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Theorem wrdind 14079
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 (𝑥 = ∅ → (𝜑𝜓))
wrdind.2 (𝑥 = 𝑦 → (𝜑𝜒))
wrdind.3 (𝑥 = (𝑦 ++ ⟨“𝑧”⟩) → (𝜑𝜃))
wrdind.4 (𝑥 = 𝐴 → (𝜑𝜏))
wrdind.5 𝜓
wrdind.6 ((𝑦 ∈ Word 𝐵𝑧𝐵) → (𝜒𝜃))
Assertion
Ref Expression
wrdind (𝐴 ∈ Word 𝐵𝜏)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝑧,𝐵   𝜒,𝑥   𝜑,𝑦,𝑧   𝜏,𝑥   𝜃,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦,𝑧)   𝜒(𝑦,𝑧)   𝜃(𝑦,𝑧)   𝜏(𝑦,𝑧)   𝐴(𝑦,𝑧)

Proof of Theorem wrdind
Dummy variables 𝑛 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lencl 13880 . . 3 (𝐴 ∈ Word 𝐵 → (♯‘𝐴) ∈ ℕ0)
2 eqeq2 2813 . . . . . 6 (𝑛 = 0 → ((♯‘𝑥) = 𝑛 ↔ (♯‘𝑥) = 0))
32imbi1d 345 . . . . 5 (𝑛 = 0 → (((♯‘𝑥) = 𝑛𝜑) ↔ ((♯‘𝑥) = 0 → 𝜑)))
43ralbidv 3165 . . . 4 (𝑛 = 0 → (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑛𝜑) ↔ ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 0 → 𝜑)))
5 eqeq2 2813 . . . . . 6 (𝑛 = 𝑚 → ((♯‘𝑥) = 𝑛 ↔ (♯‘𝑥) = 𝑚))
65imbi1d 345 . . . . 5 (𝑛 = 𝑚 → (((♯‘𝑥) = 𝑛𝜑) ↔ ((♯‘𝑥) = 𝑚𝜑)))
76ralbidv 3165 . . . 4 (𝑛 = 𝑚 → (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑛𝜑) ↔ ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑚𝜑)))
8 eqeq2 2813 . . . . . 6 (𝑛 = (𝑚 + 1) → ((♯‘𝑥) = 𝑛 ↔ (♯‘𝑥) = (𝑚 + 1)))
98imbi1d 345 . . . . 5 (𝑛 = (𝑚 + 1) → (((♯‘𝑥) = 𝑛𝜑) ↔ ((♯‘𝑥) = (𝑚 + 1) → 𝜑)))
109ralbidv 3165 . . . 4 (𝑛 = (𝑚 + 1) → (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑛𝜑) ↔ ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (𝑚 + 1) → 𝜑)))
11 eqeq2 2813 . . . . . 6 (𝑛 = (♯‘𝐴) → ((♯‘𝑥) = 𝑛 ↔ (♯‘𝑥) = (♯‘𝐴)))
1211imbi1d 345 . . . . 5 (𝑛 = (♯‘𝐴) → (((♯‘𝑥) = 𝑛𝜑) ↔ ((♯‘𝑥) = (♯‘𝐴) → 𝜑)))
1312ralbidv 3165 . . . 4 (𝑛 = (♯‘𝐴) → (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑛𝜑) ↔ ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (♯‘𝐴) → 𝜑)))
14 hasheq0 13724 . . . . . 6 (𝑥 ∈ Word 𝐵 → ((♯‘𝑥) = 0 ↔ 𝑥 = ∅))
15 wrdind.5 . . . . . . 7 𝜓
16 wrdind.1 . . . . . . 7 (𝑥 = ∅ → (𝜑𝜓))
1715, 16mpbiri 261 . . . . . 6 (𝑥 = ∅ → 𝜑)
1814, 17syl6bi 256 . . . . 5 (𝑥 ∈ Word 𝐵 → ((♯‘𝑥) = 0 → 𝜑))
1918rgen 3119 . . . 4 𝑥 ∈ Word 𝐵((♯‘𝑥) = 0 → 𝜑)
20 fveqeq2 6658 . . . . . . 7 (𝑥 = 𝑦 → ((♯‘𝑥) = 𝑚 ↔ (♯‘𝑦) = 𝑚))
21 wrdind.2 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜒))
2220, 21imbi12d 348 . . . . . 6 (𝑥 = 𝑦 → (((♯‘𝑥) = 𝑚𝜑) ↔ ((♯‘𝑦) = 𝑚𝜒)))
2322cbvralvw 3399 . . . . 5 (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑚𝜑) ↔ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒))
24 simprl 770 . . . . . . . . . . . . 13 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑥 ∈ Word 𝐵)
25 fzossfz 13055 . . . . . . . . . . . . . 14 (0..^(♯‘𝑥)) ⊆ (0...(♯‘𝑥))
26 simprr 772 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (♯‘𝑥) = (𝑚 + 1))
27 nn0p1nn 11928 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℕ)
2827ad2antrr 725 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (𝑚 + 1) ∈ ℕ)
2926, 28eqeltrd 2893 . . . . . . . . . . . . . . 15 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (♯‘𝑥) ∈ ℕ)
30 fzo0end 13128 . . . . . . . . . . . . . . 15 ((♯‘𝑥) ∈ ℕ → ((♯‘𝑥) − 1) ∈ (0..^(♯‘𝑥)))
3129, 30syl 17 . . . . . . . . . . . . . 14 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ((♯‘𝑥) − 1) ∈ (0..^(♯‘𝑥)))
3225, 31sseldi 3916 . . . . . . . . . . . . 13 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ((♯‘𝑥) − 1) ∈ (0...(♯‘𝑥)))
33 pfxlen 14040 . . . . . . . . . . . . 13 ((𝑥 ∈ Word 𝐵 ∧ ((♯‘𝑥) − 1) ∈ (0...(♯‘𝑥))) → (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = ((♯‘𝑥) − 1))
3424, 32, 33syl2anc 587 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = ((♯‘𝑥) − 1))
3526oveq1d 7154 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ((♯‘𝑥) − 1) = ((𝑚 + 1) − 1))
36 nn0cn 11899 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ0𝑚 ∈ ℂ)
3736ad2antrr 725 . . . . . . . . . . . . 13 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑚 ∈ ℂ)
38 ax-1cn 10588 . . . . . . . . . . . . 13 1 ∈ ℂ
39 pncan 10885 . . . . . . . . . . . . 13 ((𝑚 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑚 + 1) − 1) = 𝑚)
4037, 38, 39sylancl 589 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ((𝑚 + 1) − 1) = 𝑚)
4134, 35, 403eqtrd 2840 . . . . . . . . . . 11 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = 𝑚)
42 fveqeq2 6658 . . . . . . . . . . . . 13 (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → ((♯‘𝑦) = 𝑚 ↔ (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = 𝑚))
43 vex 3447 . . . . . . . . . . . . . . 15 𝑦 ∈ V
4443, 21sbcie 3763 . . . . . . . . . . . . . 14 ([𝑦 / 𝑥]𝜑𝜒)
45 dfsbcq 3725 . . . . . . . . . . . . . 14 (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → ([𝑦 / 𝑥]𝜑[(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑))
4644, 45bitr3id 288 . . . . . . . . . . . . 13 (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → (𝜒[(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑))
4742, 46imbi12d 348 . . . . . . . . . . . 12 (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → (((♯‘𝑦) = 𝑚𝜒) ↔ ((♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = 𝑚[(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑)))
48 simplr 768 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒))
49 pfxcl 14034 . . . . . . . . . . . . 13 (𝑥 ∈ Word 𝐵 → (𝑥 prefix ((♯‘𝑥) − 1)) ∈ Word 𝐵)
5049ad2antrl 727 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (𝑥 prefix ((♯‘𝑥) − 1)) ∈ Word 𝐵)
5147, 48, 50rspcdva 3576 . . . . . . . . . . 11 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ((♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = 𝑚[(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑))
5241, 51mpd 15 . . . . . . . . . 10 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → [(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑)
5329nnge1d 11677 . . . . . . . . . . . . 13 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 1 ≤ (♯‘𝑥))
54 wrdlenge1n0 13897 . . . . . . . . . . . . . 14 (𝑥 ∈ Word 𝐵 → (𝑥 ≠ ∅ ↔ 1 ≤ (♯‘𝑥)))
5554ad2antrl 727 . . . . . . . . . . . . 13 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (𝑥 ≠ ∅ ↔ 1 ≤ (♯‘𝑥)))
5653, 55mpbird 260 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑥 ≠ ∅)
57 lswcl 13915 . . . . . . . . . . . 12 ((𝑥 ∈ Word 𝐵𝑥 ≠ ∅) → (lastS‘𝑥) ∈ 𝐵)
5824, 56, 57syl2anc 587 . . . . . . . . . . 11 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (lastS‘𝑥) ∈ 𝐵)
59 oveq1 7146 . . . . . . . . . . . . . 14 (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → (𝑦 ++ ⟨“𝑧”⟩) = ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“𝑧”⟩))
6059sbceq1d 3728 . . . . . . . . . . . . 13 (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → ([(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑[((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“𝑧”⟩) / 𝑥]𝜑))
6145, 60imbi12d 348 . . . . . . . . . . . 12 (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → (([𝑦 / 𝑥]𝜑[(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑) ↔ ([(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑[((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“𝑧”⟩) / 𝑥]𝜑)))
62 s1eq 13949 . . . . . . . . . . . . . . 15 (𝑧 = (lastS‘𝑥) → ⟨“𝑧”⟩ = ⟨“(lastS‘𝑥)”⟩)
6362oveq2d 7155 . . . . . . . . . . . . . 14 (𝑧 = (lastS‘𝑥) → ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“𝑧”⟩) = ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩))
6463sbceq1d 3728 . . . . . . . . . . . . 13 (𝑧 = (lastS‘𝑥) → ([((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“𝑧”⟩) / 𝑥]𝜑[((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥]𝜑))
6564imbi2d 344 . . . . . . . . . . . 12 (𝑧 = (lastS‘𝑥) → (([(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑[((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“𝑧”⟩) / 𝑥]𝜑) ↔ ([(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑[((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥]𝜑)))
66 wrdind.6 . . . . . . . . . . . . 13 ((𝑦 ∈ Word 𝐵𝑧𝐵) → (𝜒𝜃))
67 ovex 7172 . . . . . . . . . . . . . 14 (𝑦 ++ ⟨“𝑧”⟩) ∈ V
68 wrdind.3 . . . . . . . . . . . . . 14 (𝑥 = (𝑦 ++ ⟨“𝑧”⟩) → (𝜑𝜃))
6967, 68sbcie 3763 . . . . . . . . . . . . 13 ([(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑𝜃)
7066, 44, 693imtr4g 299 . . . . . . . . . . . 12 ((𝑦 ∈ Word 𝐵𝑧𝐵) → ([𝑦 / 𝑥]𝜑[(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑))
7161, 65, 70vtocl2ga 3526 . . . . . . . . . . 11 (((𝑥 prefix ((♯‘𝑥) − 1)) ∈ Word 𝐵 ∧ (lastS‘𝑥) ∈ 𝐵) → ([(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑[((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥]𝜑))
7250, 58, 71syl2anc 587 . . . . . . . . . 10 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ([(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑[((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥]𝜑))
7352, 72mpd 15 . . . . . . . . 9 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → [((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥]𝜑)
74 wrdfin 13879 . . . . . . . . . . . . . 14 (𝑥 ∈ Word 𝐵𝑥 ∈ Fin)
7574ad2antrl 727 . . . . . . . . . . . . 13 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑥 ∈ Fin)
76 hashnncl 13727 . . . . . . . . . . . . 13 (𝑥 ∈ Fin → ((♯‘𝑥) ∈ ℕ ↔ 𝑥 ≠ ∅))
7775, 76syl 17 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ((♯‘𝑥) ∈ ℕ ↔ 𝑥 ≠ ∅))
7829, 77mpbid 235 . . . . . . . . . . 11 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑥 ≠ ∅)
79 pfxlswccat 14070 . . . . . . . . . . . 12 ((𝑥 ∈ Word 𝐵𝑥 ≠ ∅) → ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) = 𝑥)
8079eqcomd 2807 . . . . . . . . . . 11 ((𝑥 ∈ Word 𝐵𝑥 ≠ ∅) → 𝑥 = ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩))
8124, 78, 80syl2anc 587 . . . . . . . . . 10 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑥 = ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩))
82 sbceq1a 3734 . . . . . . . . . 10 (𝑥 = ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) → (𝜑[((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥]𝜑))
8381, 82syl 17 . . . . . . . . 9 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (𝜑[((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥]𝜑))
8473, 83mpbird 260 . . . . . . . 8 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝜑)
8584expr 460 . . . . . . 7 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ 𝑥 ∈ Word 𝐵) → ((♯‘𝑥) = (𝑚 + 1) → 𝜑))
8685ralrimiva 3152 . . . . . 6 ((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) → ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (𝑚 + 1) → 𝜑))
8786ex 416 . . . . 5 (𝑚 ∈ ℕ0 → (∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒) → ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (𝑚 + 1) → 𝜑)))
8823, 87syl5bi 245 . . . 4 (𝑚 ∈ ℕ0 → (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑚𝜑) → ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (𝑚 + 1) → 𝜑)))
894, 7, 10, 13, 19, 88nn0ind 12069 . . 3 ((♯‘𝐴) ∈ ℕ0 → ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (♯‘𝐴) → 𝜑))
901, 89syl 17 . 2 (𝐴 ∈ Word 𝐵 → ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (♯‘𝐴) → 𝜑))
91 eqidd 2802 . 2 (𝐴 ∈ Word 𝐵 → (♯‘𝐴) = (♯‘𝐴))
92 fveqeq2 6658 . . . 4 (𝑥 = 𝐴 → ((♯‘𝑥) = (♯‘𝐴) ↔ (♯‘𝐴) = (♯‘𝐴)))
93 wrdind.4 . . . 4 (𝑥 = 𝐴 → (𝜑𝜏))
9492, 93imbi12d 348 . . 3 (𝑥 = 𝐴 → (((♯‘𝑥) = (♯‘𝐴) → 𝜑) ↔ ((♯‘𝐴) = (♯‘𝐴) → 𝜏)))
9594rspcv 3569 . 2 (𝐴 ∈ Word 𝐵 → (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (♯‘𝐴) → 𝜑) → ((♯‘𝐴) = (♯‘𝐴) → 𝜏)))
9690, 91, 95mp2d 49 1 (𝐴 ∈ Word 𝐵𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  wne 2990  wral 3109  [wsbc 3723  c0 4246   class class class wbr 5033  cfv 6328  (class class class)co 7139  Fincfn 8496  cc 10528  0cc0 10530  1c1 10531   + caddc 10533  cle 10669  cmin 10863  cn 11629  0cn0 11889  ...cfz 12889  ..^cfzo 13032  chash 13690  Word cword 13861  lastSclsw 13909   ++ cconcat 13917  ⟨“cs1 13944   prefix cpfx 14027
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-n0 11890  df-xnn0 11960  df-z 11974  df-uz 12236  df-fz 12890  df-fzo 13033  df-hash 13691  df-word 13862  df-lsw 13910  df-concat 13918  df-s1 13945  df-substr 13998  df-pfx 14028
This theorem is referenced by:  frmdgsum  18023  gsumwrev  18490  gsmsymgrfix  18552  efginvrel2  18849  signstfvneq0  31956  signstfvc  31958  mrsubvrs  32883
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