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Theorem wrdind 14672
Description: Perform induction over the structure of a word. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Proof shortened by AV, 12-Oct-2022.)
Hypotheses
Ref Expression
wrdind.1 (π‘₯ = βˆ… β†’ (πœ‘ ↔ πœ“))
wrdind.2 (π‘₯ = 𝑦 β†’ (πœ‘ ↔ πœ’))
wrdind.3 (π‘₯ = (𝑦 ++ βŸ¨β€œπ‘§β€βŸ©) β†’ (πœ‘ ↔ πœƒ))
wrdind.4 (π‘₯ = 𝐴 β†’ (πœ‘ ↔ 𝜏))
wrdind.5 πœ“
wrdind.6 ((𝑦 ∈ Word 𝐡 ∧ 𝑧 ∈ 𝐡) β†’ (πœ’ β†’ πœƒ))
Assertion
Ref Expression
wrdind (𝐴 ∈ Word 𝐡 β†’ 𝜏)
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝑦,𝑧,𝐡   πœ’,π‘₯   πœ‘,𝑦,𝑧   𝜏,π‘₯   πœƒ,π‘₯
Allowed substitution hints:   πœ‘(π‘₯)   πœ“(π‘₯,𝑦,𝑧)   πœ’(𝑦,𝑧)   πœƒ(𝑦,𝑧)   𝜏(𝑦,𝑧)   𝐴(𝑦,𝑧)

Proof of Theorem wrdind
Dummy variables 𝑛 π‘š are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lencl 14483 . . 3 (𝐴 ∈ Word 𝐡 β†’ (β™―β€˜π΄) ∈ β„•0)
2 eqeq2 2745 . . . . . 6 (𝑛 = 0 β†’ ((β™―β€˜π‘₯) = 𝑛 ↔ (β™―β€˜π‘₯) = 0))
32imbi1d 342 . . . . 5 (𝑛 = 0 β†’ (((β™―β€˜π‘₯) = 𝑛 β†’ πœ‘) ↔ ((β™―β€˜π‘₯) = 0 β†’ πœ‘)))
43ralbidv 3178 . . . 4 (𝑛 = 0 β†’ (βˆ€π‘₯ ∈ Word 𝐡((β™―β€˜π‘₯) = 𝑛 β†’ πœ‘) ↔ βˆ€π‘₯ ∈ Word 𝐡((β™―β€˜π‘₯) = 0 β†’ πœ‘)))
5 eqeq2 2745 . . . . . 6 (𝑛 = π‘š β†’ ((β™―β€˜π‘₯) = 𝑛 ↔ (β™―β€˜π‘₯) = π‘š))
65imbi1d 342 . . . . 5 (𝑛 = π‘š β†’ (((β™―β€˜π‘₯) = 𝑛 β†’ πœ‘) ↔ ((β™―β€˜π‘₯) = π‘š β†’ πœ‘)))
76ralbidv 3178 . . . 4 (𝑛 = π‘š β†’ (βˆ€π‘₯ ∈ Word 𝐡((β™―β€˜π‘₯) = 𝑛 β†’ πœ‘) ↔ βˆ€π‘₯ ∈ Word 𝐡((β™―β€˜π‘₯) = π‘š β†’ πœ‘)))
8 eqeq2 2745 . . . . . 6 (𝑛 = (π‘š + 1) β†’ ((β™―β€˜π‘₯) = 𝑛 ↔ (β™―β€˜π‘₯) = (π‘š + 1)))
98imbi1d 342 . . . . 5 (𝑛 = (π‘š + 1) β†’ (((β™―β€˜π‘₯) = 𝑛 β†’ πœ‘) ↔ ((β™―β€˜π‘₯) = (π‘š + 1) β†’ πœ‘)))
109ralbidv 3178 . . . 4 (𝑛 = (π‘š + 1) β†’ (βˆ€π‘₯ ∈ Word 𝐡((β™―β€˜π‘₯) = 𝑛 β†’ πœ‘) ↔ βˆ€π‘₯ ∈ Word 𝐡((β™―β€˜π‘₯) = (π‘š + 1) β†’ πœ‘)))
11 eqeq2 2745 . . . . . 6 (𝑛 = (β™―β€˜π΄) β†’ ((β™―β€˜π‘₯) = 𝑛 ↔ (β™―β€˜π‘₯) = (β™―β€˜π΄)))
1211imbi1d 342 . . . . 5 (𝑛 = (β™―β€˜π΄) β†’ (((β™―β€˜π‘₯) = 𝑛 β†’ πœ‘) ↔ ((β™―β€˜π‘₯) = (β™―β€˜π΄) β†’ πœ‘)))
1312ralbidv 3178 . . . 4 (𝑛 = (β™―β€˜π΄) β†’ (βˆ€π‘₯ ∈ Word 𝐡((β™―β€˜π‘₯) = 𝑛 β†’ πœ‘) ↔ βˆ€π‘₯ ∈ Word 𝐡((β™―β€˜π‘₯) = (β™―β€˜π΄) β†’ πœ‘)))
14 hasheq0 14323 . . . . . 6 (π‘₯ ∈ Word 𝐡 β†’ ((β™―β€˜π‘₯) = 0 ↔ π‘₯ = βˆ…))
15 wrdind.5 . . . . . . 7 πœ“
16 wrdind.1 . . . . . . 7 (π‘₯ = βˆ… β†’ (πœ‘ ↔ πœ“))
1715, 16mpbiri 258 . . . . . 6 (π‘₯ = βˆ… β†’ πœ‘)
1814, 17syl6bi 253 . . . . 5 (π‘₯ ∈ Word 𝐡 β†’ ((β™―β€˜π‘₯) = 0 β†’ πœ‘))
1918rgen 3064 . . . 4 βˆ€π‘₯ ∈ Word 𝐡((β™―β€˜π‘₯) = 0 β†’ πœ‘)
20 fveqeq2 6901 . . . . . . 7 (π‘₯ = 𝑦 β†’ ((β™―β€˜π‘₯) = π‘š ↔ (β™―β€˜π‘¦) = π‘š))
21 wrdind.2 . . . . . . 7 (π‘₯ = 𝑦 β†’ (πœ‘ ↔ πœ’))
2220, 21imbi12d 345 . . . . . 6 (π‘₯ = 𝑦 β†’ (((β™―β€˜π‘₯) = π‘š β†’ πœ‘) ↔ ((β™―β€˜π‘¦) = π‘š β†’ πœ’)))
2322cbvralvw 3235 . . . . 5 (βˆ€π‘₯ ∈ Word 𝐡((β™―β€˜π‘₯) = π‘š β†’ πœ‘) ↔ βˆ€π‘¦ ∈ Word 𝐡((β™―β€˜π‘¦) = π‘š β†’ πœ’))
24 simprl 770 . . . . . . . . . . . . 13 (((π‘š ∈ β„•0 ∧ βˆ€π‘¦ ∈ Word 𝐡((β™―β€˜π‘¦) = π‘š β†’ πœ’)) ∧ (π‘₯ ∈ Word 𝐡 ∧ (β™―β€˜π‘₯) = (π‘š + 1))) β†’ π‘₯ ∈ Word 𝐡)
25 fzossfz 13651 . . . . . . . . . . . . . 14 (0..^(β™―β€˜π‘₯)) βŠ† (0...(β™―β€˜π‘₯))
26 simprr 772 . . . . . . . . . . . . . . . 16 (((π‘š ∈ β„•0 ∧ βˆ€π‘¦ ∈ Word 𝐡((β™―β€˜π‘¦) = π‘š β†’ πœ’)) ∧ (π‘₯ ∈ Word 𝐡 ∧ (β™―β€˜π‘₯) = (π‘š + 1))) β†’ (β™―β€˜π‘₯) = (π‘š + 1))
27 nn0p1nn 12511 . . . . . . . . . . . . . . . . 17 (π‘š ∈ β„•0 β†’ (π‘š + 1) ∈ β„•)
2827ad2antrr 725 . . . . . . . . . . . . . . . 16 (((π‘š ∈ β„•0 ∧ βˆ€π‘¦ ∈ Word 𝐡((β™―β€˜π‘¦) = π‘š β†’ πœ’)) ∧ (π‘₯ ∈ Word 𝐡 ∧ (β™―β€˜π‘₯) = (π‘š + 1))) β†’ (π‘š + 1) ∈ β„•)
2926, 28eqeltrd 2834 . . . . . . . . . . . . . . 15 (((π‘š ∈ β„•0 ∧ βˆ€π‘¦ ∈ Word 𝐡((β™―β€˜π‘¦) = π‘š β†’ πœ’)) ∧ (π‘₯ ∈ Word 𝐡 ∧ (β™―β€˜π‘₯) = (π‘š + 1))) β†’ (β™―β€˜π‘₯) ∈ β„•)
30 fzo0end 13724 . . . . . . . . . . . . . . 15 ((β™―β€˜π‘₯) ∈ β„• β†’ ((β™―β€˜π‘₯) βˆ’ 1) ∈ (0..^(β™―β€˜π‘₯)))
3129, 30syl 17 . . . . . . . . . . . . . 14 (((π‘š ∈ β„•0 ∧ βˆ€π‘¦ ∈ Word 𝐡((β™―β€˜π‘¦) = π‘š β†’ πœ’)) ∧ (π‘₯ ∈ Word 𝐡 ∧ (β™―β€˜π‘₯) = (π‘š + 1))) β†’ ((β™―β€˜π‘₯) βˆ’ 1) ∈ (0..^(β™―β€˜π‘₯)))
3225, 31sselid 3981 . . . . . . . . . . . . 13 (((π‘š ∈ β„•0 ∧ βˆ€π‘¦ ∈ Word 𝐡((β™―β€˜π‘¦) = π‘š β†’ πœ’)) ∧ (π‘₯ ∈ Word 𝐡 ∧ (β™―β€˜π‘₯) = (π‘š + 1))) β†’ ((β™―β€˜π‘₯) βˆ’ 1) ∈ (0...(β™―β€˜π‘₯)))
33 pfxlen 14633 . . . . . . . . . . . . 13 ((π‘₯ ∈ Word 𝐡 ∧ ((β™―β€˜π‘₯) βˆ’ 1) ∈ (0...(β™―β€˜π‘₯))) β†’ (β™―β€˜(π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1))) = ((β™―β€˜π‘₯) βˆ’ 1))
3424, 32, 33syl2anc 585 . . . . . . . . . . . 12 (((π‘š ∈ β„•0 ∧ βˆ€π‘¦ ∈ Word 𝐡((β™―β€˜π‘¦) = π‘š β†’ πœ’)) ∧ (π‘₯ ∈ Word 𝐡 ∧ (β™―β€˜π‘₯) = (π‘š + 1))) β†’ (β™―β€˜(π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1))) = ((β™―β€˜π‘₯) βˆ’ 1))
3526oveq1d 7424 . . . . . . . . . . . 12 (((π‘š ∈ β„•0 ∧ βˆ€π‘¦ ∈ Word 𝐡((β™―β€˜π‘¦) = π‘š β†’ πœ’)) ∧ (π‘₯ ∈ Word 𝐡 ∧ (β™―β€˜π‘₯) = (π‘š + 1))) β†’ ((β™―β€˜π‘₯) βˆ’ 1) = ((π‘š + 1) βˆ’ 1))
36 nn0cn 12482 . . . . . . . . . . . . . 14 (π‘š ∈ β„•0 β†’ π‘š ∈ β„‚)
3736ad2antrr 725 . . . . . . . . . . . . 13 (((π‘š ∈ β„•0 ∧ βˆ€π‘¦ ∈ Word 𝐡((β™―β€˜π‘¦) = π‘š β†’ πœ’)) ∧ (π‘₯ ∈ Word 𝐡 ∧ (β™―β€˜π‘₯) = (π‘š + 1))) β†’ π‘š ∈ β„‚)
38 ax-1cn 11168 . . . . . . . . . . . . 13 1 ∈ β„‚
39 pncan 11466 . . . . . . . . . . . . 13 ((π‘š ∈ β„‚ ∧ 1 ∈ β„‚) β†’ ((π‘š + 1) βˆ’ 1) = π‘š)
4037, 38, 39sylancl 587 . . . . . . . . . . . 12 (((π‘š ∈ β„•0 ∧ βˆ€π‘¦ ∈ Word 𝐡((β™―β€˜π‘¦) = π‘š β†’ πœ’)) ∧ (π‘₯ ∈ Word 𝐡 ∧ (β™―β€˜π‘₯) = (π‘š + 1))) β†’ ((π‘š + 1) βˆ’ 1) = π‘š)
4134, 35, 403eqtrd 2777 . . . . . . . . . . 11 (((π‘š ∈ β„•0 ∧ βˆ€π‘¦ ∈ Word 𝐡((β™―β€˜π‘¦) = π‘š β†’ πœ’)) ∧ (π‘₯ ∈ Word 𝐡 ∧ (β™―β€˜π‘₯) = (π‘š + 1))) β†’ (β™―β€˜(π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1))) = π‘š)
42 fveqeq2 6901 . . . . . . . . . . . . 13 (𝑦 = (π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) β†’ ((β™―β€˜π‘¦) = π‘š ↔ (β™―β€˜(π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1))) = π‘š))
43 vex 3479 . . . . . . . . . . . . . . 15 𝑦 ∈ V
4443, 21sbcie 3821 . . . . . . . . . . . . . 14 ([𝑦 / π‘₯]πœ‘ ↔ πœ’)
45 dfsbcq 3780 . . . . . . . . . . . . . 14 (𝑦 = (π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) β†’ ([𝑦 / π‘₯]πœ‘ ↔ [(π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) / π‘₯]πœ‘))
4644, 45bitr3id 285 . . . . . . . . . . . . 13 (𝑦 = (π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) β†’ (πœ’ ↔ [(π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) / π‘₯]πœ‘))
4742, 46imbi12d 345 . . . . . . . . . . . 12 (𝑦 = (π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) β†’ (((β™―β€˜π‘¦) = π‘š β†’ πœ’) ↔ ((β™―β€˜(π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1))) = π‘š β†’ [(π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) / π‘₯]πœ‘)))
48 simplr 768 . . . . . . . . . . . 12 (((π‘š ∈ β„•0 ∧ βˆ€π‘¦ ∈ Word 𝐡((β™―β€˜π‘¦) = π‘š β†’ πœ’)) ∧ (π‘₯ ∈ Word 𝐡 ∧ (β™―β€˜π‘₯) = (π‘š + 1))) β†’ βˆ€π‘¦ ∈ Word 𝐡((β™―β€˜π‘¦) = π‘š β†’ πœ’))
49 pfxcl 14627 . . . . . . . . . . . . 13 (π‘₯ ∈ Word 𝐡 β†’ (π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) ∈ Word 𝐡)
5049ad2antrl 727 . . . . . . . . . . . 12 (((π‘š ∈ β„•0 ∧ βˆ€π‘¦ ∈ Word 𝐡((β™―β€˜π‘¦) = π‘š β†’ πœ’)) ∧ (π‘₯ ∈ Word 𝐡 ∧ (β™―β€˜π‘₯) = (π‘š + 1))) β†’ (π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) ∈ Word 𝐡)
5147, 48, 50rspcdva 3614 . . . . . . . . . . 11 (((π‘š ∈ β„•0 ∧ βˆ€π‘¦ ∈ Word 𝐡((β™―β€˜π‘¦) = π‘š β†’ πœ’)) ∧ (π‘₯ ∈ Word 𝐡 ∧ (β™―β€˜π‘₯) = (π‘š + 1))) β†’ ((β™―β€˜(π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1))) = π‘š β†’ [(π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) / π‘₯]πœ‘))
5241, 51mpd 15 . . . . . . . . . 10 (((π‘š ∈ β„•0 ∧ βˆ€π‘¦ ∈ Word 𝐡((β™―β€˜π‘¦) = π‘š β†’ πœ’)) ∧ (π‘₯ ∈ Word 𝐡 ∧ (β™―β€˜π‘₯) = (π‘š + 1))) β†’ [(π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) / π‘₯]πœ‘)
5329nnge1d 12260 . . . . . . . . . . . . 13 (((π‘š ∈ β„•0 ∧ βˆ€π‘¦ ∈ Word 𝐡((β™―β€˜π‘¦) = π‘š β†’ πœ’)) ∧ (π‘₯ ∈ Word 𝐡 ∧ (β™―β€˜π‘₯) = (π‘š + 1))) β†’ 1 ≀ (β™―β€˜π‘₯))
54 wrdlenge1n0 14500 . . . . . . . . . . . . . 14 (π‘₯ ∈ Word 𝐡 β†’ (π‘₯ β‰  βˆ… ↔ 1 ≀ (β™―β€˜π‘₯)))
5554ad2antrl 727 . . . . . . . . . . . . 13 (((π‘š ∈ β„•0 ∧ βˆ€π‘¦ ∈ Word 𝐡((β™―β€˜π‘¦) = π‘š β†’ πœ’)) ∧ (π‘₯ ∈ Word 𝐡 ∧ (β™―β€˜π‘₯) = (π‘š + 1))) β†’ (π‘₯ β‰  βˆ… ↔ 1 ≀ (β™―β€˜π‘₯)))
5653, 55mpbird 257 . . . . . . . . . . . 12 (((π‘š ∈ β„•0 ∧ βˆ€π‘¦ ∈ Word 𝐡((β™―β€˜π‘¦) = π‘š β†’ πœ’)) ∧ (π‘₯ ∈ Word 𝐡 ∧ (β™―β€˜π‘₯) = (π‘š + 1))) β†’ π‘₯ β‰  βˆ…)
57 lswcl 14518 . . . . . . . . . . . 12 ((π‘₯ ∈ Word 𝐡 ∧ π‘₯ β‰  βˆ…) β†’ (lastSβ€˜π‘₯) ∈ 𝐡)
5824, 56, 57syl2anc 585 . . . . . . . . . . 11 (((π‘š ∈ β„•0 ∧ βˆ€π‘¦ ∈ Word 𝐡((β™―β€˜π‘¦) = π‘š β†’ πœ’)) ∧ (π‘₯ ∈ Word 𝐡 ∧ (β™―β€˜π‘₯) = (π‘š + 1))) β†’ (lastSβ€˜π‘₯) ∈ 𝐡)
59 oveq1 7416 . . . . . . . . . . . . . 14 (𝑦 = (π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) β†’ (𝑦 ++ βŸ¨β€œπ‘§β€βŸ©) = ((π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) ++ βŸ¨β€œπ‘§β€βŸ©))
6059sbceq1d 3783 . . . . . . . . . . . . 13 (𝑦 = (π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) β†’ ([(𝑦 ++ βŸ¨β€œπ‘§β€βŸ©) / π‘₯]πœ‘ ↔ [((π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) ++ βŸ¨β€œπ‘§β€βŸ©) / π‘₯]πœ‘))
6145, 60imbi12d 345 . . . . . . . . . . . 12 (𝑦 = (π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) β†’ (([𝑦 / π‘₯]πœ‘ β†’ [(𝑦 ++ βŸ¨β€œπ‘§β€βŸ©) / π‘₯]πœ‘) ↔ ([(π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) / π‘₯]πœ‘ β†’ [((π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) ++ βŸ¨β€œπ‘§β€βŸ©) / π‘₯]πœ‘)))
62 s1eq 14550 . . . . . . . . . . . . . . 15 (𝑧 = (lastSβ€˜π‘₯) β†’ βŸ¨β€œπ‘§β€βŸ© = βŸ¨β€œ(lastSβ€˜π‘₯)β€βŸ©)
6362oveq2d 7425 . . . . . . . . . . . . . 14 (𝑧 = (lastSβ€˜π‘₯) β†’ ((π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) ++ βŸ¨β€œπ‘§β€βŸ©) = ((π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) ++ βŸ¨β€œ(lastSβ€˜π‘₯)β€βŸ©))
6463sbceq1d 3783 . . . . . . . . . . . . 13 (𝑧 = (lastSβ€˜π‘₯) β†’ ([((π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) ++ βŸ¨β€œπ‘§β€βŸ©) / π‘₯]πœ‘ ↔ [((π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) ++ βŸ¨β€œ(lastSβ€˜π‘₯)β€βŸ©) / π‘₯]πœ‘))
6564imbi2d 341 . . . . . . . . . . . 12 (𝑧 = (lastSβ€˜π‘₯) β†’ (([(π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) / π‘₯]πœ‘ β†’ [((π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) ++ βŸ¨β€œπ‘§β€βŸ©) / π‘₯]πœ‘) ↔ ([(π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) / π‘₯]πœ‘ β†’ [((π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) ++ βŸ¨β€œ(lastSβ€˜π‘₯)β€βŸ©) / π‘₯]πœ‘)))
66 wrdind.6 . . . . . . . . . . . . 13 ((𝑦 ∈ Word 𝐡 ∧ 𝑧 ∈ 𝐡) β†’ (πœ’ β†’ πœƒ))
67 ovex 7442 . . . . . . . . . . . . . 14 (𝑦 ++ βŸ¨β€œπ‘§β€βŸ©) ∈ V
68 wrdind.3 . . . . . . . . . . . . . 14 (π‘₯ = (𝑦 ++ βŸ¨β€œπ‘§β€βŸ©) β†’ (πœ‘ ↔ πœƒ))
6967, 68sbcie 3821 . . . . . . . . . . . . 13 ([(𝑦 ++ βŸ¨β€œπ‘§β€βŸ©) / π‘₯]πœ‘ ↔ πœƒ)
7066, 44, 693imtr4g 296 . . . . . . . . . . . 12 ((𝑦 ∈ Word 𝐡 ∧ 𝑧 ∈ 𝐡) β†’ ([𝑦 / π‘₯]πœ‘ β†’ [(𝑦 ++ βŸ¨β€œπ‘§β€βŸ©) / π‘₯]πœ‘))
7161, 65, 70vtocl2ga 3567 . . . . . . . . . . 11 (((π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) ∈ Word 𝐡 ∧ (lastSβ€˜π‘₯) ∈ 𝐡) β†’ ([(π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) / π‘₯]πœ‘ β†’ [((π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) ++ βŸ¨β€œ(lastSβ€˜π‘₯)β€βŸ©) / π‘₯]πœ‘))
7250, 58, 71syl2anc 585 . . . . . . . . . 10 (((π‘š ∈ β„•0 ∧ βˆ€π‘¦ ∈ Word 𝐡((β™―β€˜π‘¦) = π‘š β†’ πœ’)) ∧ (π‘₯ ∈ Word 𝐡 ∧ (β™―β€˜π‘₯) = (π‘š + 1))) β†’ ([(π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) / π‘₯]πœ‘ β†’ [((π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) ++ βŸ¨β€œ(lastSβ€˜π‘₯)β€βŸ©) / π‘₯]πœ‘))
7352, 72mpd 15 . . . . . . . . 9 (((π‘š ∈ β„•0 ∧ βˆ€π‘¦ ∈ Word 𝐡((β™―β€˜π‘¦) = π‘š β†’ πœ’)) ∧ (π‘₯ ∈ Word 𝐡 ∧ (β™―β€˜π‘₯) = (π‘š + 1))) β†’ [((π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) ++ βŸ¨β€œ(lastSβ€˜π‘₯)β€βŸ©) / π‘₯]πœ‘)
74 wrdfin 14482 . . . . . . . . . . . . . 14 (π‘₯ ∈ Word 𝐡 β†’ π‘₯ ∈ Fin)
7574ad2antrl 727 . . . . . . . . . . . . 13 (((π‘š ∈ β„•0 ∧ βˆ€π‘¦ ∈ Word 𝐡((β™―β€˜π‘¦) = π‘š β†’ πœ’)) ∧ (π‘₯ ∈ Word 𝐡 ∧ (β™―β€˜π‘₯) = (π‘š + 1))) β†’ π‘₯ ∈ Fin)
76 hashnncl 14326 . . . . . . . . . . . . 13 (π‘₯ ∈ Fin β†’ ((β™―β€˜π‘₯) ∈ β„• ↔ π‘₯ β‰  βˆ…))
7775, 76syl 17 . . . . . . . . . . . 12 (((π‘š ∈ β„•0 ∧ βˆ€π‘¦ ∈ Word 𝐡((β™―β€˜π‘¦) = π‘š β†’ πœ’)) ∧ (π‘₯ ∈ Word 𝐡 ∧ (β™―β€˜π‘₯) = (π‘š + 1))) β†’ ((β™―β€˜π‘₯) ∈ β„• ↔ π‘₯ β‰  βˆ…))
7829, 77mpbid 231 . . . . . . . . . . 11 (((π‘š ∈ β„•0 ∧ βˆ€π‘¦ ∈ Word 𝐡((β™―β€˜π‘¦) = π‘š β†’ πœ’)) ∧ (π‘₯ ∈ Word 𝐡 ∧ (β™―β€˜π‘₯) = (π‘š + 1))) β†’ π‘₯ β‰  βˆ…)
79 pfxlswccat 14663 . . . . . . . . . . . 12 ((π‘₯ ∈ Word 𝐡 ∧ π‘₯ β‰  βˆ…) β†’ ((π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) ++ βŸ¨β€œ(lastSβ€˜π‘₯)β€βŸ©) = π‘₯)
8079eqcomd 2739 . . . . . . . . . . 11 ((π‘₯ ∈ Word 𝐡 ∧ π‘₯ β‰  βˆ…) β†’ π‘₯ = ((π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) ++ βŸ¨β€œ(lastSβ€˜π‘₯)β€βŸ©))
8124, 78, 80syl2anc 585 . . . . . . . . . 10 (((π‘š ∈ β„•0 ∧ βˆ€π‘¦ ∈ Word 𝐡((β™―β€˜π‘¦) = π‘š β†’ πœ’)) ∧ (π‘₯ ∈ Word 𝐡 ∧ (β™―β€˜π‘₯) = (π‘š + 1))) β†’ π‘₯ = ((π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) ++ βŸ¨β€œ(lastSβ€˜π‘₯)β€βŸ©))
82 sbceq1a 3789 . . . . . . . . . 10 (π‘₯ = ((π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) ++ βŸ¨β€œ(lastSβ€˜π‘₯)β€βŸ©) β†’ (πœ‘ ↔ [((π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) ++ βŸ¨β€œ(lastSβ€˜π‘₯)β€βŸ©) / π‘₯]πœ‘))
8381, 82syl 17 . . . . . . . . 9 (((π‘š ∈ β„•0 ∧ βˆ€π‘¦ ∈ Word 𝐡((β™―β€˜π‘¦) = π‘š β†’ πœ’)) ∧ (π‘₯ ∈ Word 𝐡 ∧ (β™―β€˜π‘₯) = (π‘š + 1))) β†’ (πœ‘ ↔ [((π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) ++ βŸ¨β€œ(lastSβ€˜π‘₯)β€βŸ©) / π‘₯]πœ‘))
8473, 83mpbird 257 . . . . . . . 8 (((π‘š ∈ β„•0 ∧ βˆ€π‘¦ ∈ Word 𝐡((β™―β€˜π‘¦) = π‘š β†’ πœ’)) ∧ (π‘₯ ∈ Word 𝐡 ∧ (β™―β€˜π‘₯) = (π‘š + 1))) β†’ πœ‘)
8584expr 458 . . . . . . 7 (((π‘š ∈ β„•0 ∧ βˆ€π‘¦ ∈ Word 𝐡((β™―β€˜π‘¦) = π‘š β†’ πœ’)) ∧ π‘₯ ∈ Word 𝐡) β†’ ((β™―β€˜π‘₯) = (π‘š + 1) β†’ πœ‘))
8685ralrimiva 3147 . . . . . 6 ((π‘š ∈ β„•0 ∧ βˆ€π‘¦ ∈ Word 𝐡((β™―β€˜π‘¦) = π‘š β†’ πœ’)) β†’ βˆ€π‘₯ ∈ Word 𝐡((β™―β€˜π‘₯) = (π‘š + 1) β†’ πœ‘))
8786ex 414 . . . . 5 (π‘š ∈ β„•0 β†’ (βˆ€π‘¦ ∈ Word 𝐡((β™―β€˜π‘¦) = π‘š β†’ πœ’) β†’ βˆ€π‘₯ ∈ Word 𝐡((β™―β€˜π‘₯) = (π‘š + 1) β†’ πœ‘)))
8823, 87biimtrid 241 . . . 4 (π‘š ∈ β„•0 β†’ (βˆ€π‘₯ ∈ Word 𝐡((β™―β€˜π‘₯) = π‘š β†’ πœ‘) β†’ βˆ€π‘₯ ∈ Word 𝐡((β™―β€˜π‘₯) = (π‘š + 1) β†’ πœ‘)))
894, 7, 10, 13, 19, 88nn0ind 12657 . . 3 ((β™―β€˜π΄) ∈ β„•0 β†’ βˆ€π‘₯ ∈ Word 𝐡((β™―β€˜π‘₯) = (β™―β€˜π΄) β†’ πœ‘))
901, 89syl 17 . 2 (𝐴 ∈ Word 𝐡 β†’ βˆ€π‘₯ ∈ Word 𝐡((β™―β€˜π‘₯) = (β™―β€˜π΄) β†’ πœ‘))
91 eqidd 2734 . 2 (𝐴 ∈ Word 𝐡 β†’ (β™―β€˜π΄) = (β™―β€˜π΄))
92 fveqeq2 6901 . . . 4 (π‘₯ = 𝐴 β†’ ((β™―β€˜π‘₯) = (β™―β€˜π΄) ↔ (β™―β€˜π΄) = (β™―β€˜π΄)))
93 wrdind.4 . . . 4 (π‘₯ = 𝐴 β†’ (πœ‘ ↔ 𝜏))
9492, 93imbi12d 345 . . 3 (π‘₯ = 𝐴 β†’ (((β™―β€˜π‘₯) = (β™―β€˜π΄) β†’ πœ‘) ↔ ((β™―β€˜π΄) = (β™―β€˜π΄) β†’ 𝜏)))
9594rspcv 3609 . 2 (𝐴 ∈ Word 𝐡 β†’ (βˆ€π‘₯ ∈ Word 𝐡((β™―β€˜π‘₯) = (β™―β€˜π΄) β†’ πœ‘) β†’ ((β™―β€˜π΄) = (β™―β€˜π΄) β†’ 𝜏)))
9690, 91, 95mp2d 49 1 (𝐴 ∈ Word 𝐡 β†’ 𝜏)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  [wsbc 3778  βˆ…c0 4323   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409  Fincfn 8939  β„‚cc 11108  0cc0 11110  1c1 11111   + caddc 11113   ≀ cle 11249   βˆ’ cmin 11444  β„•cn 12212  β„•0cn0 12472  ...cfz 13484  ..^cfzo 13627  β™―chash 14290  Word cword 14464  lastSclsw 14512   ++ cconcat 14520  βŸ¨β€œcs1 14545   prefix cpfx 14620
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-xnn0 12545  df-z 12559  df-uz 12823  df-fz 13485  df-fzo 13628  df-hash 14291  df-word 14465  df-lsw 14513  df-concat 14521  df-s1 14546  df-substr 14591  df-pfx 14621
This theorem is referenced by:  frmdgsum  18743  gsumwrev  19233  gsmsymgrfix  19296  efginvrel2  19595  signstfvneq0  33583  signstfvc  33585  mrsubvrs  34513
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