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Theorem clwlkclwwlklem2a 28984
Description: Lemma for clwlkclwwlklem2 28986. (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 484 . . . . . . . . . 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 6800 . . . . . . . . . . . . . 14 (𝐸:dom 𝐸–1-1→𝑅 β†’ 𝐸:dom 𝐸–1-1-ontoβ†’ran 𝐸)
323ad2ant1 1134 . . . . . . . . . . . . 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 482 . . . . . . . . . . . 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 727 . . . . . . . . . . 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 13620 . . . . . . . . . . . . . . . . . . . . 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 14428 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑃 ∈ Word 𝑉 β†’ (β™―β€˜π‘ƒ) ∈ β„•0)
8 simpl 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((π‘₯ ∈ β„•0 ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ π‘₯ ∈ β„•0)
98adantr 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12519 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (π‘₯ ∈ β„•0 ↔ (π‘₯ ∈ β„€ ∧ 0 ≀ π‘₯))
11 0red 11165 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((π‘₯ ∈ β„€ ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ 0 ∈ ℝ)
12 zre 12510 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (π‘₯ ∈ β„€ β†’ π‘₯ ∈ ℝ)
1312adantr 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((π‘₯ ∈ β„€ ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ π‘₯ ∈ ℝ)
14 nn0re 12429 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (β™―β€˜π‘ƒ) ∈ ℝ)
15 2re 12234 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2 ∈ ℝ
1615a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ 2 ∈ ℝ)
1714, 16resubcld 11590 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ ((β™―β€˜π‘ƒ) βˆ’ 2) ∈ ℝ)
1817adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((π‘₯ ∈ β„€ ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ ((β™―β€˜π‘ƒ) βˆ’ 2) ∈ ℝ)
19 lelttr 11252 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12531 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (β™―β€˜π‘ƒ) ∈ β„€)
22 2z 12542 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2 ∈ β„€
2322a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ 2 ∈ β„€)
2421, 23zsubcld 12619 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ ((β™―β€˜π‘ƒ) βˆ’ 2) ∈ β„€)
2524anim1i 616 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12516 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12430 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (β™―β€˜π‘ƒ) ∈ β„‚)
29 peano2cnm 11474 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11508 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) = ((β™―β€˜π‘ƒ) βˆ’ 1))
3231oveq1d 7377 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11157 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ 1 ∈ β„‚)
3428, 33, 33subsub4d 11550 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12285 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 (1 + 1) = 2
3635a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (1 + 1) = 2)
3736oveq2d 7378 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ ((β™―β€˜π‘ƒ) βˆ’ (1 + 1)) = ((β™―β€˜π‘ƒ) βˆ’ 2))
3834, 37eqtrd 2777 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 1) = ((β™―β€˜π‘ƒ) βˆ’ 2))
3932, 38eqtrd 2777 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2823 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 414 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 432 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 409 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12223 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2 = (1 + 1)
5554a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ 2 = (1 + 1))
5655oveq2d 7378 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ ((β™―β€˜π‘ƒ) βˆ’ 2) = ((β™―β€˜π‘ƒ) βˆ’ (1 + 1)))
5731eqcomd 2743 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) = (((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0))
5857oveq1d 7377 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2783 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5122 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 249 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 409 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 13620 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1344 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 422 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 414 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . . . . . . . . . . . . . . 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 1132 . . . . . . . . . . . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . . . . . . . . . . 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 1131 . . . . . . . . . . . . . . . . . . . . 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 255 . . . . . . . . . . . . . . . . . . . 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 419 . . . . . . . . . . . . . . . . . 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 6847 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = π‘₯ β†’ (π‘ƒβ€˜π‘–) = (π‘ƒβ€˜π‘₯))
82 fvoveq1 7385 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = π‘₯ β†’ (π‘ƒβ€˜(𝑖 + 1)) = (π‘ƒβ€˜(π‘₯ + 1)))
8381, 82preq12d 4707 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = π‘₯ β†’ {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} = {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})
8483eleq1d 2823 . . . . . . . . . . . . . . . . . . 19 (𝑖 = π‘₯ β†’ ({(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ↔ {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))} ∈ ran 𝐸))
8584adantl 483 . . . . . . . . . . . . . . . . . 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 3576 . . . . . . . . . . . . . . . . 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 414 . . . . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . 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 409 . . . . . . . . . . . . 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 454 . . . . . . . . . . . 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 409 . . . . . . . . . . 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 7236 . . . . . . . . . . 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 585 . . . . . . . . . 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 513 . . . . . . . . 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 872 . . . . . . . 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 484 . . . . . . . . . 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 727 . . . . . . . . . . 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 12531 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (π‘₯ ∈ β„•0 β†’ π‘₯ ∈ β„€)
100 peano2zm 12553 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 614 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((π‘₯ ∈ β„•0 ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ (π‘₯ ∈ β„€ ∧ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ β„€))
103 zltlem1 12563 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5122 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12429 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (π‘₯ ∈ β„•0 β†’ π‘₯ ∈ ℝ)
112111adantr 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((π‘₯ ∈ β„•0 ∧ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 1)) β†’ π‘₯ ∈ ℝ)
113112, 17anim12i 614 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11240 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 616 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11245 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 421 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1132 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 409 . . . . . . . . . . . . . . . . . . . . . . . . . 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 1135 . . . . . . . . . . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . . . . . . . . . . . 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 6851 . . . . . . . . . . . . . . . . . . . . . 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 4705 . . . . . . . . . . . . . . . . . . . . 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 2823 . . . . . . . . . . . . . . . . . . . 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 422 . . . . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . 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 419 . . . . . . . . . . . 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 409 . . . . . . . . . . 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 7236 . . . . . . . . . . 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 585 . . . . . . . . . 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 513 . . . . . . . . 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 873 . . . . . . . 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 811 . . . . . . 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 4535 . . . . . . 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 7067 . . . . 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 14413 . . . . 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 14414 . . . . . . . 8 (𝑃 ∈ Word 𝑉 β†’ 𝑃:(0..^(β™―β€˜π‘ƒ))βŸΆπ‘‰)
155154adantr 482 . . . . . . 7 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ 𝑃:(0..^(β™―β€˜π‘ƒ))βŸΆπ‘‰)
156150clwlkclwwlklem2a2 28979 . . . . . . . . 9 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (β™―β€˜πΉ) = ((β™―β€˜π‘ƒ) βˆ’ 1))
157 fzoval 13580 . . . . . . . . . . 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 7370 . . . . . . . . . . 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 2745 . . . . . . . . . 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 2797 . . . . . . . . 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 592 . . . . . . . 8 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (0..^(β™―β€˜π‘ƒ)) = (0...(β™―β€˜πΉ)))
163162feq2d 6659 . . . . . . 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 1131 . . . . 5 ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰)
166165adantr 482 . . . 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 28978 . . . . . . 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 1131 . . . . . 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 408 . . . . 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 1131 . . . . . . . 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 482 . . . . . . 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 28983 . . . . . . . . . 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 457 . . . . . . . . 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 3165 . . . . . . . 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 7370 . . . . . . . . . 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 3316 . . . . . . . . 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 341 . . . . . . . 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, 177syl5ibr 246 . . . . . . 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 716 . . . . 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 1129 . . 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 28980 . . . . . . . . . 10 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (π‘ƒβ€˜(β™―β€˜πΉ)) = (lastSβ€˜π‘ƒ))
1841833adant1 1131 . . . . . . . . 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 2743 . . . . . . . 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 2748 . . . . . . 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 249 . . . . . 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 2745 . . . . 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 482 . . . 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 409 . . 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 513 . 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 414 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 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  ifcif 4491  {cpr 4593   class class class wbr 5110   ↦ cmpt 5193  β—‘ccnv 5637  dom cdm 5638  ran crn 5639  βŸΆwf 6497  β€“1-1β†’wf1 6498  β€“1-1-ontoβ†’wf1o 6500  β€˜cfv 6501  (class class class)co 7362  β„‚cc 11056  β„cr 11057  0cc0 11058  1c1 11059   + caddc 11061   < clt 11196   ≀ cle 11197   βˆ’ cmin 11392  β„•cn 12160  2c2 12215  β„•0cn0 12420  β„€cz 12506  ...cfz 13431  ..^cfzo 13574  β™―chash 14237  Word cword 14409  lastSclsw 14457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-2 12223  df-n0 12421  df-z 12507  df-uz 12771  df-fz 13432  df-fzo 13575  df-hash 14238  df-word 14410  df-lsw 14458
This theorem is referenced by:  clwlkclwwlklem1  28985
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