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Theorem wrdind 14610
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 14421 . . 3 (𝐴 ∈ Word 𝐡 β†’ (β™―β€˜π΄) ∈ β„•0)
2 eqeq2 2748 . . . . . 6 (𝑛 = 0 β†’ ((β™―β€˜π‘₯) = 𝑛 ↔ (β™―β€˜π‘₯) = 0))
32imbi1d 341 . . . . 5 (𝑛 = 0 β†’ (((β™―β€˜π‘₯) = 𝑛 β†’ πœ‘) ↔ ((β™―β€˜π‘₯) = 0 β†’ πœ‘)))
43ralbidv 3174 . . . 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 2748 . . . . . 6 (𝑛 = π‘š β†’ ((β™―β€˜π‘₯) = 𝑛 ↔ (β™―β€˜π‘₯) = π‘š))
65imbi1d 341 . . . . 5 (𝑛 = π‘š β†’ (((β™―β€˜π‘₯) = 𝑛 β†’ πœ‘) ↔ ((β™―β€˜π‘₯) = π‘š β†’ πœ‘)))
76ralbidv 3174 . . . 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 2748 . . . . . 6 (𝑛 = (π‘š + 1) β†’ ((β™―β€˜π‘₯) = 𝑛 ↔ (β™―β€˜π‘₯) = (π‘š + 1)))
98imbi1d 341 . . . . 5 (𝑛 = (π‘š + 1) β†’ (((β™―β€˜π‘₯) = 𝑛 β†’ πœ‘) ↔ ((β™―β€˜π‘₯) = (π‘š + 1) β†’ πœ‘)))
109ralbidv 3174 . . . 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 2748 . . . . . 6 (𝑛 = (β™―β€˜π΄) β†’ ((β™―β€˜π‘₯) = 𝑛 ↔ (β™―β€˜π‘₯) = (β™―β€˜π΄)))
1211imbi1d 341 . . . . 5 (𝑛 = (β™―β€˜π΄) β†’ (((β™―β€˜π‘₯) = 𝑛 β†’ πœ‘) ↔ ((β™―β€˜π‘₯) = (β™―β€˜π΄) β†’ πœ‘)))
1312ralbidv 3174 . . . 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 14263 . . . . . 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 257 . . . . . 6 (π‘₯ = βˆ… β†’ πœ‘)
1814, 17syl6bi 252 . . . . 5 (π‘₯ ∈ Word 𝐡 β†’ ((β™―β€˜π‘₯) = 0 β†’ πœ‘))
1918rgen 3066 . . . 4 βˆ€π‘₯ ∈ Word 𝐡((β™―β€˜π‘₯) = 0 β†’ πœ‘)
20 fveqeq2 6851 . . . . . . 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 344 . . . . . 6 (π‘₯ = 𝑦 β†’ (((β™―β€˜π‘₯) = π‘š β†’ πœ‘) ↔ ((β™―β€˜π‘¦) = π‘š β†’ πœ’)))
2322cbvralvw 3225 . . . . 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 769 . . . . . . . . . . . . 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 13591 . . . . . . . . . . . . . 14 (0..^(β™―β€˜π‘₯)) βŠ† (0...(β™―β€˜π‘₯))
26 simprr 771 . . . . . . . . . . . . . . . 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 12452 . . . . . . . . . . . . . . . . 17 (π‘š ∈ β„•0 β†’ (π‘š + 1) ∈ β„•)
2827ad2antrr 724 . . . . . . . . . . . . . . . 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 2837 . . . . . . . . . . . . . . 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 13664 . . . . . . . . . . . . . . 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 3942 . . . . . . . . . . . . 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 14571 . . . . . . . . . . . . 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 584 . . . . . . . . . . . 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 7372 . . . . . . . . . . . 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 12423 . . . . . . . . . . . . . 14 (π‘š ∈ β„•0 β†’ π‘š ∈ β„‚)
3736ad2antrr 724 . . . . . . . . . . . . 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 11109 . . . . . . . . . . . . 13 1 ∈ β„‚
39 pncan 11407 . . . . . . . . . . . . 13 ((π‘š ∈ β„‚ ∧ 1 ∈ β„‚) β†’ ((π‘š + 1) βˆ’ 1) = π‘š)
4037, 38, 39sylancl 586 . . . . . . . . . . . 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 2780 . . . . . . . . . . 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 6851 . . . . . . . . . . . . 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 3449 . . . . . . . . . . . . . . 15 𝑦 ∈ V
4443, 21sbcie 3782 . . . . . . . . . . . . . 14 ([𝑦 / π‘₯]πœ‘ ↔ πœ’)
45 dfsbcq 3741 . . . . . . . . . . . . . 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 284 . . . . . . . . . . . . 13 (𝑦 = (π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) β†’ (πœ’ ↔ [(π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) / π‘₯]πœ‘))
4742, 46imbi12d 344 . . . . . . . . . . . 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 767 . . . . . . . . . . . 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 14565 . . . . . . . . . . . . 13 (π‘₯ ∈ Word 𝐡 β†’ (π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) ∈ Word 𝐡)
5049ad2antrl 726 . . . . . . . . . . . 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 3582 . . . . . . . . . . 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 12201 . . . . . . . . . . . . 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 14438 . . . . . . . . . . . . . 14 (π‘₯ ∈ Word 𝐡 β†’ (π‘₯ β‰  βˆ… ↔ 1 ≀ (β™―β€˜π‘₯)))
5554ad2antrl 726 . . . . . . . . . . . . 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 256 . . . . . . . . . . . 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 14456 . . . . . . . . . . . 12 ((π‘₯ ∈ Word 𝐡 ∧ π‘₯ β‰  βˆ…) β†’ (lastSβ€˜π‘₯) ∈ 𝐡)
5824, 56, 57syl2anc 584 . . . . . . . . . . 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 7364 . . . . . . . . . . . . . 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 3744 . . . . . . . . . . . . 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 344 . . . . . . . . . . . 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 14488 . . . . . . . . . . . . . . 15 (𝑧 = (lastSβ€˜π‘₯) β†’ βŸ¨β€œπ‘§β€βŸ© = βŸ¨β€œ(lastSβ€˜π‘₯)β€βŸ©)
6362oveq2d 7373 . . . . . . . . . . . . . 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 3744 . . . . . . . . . . . . 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 340 . . . . . . . . . . . 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 7390 . . . . . . . . . . . . . 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 3782 . . . . . . . . . . . . 13 ([(𝑦 ++ βŸ¨β€œπ‘§β€βŸ©) / π‘₯]πœ‘ ↔ πœƒ)
7066, 44, 693imtr4g 295 . . . . . . . . . . . 12 ((𝑦 ∈ Word 𝐡 ∧ 𝑧 ∈ 𝐡) β†’ ([𝑦 / π‘₯]πœ‘ β†’ [(𝑦 ++ βŸ¨β€œπ‘§β€βŸ©) / π‘₯]πœ‘))
7161, 65, 70vtocl2ga 3535 . . . . . . . . . . 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 584 . . . . . . . . . 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 14420 . . . . . . . . . . . . . 14 (π‘₯ ∈ Word 𝐡 β†’ π‘₯ ∈ Fin)
7574ad2antrl 726 . . . . . . . . . . . . 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 14266 . . . . . . . . . . . . 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 14601 . . . . . . . . . . . 12 ((π‘₯ ∈ Word 𝐡 ∧ π‘₯ β‰  βˆ…) β†’ ((π‘₯ prefix ((β™―β€˜π‘₯) βˆ’ 1)) ++ βŸ¨β€œ(lastSβ€˜π‘₯)β€βŸ©) = π‘₯)
8079eqcomd 2742 . . . . . . . . . . 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 584 . . . . . . . . . 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 3750 . . . . . . . . . 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 256 . . . . . . . 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 457 . . . . . . 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 3143 . . . . . 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 413 . . . . 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 12598 . . 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 2737 . 2 (𝐴 ∈ Word 𝐡 β†’ (β™―β€˜π΄) = (β™―β€˜π΄))
92 fveqeq2 6851 . . . 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 344 . . 3 (π‘₯ = 𝐴 β†’ (((β™―β€˜π‘₯) = (β™―β€˜π΄) β†’ πœ‘) ↔ ((β™―β€˜π΄) = (β™―β€˜π΄) β†’ 𝜏)))
9594rspcv 3577 . 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 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2943  βˆ€wral 3064  [wsbc 3739  βˆ…c0 4282   class class class wbr 5105  β€˜cfv 6496  (class class class)co 7357  Fincfn 8883  β„‚cc 11049  0cc0 11051  1c1 11052   + caddc 11054   ≀ cle 11190   βˆ’ cmin 11385  β„•cn 12153  β„•0cn0 12413  ...cfz 13424  ..^cfzo 13567  β™―chash 14230  Word cword 14402  lastSclsw 14450   ++ cconcat 14458  βŸ¨β€œcs1 14483   prefix cpfx 14558
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-hash 14231  df-word 14403  df-lsw 14451  df-concat 14459  df-s1 14484  df-substr 14529  df-pfx 14559
This theorem is referenced by:  frmdgsum  18672  gsumwrev  19147  gsmsymgrfix  19210  efginvrel2  19509  signstfvneq0  33184  signstfvc  33186  mrsubvrs  34116
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