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Theorem clwlkclwwlklem2a 29240
Description: Lemma for clwlkclwwlklem2 29242. (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 483 . . . . . . . . . 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 6841 . . . . . . . . . . . . . 14 (𝐸:dom 𝐸–1-1→𝑅 β†’ 𝐸:dom 𝐸–1-1-ontoβ†’ran 𝐸)
323ad2ant1 1133 . . . . . . . . . . . . 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 481 . . . . . . . . . . . 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 13669 . . . . . . . . . . . . . . . . . . . . 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 14479 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑃 ∈ Word 𝑉 β†’ (β™―β€˜π‘ƒ) ∈ β„•0)
8 simpl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((π‘₯ ∈ β„•0 ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ π‘₯ ∈ β„•0)
98adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12567 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (π‘₯ ∈ β„•0 ↔ (π‘₯ ∈ β„€ ∧ 0 ≀ π‘₯))
11 0red 11213 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((π‘₯ ∈ β„€ ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ 0 ∈ ℝ)
12 zre 12558 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (π‘₯ ∈ β„€ β†’ π‘₯ ∈ ℝ)
1312adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((π‘₯ ∈ β„€ ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ π‘₯ ∈ ℝ)
14 nn0re 12477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (β™―β€˜π‘ƒ) ∈ ℝ)
15 2re 12282 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2 ∈ ℝ
1615a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ 2 ∈ ℝ)
1714, 16resubcld 11638 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ ((β™―β€˜π‘ƒ) βˆ’ 2) ∈ ℝ)
1817adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((π‘₯ ∈ β„€ ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ ((β™―β€˜π‘ƒ) βˆ’ 2) ∈ ℝ)
19 lelttr 11300 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12579 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (β™―β€˜π‘ƒ) ∈ β„€)
22 2z 12590 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2 ∈ β„€
2322a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ 2 ∈ β„€)
2421, 23zsubcld 12667 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ ((β™―β€˜π‘ƒ) βˆ’ 2) ∈ β„€)
2524anim1i 615 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12564 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12478 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (β™―β€˜π‘ƒ) ∈ β„‚)
29 peano2cnm 11522 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11556 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0) = ((β™―β€˜π‘ƒ) βˆ’ 1))
3231oveq1d 7420 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11205 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ 1 ∈ β„‚)
3428, 33, 33subsub4d 11598 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12333 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 (1 + 1) = 2
3635a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (1 + 1) = 2)
3736oveq2d 7421 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ ((β™―β€˜π‘ƒ) βˆ’ (1 + 1)) = ((β™―β€˜π‘ƒ) βˆ’ 2))
3834, 37eqtrd 2772 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ (((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 1) = ((β™―β€˜π‘ƒ) βˆ’ 2))
3932, 38eqtrd 2772 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2818 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 431 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12271 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2 = (1 + 1)
5554a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ 2 = (1 + 1))
5655oveq2d 7421 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ ((β™―β€˜π‘ƒ) βˆ’ 2) = ((β™―β€˜π‘ƒ) βˆ’ (1 + 1)))
5731eqcomd 2738 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((β™―β€˜π‘ƒ) ∈ β„•0 β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) = (((β™―β€˜π‘ƒ) βˆ’ 1) βˆ’ 0))
5857oveq1d 7420 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2778 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5159 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 13669 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1343 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 421 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . . . . . . . . . . . . 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 1131 . . . . . . . . . . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . . . . . . . . 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 1130 . . . . . . . . . . . . . . . . . . . . 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 418 . . . . . . . . . . . . . . . . . 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 6888 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = π‘₯ β†’ (π‘ƒβ€˜π‘–) = (π‘ƒβ€˜π‘₯))
82 fvoveq1 7428 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = π‘₯ β†’ (π‘ƒβ€˜(𝑖 + 1)) = (π‘ƒβ€˜(π‘₯ + 1)))
8381, 82preq12d 4744 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = π‘₯ β†’ {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} = {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})
8483eleq1d 2818 . . . . . . . . . . . . . . . . . . 19 (𝑖 = π‘₯ β†’ ({(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 ↔ {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))} ∈ ran 𝐸))
8584adantl 482 . . . . . . . . . . . . . . . . . 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 3604 . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . 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 453 . . . . . . . . . . . 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 408 . . . . . . . . . . 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 7279 . . . . . . . . . . 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 584 . . . . . . . . . 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 512 . . . . . . . . 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 483 . . . . . . . . . 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 12579 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (π‘₯ ∈ β„•0 β†’ π‘₯ ∈ β„€)
100 peano2zm 12601 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 613 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((π‘₯ ∈ β„•0 ∧ (β™―β€˜π‘ƒ) ∈ β„•0) β†’ (π‘₯ ∈ β„€ ∧ ((β™―β€˜π‘ƒ) βˆ’ 1) ∈ β„€))
103 zltlem1 12611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5159 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (π‘₯ ∈ β„•0 β†’ π‘₯ ∈ ℝ)
112111adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((π‘₯ ∈ β„•0 ∧ π‘₯ < ((β™―β€˜π‘ƒ) βˆ’ 1)) β†’ π‘₯ ∈ ℝ)
113112, 17anim12i 613 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11288 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 615 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11293 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 420 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1131 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . . . . . . . . . . . . . . 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 1134 . . . . . . . . . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . . . . . . . . . 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 6892 . . . . . . . . . . . . . . . . . . . . . 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 4742 . . . . . . . . . . . . . . . . . . . . 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 2818 . . . . . . . . . . . . . . . . . . . 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 421 . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . 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 418 . . . . . . . . . . . 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 408 . . . . . . . . . . 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 7279 . . . . . . . . . . 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 584 . . . . . . . . . 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 512 . . . . . . . . 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 4571 . . . . . . 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 7110 . . . . 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 14464 . . . . 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 14465 . . . . . . . 8 (𝑃 ∈ Word 𝑉 β†’ 𝑃:(0..^(β™―β€˜π‘ƒ))βŸΆπ‘‰)
155154adantr 481 . . . . . . 7 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ 𝑃:(0..^(β™―β€˜π‘ƒ))βŸΆπ‘‰)
156150clwlkclwwlklem2a2 29235 . . . . . . . . 9 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (β™―β€˜πΉ) = ((β™―β€˜π‘ƒ) βˆ’ 1))
157 fzoval 13629 . . . . . . . . . . 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 7413 . . . . . . . . . . 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 2740 . . . . . . . . . 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 2792 . . . . . . . . 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 591 . . . . . . . 8 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (0..^(β™―β€˜π‘ƒ)) = (0...(β™―β€˜πΉ)))
163162feq2d 6700 . . . . . . 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 1130 . . . . 5 ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰)
166165adantr 481 . . . 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 29234 . . . . . . 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 1130 . . . . . 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 407 . . . . 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 1130 . . . . . . . 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 481 . . . . . . 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 29239 . . . . . . . . . 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 456 . . . . . . . . 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 3167 . . . . . . . 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 7413 . . . . . . . . . 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 3325 . . . . . . . . 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 340 . . . . . . . 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 1128 . . 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 29236 . . . . . . . . . 10 ((𝑃 ∈ Word 𝑉 ∧ 2 ≀ (β™―β€˜π‘ƒ)) β†’ (π‘ƒβ€˜(β™―β€˜πΉ)) = (lastSβ€˜π‘ƒ))
1841833adant1 1130 . . . . . . . . 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 2738 . . . . . . . 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 2743 . . . . . . 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 2740 . . . . 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 481 . . . 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 408 . . 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 512 . 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 413 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 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  ifcif 4527  {cpr 4629   class class class wbr 5147   ↦ cmpt 5230  β—‘ccnv 5674  dom cdm 5675  ran crn 5676  βŸΆwf 6536  β€“1-1β†’wf1 6537  β€“1-1-ontoβ†’wf1o 6539  β€˜cfv 6540  (class class class)co 7405  β„‚cc 11104  β„cr 11105  0cc0 11106  1c1 11107   + caddc 11109   < clt 11244   ≀ cle 11245   βˆ’ cmin 11440  β„•cn 12208  2c2 12263  β„•0cn0 12468  β„€cz 12554  ...cfz 13480  ..^cfzo 13623  β™―chash 14286  Word cword 14460  lastSclsw 14508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-n0 12469  df-z 12555  df-uz 12819  df-fz 13481  df-fzo 13624  df-hash 14287  df-word 14461  df-lsw 14509
This theorem is referenced by:  clwlkclwwlklem1  29241
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