417  β„‚cc 11136  β„cr 11137  0cc0 11138  1c1 11139   + caddc 11141   < clt 11278   ≀ cle 11279   βˆ’ cmin 11474  β„•cn 12242  2c2 12297  β„•0cn0 12502  β„€cz 12588  ...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:  clwlkclwwlklem1  29865
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